text: 16 * (cid:12) (cid:12) + (cid:12)c(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) kφ2 E2n+1 ρ12 E2n+1 < 8 +4π2 kλφ22 . (73) ? The ratio ∆ of this term with the ground state energy eigenvalue reads D (cid:12) (cid:12) E (cid:12)c(cid:12) (cid:12) (cid:12) 2 1 ∆ = kφ E2n+1 ρ2 E2n+1 8 + π2 2 (74) E < 4 kφ?, min and it tells us if the additional term is actually a small perturbation or not. For instance, we can require ∆ 10−1, so getting < 0 15 (75) k < . . φ? We see how the smaller the value of , the more accurate the energy spectrum (63) we got in polymer k φ? representation. VI. CONCLUSION We presented a self-consistent cosmological picture, based on the implementation of an evolutionary quantum gravity approach to a reliable model of the primordial isotropic Universe. Indeed, we construct a nonsingular cosmology, corresponding to a cyclic Universe, having a quantum big bounce in the past, associated to the details of its quantum dynamics, and a late turning point, due to the presence of a small negative cosmological constant. The main issue of the analysis above consists in determining, via a polymer quantum approach, a ground state of the Universe, associated to a positive dust energy density and to a high occupation number. For such a vacuum state of the model, it is possible to construct a reliable classical limit, which reconciles the early evolutionary Universe phase with a standard preinﬂationary scenario (with a decay from a false vacuum). Furthermore, the emerging dust energy density is strongly redshifted by the Universe de Sitter phase and thus, the postinﬂationary dynamics is indistinguishable from the Standard Model one. Finally, we observe that the value of the negative cosmological constant does not enter the ground state energy of the Universe and so it can be easily stated as an amount which is unable to aﬀect the present Universe dynamics. We can consequently conclude that an evolutionary quantum cosmology exists, able to solve the singular nature of the big bang, without any dynamical discrepancy with respect to a standard Friedmann Universe and with the additional feature of a natural picture of the model classical limit. Despite the fact that the present study oﬀers an interesting cosmological paradigm for the matter-time dualism and the related evolutionary quantum gravity, we stress how the problem of a unique deﬁnition of a physical clock for the dynamics of the gravitational ﬁeld in canonical quantum gravity remains one of the most challenging topics in this area and deserves attention for both its fundamental developments and speciﬁc applications. We conclude by observing how the relevance of the present model must be also recognized in its possibly very general character. In fact, the anisotropic Universe degrees of freedom are dynamically equivalent to a scalar ﬁeld and the associated potential (due to the cosmological model spatial curvature) is quadratic in the limit of a quasi-isotropic Universe. Then, we could ﬁrst extend the present model to the homogeneous Bianchi Universe (in particular to the Bianchi IX model, generalizing the closed isotropic Universe) and then to a generic inhomogeneous Universe, via a quantum version of the so-called Belinski-Lipschitz-Kalatnikov conjecture [40, 41]. Although, diﬀerently from the scalar ﬁeld dynamics, there is no ﬁrm evidence that the limit of small anisotropies is prescribed by the Universe evolution, the perspective to replace the scalar ﬁeld with the cosmological gravitational ﬁeld degrees of freedom, opens very general and intriguing scenarios for the implementation of the idea traced here. VII. APPENDIX: BORN-OPPENHEIMER APPROXIMATION We demonstrate that the solutions of the evolutionary equation (11) in the large limit can be con- ρ structed as the product of the two harmonic oscillator solutions (14) and (18). In particular, by inserting the expression (21) into Eq. (11) one gets ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: ratio

text: of

text: this

text: term

text: with

text: the

text: ground

text: state

text: energy

text: eigenvalue

text: reads

text: D

text: E

text: kφ

text: ρ2

text: π2

text: E

text: kφ?,

text: min

text: and

text: it

text: tells

text: us

text: if

text: the

text: additional

text: term

text: is

text: actually

text: a

text: small

text: perturbation

text: or

text: not.

text: For

text: instance,

text: we

text: can

text: require

text: so

text: getting

text: k

text: φ?

text: We

text: see

text: how

text: the

text: smaller

text: the

text: value

text: of

text: the

text: more

text: accurate

text: the

text: energy

text: spectrum

text: we

text: got

text: in

text: polymer

text: k

text: φ?

text: representation.

text: VI.

text: CONCLUSION

text: We

text: presented

text: a

text: self-consistent

text: cosmological

text: picture,

text: based

text: on

text: the

text: implementation

text: of

text: an

text: evolutionary

text: quantum

text: gravity

text: approach

text: to

text: a

text: reliable

text: model

text: of

text: the

text: primordial

text: isotropic

text: Universe.

text: Indeed,

text: we

text: construct

text: a

text: nonsingular

text: cosmology,

text: corresponding

text: to

text: a

text: cyclic

text: Universe,

text: having

text: a

text: quantum

text: big

text: bounce

text: in

text: the

text: past,

text: associated

text: to

text: the

text: details

text: of

text: its

text: quantum

text: dynamics,

text: and

text: a

text: late

text: turning

text: point,

text: due

text: to

text: the

text: presence

text: of

text: a

text: small

text: negative

text: cosmological

text: constant.

text: The

text: main

text: issue

text: of

text: the

text: analysis

text: above

text: consists

text: in

text: determining,

text: via

text: a

text: polymer

text: quantum

text: approach,

text: a

text: ground

text: state

text: of

text: the

text: Universe,

text: associated

text: to

text: a

text: positive

text: dust

text: energy

text: density

text: and

text: to

text: a

text: high

text: occupation

text: number.

text: For

text: such

text: a

text: vacuum

text: state

text: of

text: the

text: model,

text: it

text: is

text: possible

text: to

text: construct

text: a

text: reliable

text: classical

text: limit,

text: which

text: reconciles

text: the

text: early

text: evolutionary

text: Universe

text: phase

text: with

text: a

text: standard

text: preinﬂationary

text: scenario

text: (with

text: a

text: decay

text: from

text: a

text: false

text: vacuum).

text: Furthermore,

text: the

text: emerging

text: dust

text: energy

text: density

text: is

text: strongly

text: redshifted

text: by

text: the

text: Universe

text: de

text: Sitter

text: phase

text: and

text: thus,

text: the

text: postinﬂationary

text: dynamics

text: is

text: indistinguishable

text: from

text: the

text: Standard

text: Model

text: one.

text: Finally,

text: we

text: observe

text: that

text: the

text: value

text: of

text: the

text: negative

text: cosmological

text: constant

text: does

text: not

text: enter

text: the

text: ground

text: state

text: energy

text: of

text: the

text: Universe

text: and

text: so

text: it

text: can

text: be

text: easily

text: stated

text: as

text: an

text: amount

text: which

text: is

text: unable

text: to

text: aﬀect

text: the

text: present

text: Universe

text: dynamics.

text: We

text: can

text: consequently

text: conclude

text: that

text: an

text: evolutionary

text: quantum

text: cosmology

text: exists,

text: able

text: to

text: solve

text: the

text: singular

text: nature

text: of

text: the

text: big

text: bang,

text: without

text: any

text: dynamical

text: discrepancy

text: with

text: respect

text: to

text: a

text: standard

text: Friedmann

text: Universe

text: and

text: with

text: the

text: additional

text: feature

text: of

text: a

text: natural

text: picture

text: of

text: the

text: model

text: classical

text: limit.

text: Despite

text: the

text: fact

text: that

text: the

text: present

text: study

text: oﬀers

text: an

text: interesting

text: cosmological

text: paradigm

text: for

text: the

text: matter-time

text: dualism

text: and

text: the

text: related

text: evolutionary

text: quantum

text: gravity,

text: we

text: stress

text: how

text: the

text: problem

text: of

text: a

text: unique

text: deﬁnition

text: of

text: a

text: physical

text: clock

text: for

text: the

text: dynamics

text: of

text: the

text: gravitational

text: ﬁeld

text: in

text: canonical

text: quantum

text: gravity

text: remains

text: one

text: of

text: the

text: most

text: challenging

text: topics

text: in

text: this

text: area

text: and

text: deserves

text: attention

text: for

text: both

text: its

text: fundamental

text: developments

text: and

text: speciﬁc

text: applications.

text: We

text: conclude

text: by

text: observing

text: how

text: the

text: relevance

text: of

text: the

text: present

text: model

text: must

text: be

text: also

text: recognized

text: in

text: its

text: possibly

text: very

text: general

text: character.

text: In

text: fact,

text: the

text: anisotropic

text: Universe

text: degrees

text: of

text: freedom

text: are

text: dynamically

text: equivalent

text: to

text: a

text: scalar

text: ﬁeld

text: and

text: the

text: associated

text: potential

text: (due

text: to

text: the

text: cosmological

text: model

text: spatial

text: curvature)

text: is

text: quadratic

text: in

text: the

text: limit

text: of

text: a

text: quasi-isotropic

text: Universe.

text: Then,

text: we

text: could

text: ﬁrst

text: extend

text: the

text: present

text: model

text: to

text: the

text: homogeneous

text: Bianchi

text: Universe

text: (in

text: particular

text: to

text: the

text: Bianchi

text: IX

text: model,

text: generalizing

text: the

text: closed

text: isotropic

text: Universe)

text: and

text: then

text: to

text: a

text: generic

text: inhomogeneous

text: Universe,

text: via

text: a

text: quantum

text: version

text: of

text: the

text: so-called

text: Belinski-Lipschitz-Kalatnikov

text: conjecture

text: Although,

text: diﬀerently

text: from

text: the

text: scalar

text: ﬁeld

text: dynamics,

text: there

text: is

text: no

text: ﬁrm

text: evidence

text: that

text: the

text: limit

text: of

text: small

text: anisotropies

text: is

text: prescribed

text: by

text: the

text: Universe

text: evolution,

text: the

text: perspective

text: to

text: replace

text: the

text: scalar

text: ﬁeld

text: with

text: the

text: cosmological

text: gravitational

text: ﬁeld

text: degrees

text: of

text: freedom,

text: opens

text: very

text: general

text: and

text: intriguing

text: scenarios

text: for

text: the

text: implementation

text: of

text: the

text: idea

text: traced

text: here.

text: VII.

text: APPENDIX:

text: BORN-OPPENHEIMER

text: APPROXIMATION

text: We

text: demonstrate

text: that

text: the

text: solutions

text: of

text: the

text: evolutionary

text: equation

text: in

text: the

text: large

text: limit

text: can

text: be

text: con-

text: ρ

text: structed

text: as

text: the

text: product

text: of

text: the

text: two

text: harmonic

text: oscillator

text: solutions

text: and

text: In

text: particular,

text: by

text: inserting

text: the

text: expression

text: into

text: Eq.

text: one

text: gets

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 3 Λ 1 1 2 H = − πG 2 − 2 + 2 + m 2 2 (8) 2 pρ 8πGρ 2(cid:126)ρ2 pφ 2 (cid:126) ρ φ . The canonical quantization of the associated dynamical system is obtained by replacing the conﬁgu- ration variables and with multiplicative operators deﬁned in a suitable (pre-)Hilbert space and the ρ φ momenta with the proper derivative operators, i.e. → − (cid:126) ∂ → − (cid:126) ∂ (9) p i , p i , ρ ∂ρ φ ∂φ such that the Wheeler-DeWitt operator Hˆ becomes (cid:20) (cid:21) 3 4 2 Λ (cid:126) 2 1 2 HˆΨ (ρ, φ) = 2π‘P ∂ 2 − 8 ρ2 − 2 2 ∂ 2 + 2 m(cid:126) ρ2φ2 Ψ (ρ, φ) , (10) G ∂ρ πG ρ ∂φ √ = (cid:126) being the Planck length. The evolutionary quantum equation associated with (10) reduces ‘ G P to an eigenvalue equation for the Hamiltonian (8) HˆΨ ( ) = Ψ ( ) (11) ρ, φ E ρ, φ . A. Quasiclassical limit Let us address a Born-Oppenheimer approximation (to be veriﬁed a posteriori), in which we construct the wave function as the product of the following two terms Ψ ( ) = ( ) ( ) (12) ρ, φ ζ ρ χ ρ, φ , in which describes the scalar ﬁeld wave function and depends parametrically on , while is the χ ρ ζ Universe wave function. Hence, we impose (cid:20) (cid:21) 2 Λ 2 −∂∂ρ2 + 12π2‘4 ρ2 ζn (ρ) = 3πG‘4 Enρζn (ρ) . (13) P P The equation above can be written√as the Schrödinger equation describing a harmonic oscillator with mass = 1 and frequency Ω = 1 3Λ. A solution of the eigenvalue problem (13) is thus given by M 3πG 2 (cid:16) (cid:17) √ ζ (ρ) = N Λ1/8e− Λ2?ρ2 H qΛ1/4ρ , (14) n n ? n ? where ( ) is the Hermite polynomial of degree , = 1 is a normalization constant and H x n N n n π1/2 2n n! Λ ≡ Λ (15) ? 12π2‘4 . P The eigenvalues Eρ reads (cid:18) (cid:19) n √ 1 1 Enρ = (cid:126)2 3Λ n + 2 , (16) where ∈ N is restricted to be odd [? ]. n In the same way, we impose ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

label: equation

text: Λ

text: H

text: πG

text: m

text: pρ

text: 8πGρ

text: pφ

text: ρ

text: φ

text: The

text: canonical

text: quantization

text: of

text: the

text: associated

text: dynamical

text: system

text: is

text: obtained

text: by

text: replacing

text: the

text: conﬁgu-

text: ration

text: variables

text: and

text: with

text: multiplicative

text: operators

text: deﬁned

text: in

text: a

text: suitable

text: (pre-)Hilbert

text: space

text: and

text: the

text: ρ

text: φ

text: momenta

text: with

text: the

text: proper

text: derivative

text: operators,

text: i.e.

text: p

text: i

text: p

text: i

text: ρ

text: ∂ρ

text: φ

text: ∂φ

text: such

text: that

text: the

text: Wheeler-DeWitt

text: operator

text: Hˆ

text: becomes

text: Λ

text: HˆΨ

text: (ρ,

text: φ)

text: 2π‘P

text: ρ2

text: ρ2φ2

text: Ψ

text: (ρ,

text: φ)

text: G

text: ∂ρ

text: πG

text: ρ

text: ∂φ

text: being

text: the

text: Planck

text: length.

text: The

text: evolutionary

text: quantum

text: equation

text: associated

text: with

text: reduces

text: G

text: P

text: to

text: an

text: eigenvalue

text: equation

text: for

text: the

text: Hamiltonian

text: HˆΨ

text: Ψ

text: ρ,

text: φ

text: E

text: ρ,

text: φ

text: A.

text: Quasiclassical

text: limit

text: Let

text: us

text: address

text: a

text: Born-Oppenheimer

text: approximation

text: (to

text: be

text: veriﬁed

text: a

text: posteriori),

text: in

text: which

text: we

text: construct

text: the

text: wave

text: function

text: as

text: the

text: product

text: of

text: the

text: following

text: two

text: terms

text: Ψ

text: ρ,

text: φ

text: ζ

text: ρ

text: χ

text: ρ,

text: φ

text: in

text: which

text: describes

text: the

text: scalar

text: ﬁeld

text: wave

text: function

text: and

text: depends

text: parametrically

text: on

text: while

text: is

text: the

text: χ

text: ρ

text: ζ

text: Universe

text: wave

text: function.

text: Hence,

text: we

text: impose

text: Λ

text: −∂∂ρ2

text: ρ2

text: ζn

text: (ρ)

text: Enρζn

text: (ρ)

text: P

text: P

text: The

text: equation

text: above

text: can

text: be

text: written√as

text: the

text: Schrödinger

text: equation

text: describing

text: a

text: harmonic

text: oscillator

text: with

text: mass

text: and

text: frequency

text: Ω

text: A

text: solution

text: of

text: the

text: eigenvalue

text: problem

text: is

text: thus

text: given

text: by

text: M

text: ζ

text: (ρ)

text: N

text: H

text: n

text: n

text: n

text: where

text: is

text: the

text: Hermite

text: polynomial

text: of

text: degree

text: is

text: a

text: normalization

text: constant

text: and

text: H

text: x

text: n

text: N

text: n

text: n

text: n!

text: Λ

text: Λ

text: P

text: The

text: eigenvalues

text: Eρ

text: reads

text: n

text: Enρ

text: 3Λ

text: n

text: where

text: N

text: is

text: restricted

text: to

text: be

text: odd

text: n

text: In

text: the

text: same

text: way,

text: we

text: impose

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 1.0 0.5 Φ 0.0 (cid:45)0.5 0 10 20 30 40 t F(cid:126)pωφig√urBe22−6:A2C. lassical trajectory of φ in function of time. Time is in units of 120πω, the scalar ﬁeld in units of It is worth noting that the presence of a negative cosmological constant ensures the existence of a turning point in the Universe late classical evolution. The associated quantum equation in evolutionary quantum gravity reads (cid:20) (cid:21) 3 4 2 Λ (cid:126) 2 2π‘P ∂ 2 − 8 ρ2 − 2 2 ∂ 2 Ψ (ρ, φ) = E Ψ (ρ, φ) , (32) G ∂ρ πG ρ ∂φ We look for a plane-wave solution for the scalar ﬁeld, i.e. 1 Ψ (ρ, φ) = √ eikφ(cid:126)φ ζ (ρ) , (33) 2 π such that one gets the following equation for (cid:20) ζ (cid:21) ∂2 − Λ ρ2 + kφ2? ζ (ρ) = E ζ (ρ) . (34) ∂ρ2 ? ρ2 ? with 2 2 = G 2 = G (35) kφ? 3π(cid:126)‘4 kφ , E? 3π‘4 E. P P A solution of Eq.(34) can be constructed by considering the following wave function √ ζ = f (ρ) e− Λ2?ρ2 ρλ, (36) where [? ] q − 1 + 1 4 2 k = φ? (37) λ 2 . ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Φ

text: t

label: caption

text: lassical

text: trajectory

text: of

text: φ

text: in

text: function

text: of

text: time.

text: Time

text: is

text: in

text: units

text: of

text: the

text: scalar

text: ﬁeld

text: in

text: units

text: of

text: It

text: is

text: worth

text: noting

text: that

text: the

text: presence

text: of

text: a

text: negative

text: cosmological

text: constant

text: ensures

text: the

text: existence

text: of

text: a

text: turning

text: point

text: in

text: the

text: Universe

text: late

text: classical

text: evolution.

text: The

text: associated

text: quantum

text: equation

text: in

text: evolutionary

text: quantum

text: gravity

text: reads

text: Λ

text: 2π‘P

text: ρ2

text: Ψ

text: (ρ,

text: φ)

text: E

text: Ψ

text: (ρ,

text: φ)

text: G

text: ∂ρ

text: πG

text: ρ

text: ∂φ

text: We

text: look

text: for

text: a

text: plane-wave

text: solution

text: for

text: the

text: scalar

text: ﬁeld,

text: i.e.

text: Ψ

text: (ρ,

text: φ)

text: ζ

text: (ρ)

text: π

text: such

text: that

text: one

text: gets

text: the

text: following

text: equation

text: for

text: ζ

text: ∂2

text: Λ

text: ρ2

text: ζ

text: (ρ)

text: E

text: ζ

text: (ρ)

text: ∂ρ2

text: ρ2

text: with

text: G

text: G

text: kφ?

text: kφ

text: E?

text: 3π‘4

text: E.

text: P

text: P

text: A

text: solution

text: of

text: can

text: be

text: constructed

text: by

text: considering

text: the

text: following

text: wave

text: function

text: ζ

text: f

text: (ρ)

text: e−

text: ρλ,

text: where

text: q

text: k

text: φ?

text: λ

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Developing a model for neutron star oscillations following starquakes 11 − where ηr is the radial component of ηi and δ(r R) is the Dirac delta function. We can see that this vanishes everywhere except at the surface. By substituting this into Poisson’s equation (56), the gravitational perturbation δΦSD can be shown to satisfy the jump conditions (cid:104) (cid:105) (cid:20) (cid:21) δΦSD R+(cid:15) = 0, (59) − R (cid:15) d (δΦSD) R+(cid:15) = 4πGρηSD(R, θ, φ) (60) dr − r R (cid:15) at the surface. By matching solutions ﬁnite inside an(cid:16)d outside the st(cid:17)ar, we ﬁnd that ∆ΦSD = 8πGρr η(cid:48)SC + UCD(r) P (cos θ). (61) 15 r 2 We also have boundary conditions on the stress tensor at the surface. Particles on the surface of Star S, where τ = 0, ir should still satisfy the same condition (‘zero traction’) after the perturbation, leading to the conditions ∆τSD(R) = 0. (62) ir This gives two independent boundary conditions for the (rr) and (rθ) components. To solve these equations, we will need to make use of the equilibrium conﬁguration of Star B, which was unstrained at angular velocity Ω and is now spinning at A angular velocity Ω . The displacement ﬁeld ξAB is found in Baym & Pines (1971) to be of the form B ∇ ξAB = UAB(r)P (cos θ)rˆ+ V AB(r) P , (63) 2 2 where P (cos θ) is the second Legendre polynomial and the r(cid:0)adial functio(cid:1)ns UAB and V AB are given by 2 UAB(r) = 1 tX (cid:18)8R2r − 3r3 ,(cid:19) (64) 20R2 1 + b V AB(r) = 1 tX 4R2r2 − 5r4 . (65) 20R2 1 + b 2 These have been written in terms of our small parameters b, t and X (13), where for a homogeneous elastic star b and t have the exact values 57µ b = , (66) 8πGρ2R2 5 t = Ω2 . (67) 2πGρ B We can also ﬁnd the surface shape of the Star B (and therefore Star C) by calculating a map from the surface of Star S to that of Star B, (cid:18) (cid:19) (ηr)SB (R) = −tR 1 1 + b Ω2A P (cos θ). (68) 4 1 + b Ω2 2 B For the rest of the calculation, we follow a similar method to that of Franco, Link & Epstein (2000), deﬁning − − ∇ µhSD = ∆pSD ρ∆ΦSD + ρ (Ω2 r2 cos2 θ). (69) i D ∇ so that 2hSD = 0. The solutions with the same symmetries as the centrifugal potential satisfy hSD = H r2P (cos θ) + H , (70) 2 2 0 for H and H constant. We decompose the displacement vector ξCD as 2 0 i (cid:13) c 0000 RAS, MNRAS 000, 000–000 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Developing

label: title

text: a

text: model

text: for

text: neutron

text: star

text: oscillations

text: following

text: starquakes

text: where

text: ηr

text: is

text: the

text: radial

text: component

text: of

text: ηi

text: and

text: δ(r

text: R)

text: is

text: the

text: Dirac

text: delta

text: function.

text: We

text: can

text: see

text: that

text: this

text: vanishes

text: everywhere

text: except

text: at

text: the

text: surface.

text: By

text: substituting

text: this

text: into

text: Poisson’s

text: equation

text: the

text: gravitational

text: perturbation

text: δΦSD

text: can

text: be

text: shown

text: to

text: satisfy

text: the

text: jump

text: conditions

text: δΦSD

text: R

text: d

text: (δΦSD)

text: θ,

text: φ)

text: dr

text: r

text: R

text: at

text: the

text: surface.

text: By

text: matching

text: solutions

text: ﬁnite

text: inside

text: outside

text: the

text: we

text: ﬁnd

text: that

text: ∆ΦSD

text: UCD(r)

text: P

text: (cos

text: θ).

text: r

text: We

text: also

text: have

text: boundary

text: conditions

text: on

text: the

text: stress

text: tensor

text: at

text: the

text: surface.

text: Particles

text: on

text: the

text: surface

text: of

text: Star

text: S,

text: where

text: τ

text: ir

text: should

text: still

text: satisfy

text: the

text: same

text: condition

text: (‘zero

text: traction’)

text: after

text: the

text: perturbation,

text: leading

text: to

text: the

text: conditions

text: ∆τSD(R)

text: ir

text: This

text: gives

text: two

text: independent

text: boundary

text: conditions

text: for

text: the

text: (rr)

text: and

text: (rθ)

text: components.

text: To

text: solve

text: these

text: equations,

text: we

text: will

text: need

text: to

text: make

text: use

text: of

text: the

text: equilibrium

text: conﬁguration

text: of

text: Star

text: B,

text: which

text: was

text: unstrained

text: at

text: angular

text: velocity

text: Ω

text: and

text: is

text: now

text: spinning

text: at

text: A

text: angular

text: velocity

text: Ω

text: The

text: displacement

text: ﬁeld

text: ξAB

text: is

text: found

text: in

text: Baym

text: Pines

text: to

text: be

text: of

text: the

text: form

text: B

text: ξAB

text: UAB(r)P

text: (cos

text: θ)rˆ+

text: V

text: AB(r)

text: P

text: where

text: P

text: (cos

text: θ)

text: is

text: the

text: second

text: Legendre

text: polynomial

text: and

text: the

text: UAB

text: and

text: V

text: AB

text: are

text: given

text: by

text: UAB(r)

text: tX

text: b

text: V

text: AB(r)

text: tX

text: b

text: These

text: have

text: been

text: written

text: in

text: terms

text: of

text: our

text: small

text: parameters

text: b,

text: t

text: and

text: X

text: where

text: for

text: a

text: homogeneous

text: elastic

text: star

text: b

text: and

text: t

text: have

text: the

text: exact

text: values

text: b

text: t

text: Ω2

text: 2πGρ

text: B

text: We

text: can

text: also

text: ﬁnd

text: the

text: surface

text: shape

text: of

text: the

text: Star

text: B

text: (and

text: therefore

text: Star

text: C)

text: by

text: calculating

text: a

text: map

text: from

text: the

text: surface

text: of

text: Star

text: S

text: to

text: that

text: of

text: Star

text: B,

text: (ηr)SB

text: (R)

text: −tR

text: b

text: P

text: (cos

text: θ).

text: b

text: Ω2

text: B

text: For

text: the

text: rest

text: of

text: the

text: calculation,

text: we

text: follow

text: a

text: similar

text: method

text: to

text: that

text: of

text: Franco,

text: Link

text: Epstein

text: deﬁning

text: µhSD

text: ∆pSD

text: ρ∆ΦSD

text: ρ

text: θ).

text: i

text: D

text: so

text: that

text: The

text: solutions

text: with

text: the

text: same

text: symmetries

text: as

text: the

text: centrifugal

text: potential

text: satisfy

text: hSD

text: H

text: (cos

text: θ)

text: H

text: for

text: H

text: and

text: H

text: constant.

text: We

text: decompose

text: the

text: displacement

text: vector

text: ξCD

text: as

text: i

text: c

text: RAS,

text: MNRAS

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text: ##LTLine##

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text: 20 L. Keer et al. realistic model – we have lost the key observational feature of the glitch, the change in rotation rate of the star – there are good reasons to discuss this ﬁrst. For a start, it is easier to interpret the results for this simpler case, and certain features of these will carry over to the more realistic case where the star is still rotating before the glitch. We can also check that the results for the rotating problem converge to the nonrotating one as rotation becomes small. Another reason for considering the nonrotating problem ﬁrst is that the rotation of the star introduces signiﬁcant com- plications to the projection scheme. One of these is that the eigenfunctions of the rotating star are no longer orthogonal. Another is the existence of the zero eigenvalue l = 1 toroidal mode (118) that our velocity initial data is built from. As this mode is zero frequency, it will not have the eiωt time-dependence of the other eigenfunctions and we will have to consider it separately. We will discuss these problems in Section . 6.1 Special case: glitch at zero spin For the special case where Ω = 0, the velocity ﬁeld part of the initial data (118) is zero. For the displacement ﬁeld, B although there is no change in angular velocity of the star at the glitch, we still have a ﬁnite value of the parameter z (20), | 5 z = bΩ2 , (119) ΩB=0 2πGρ A and the displacement ﬁeld initial data takes the same form (117) as for the general, rotating case. Our only free parameter in this Ω = 0 case is z, which we have set to unity. B The eigenfunctions we are projecting against are those of the nonrotating star (87), (88), with time-dependence ≡ ξα (x, t) ξα (x)eiωαt. (120) (0) (0) The initial data is then a sum over these modes, with each eigenfunction ξα excited by some amplitude bα. The initial data is real, so to ensure that the sum over the eig(cid:88)enfu(cid:104)nctions is also real we(cid:105)write it as ξID(x) = 1 (cid:88) (cid:104)bαξα (x) + b∗αξ∗α(x) , (cid:105) (121) 2 (0) (0) α ξ˙ID(x) = 1 iωαbαξα (x) − iωαb∗αξ∗α(x) . (122) 2 (0) (0) α The complex conjugate is not really necessary at the moment, where we are projecting against the real eigenfunctions of a nonrotating star. However, it will be required later when we extend to the rotating case and the eigenfunctions have an imaginary part, so for consistency we keep it in here. We will scale our eigenfunctions so that they are orthonormal with respect to the inner product (110) deﬁned in Section (cid:68): (cid:69) ξβ , ξα = δαβ. (123) (0) (0) By taking the inner products of ξID and ξ˙ID wi(cid:16)th(cid:68) an eigen(cid:69)function(cid:68)ξα, we ﬁ(cid:69)n(cid:17)d that the amplitudes bα satisfy bα = − i ξα , ξ˙ID + iωα ξα , ξID . (124) ωα (0) (0) Figure 7 shows the results of the projection for diﬀerent values of b, ranging from the high value of b = 0.1 7(a) down − − to the physical range of b = 10 5 7(e) and b = 10 6 7(f). The x-axis shows the radial number of the mode, while the y-axis shows the amplitude of that mode. In all of these plots, the hybrid ﬂuid-elastic mode discussed in Section has the greatest amplitude; this is the mode with the frequency closest to the fundamental Kelvin mode of a purely ﬂuid star. For the higher values of b, the lowest order modes are also excited to signiﬁcant amplitudes: these are the shear modes with the lowest ∼ − number of radial nodes. However, for physical values of b (b 10 5), only the ﬂuid-elastic mode is appreciably excited. Finally, as a check we can also show that we are reproducing the initial data correctly by reconstructing the sum over the eigenfunctions (120) with our calculated amplitudes bα. The results of this are shown in Appendix . 6.2 The general case: glitch at arbitrary spin We will now move on to the projection of the initial data in the general case, where Star B is rotating before the glitch. Here we will have to deal with the zero eigenvalue l = 1 toroidal mode (118) that our velocity initial data is built from. As this mode is zero frequency, it will not have the eiωt time-dependence of the other modes and we will have to consider it (cid:13) c 0000 RAS, MNRAS 000, 000–000 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L.

text: Keer

text: et

text: al.

text: realistic

text: model

text: we

text: have

text: lost

text: the

text: key

text: observational

text: feature

text: of

text: the

text: glitch,

text: the

text: change

text: in

text: rotation

text: rate

text: of

text: the

text: star

text: there

text: are

text: good

text: reasons

text: to

text: discuss

text: this

text: ﬁrst.

text: For

text: a

text: start,

text: it

text: is

text: easier

text: to

text: interpret

text: the

text: results

text: for

text: this

text: simpler

text: case,

text: and

text: certain

text: features

text: of

text: these

text: will

text: carry

text: over

text: to

text: the

text: more

text: realistic

text: case

text: where

text: the

text: star

text: is

text: still

text: rotating

text: before

text: the

text: glitch.

text: We

text: can

text: also

text: check

text: that

text: the

text: results

text: for

text: the

text: rotating

text: problem

text: converge

text: to

text: the

text: nonrotating

text: one

text: as

text: rotation

text: becomes

text: small.

text: Another

text: reason

text: for

text: considering

text: the

text: nonrotating

text: problem

text: ﬁrst

text: is

text: that

text: the

text: rotation

text: of

text: the

text: star

text: introduces

text: signiﬁcant

text: com-

text: plications

text: to

text: the

text: projection

text: scheme.

text: One

text: of

text: these

text: is

text: that

text: the

text: eigenfunctions

text: of

text: the

text: rotating

text: star

text: are

text: no

text: longer

text: orthogonal.

text: Another

text: is

text: the

text: existence

text: of

text: the

text: zero

text: eigenvalue

text: l

text: toroidal

text: mode

text: that

text: our

text: velocity

text: initial

text: data

text: is

text: built

text: from.

text: As

text: this

text: mode

text: is

text: zero

text: frequency,

text: it

text: will

text: not

text: have

text: the

text: eiωt

text: time-dependence

text: of

text: the

text: other

text: eigenfunctions

text: and

text: we

text: will

text: have

text: to

text: consider

text: it

text: separately.

text: We

text: will

text: discuss

text: these

text: problems

text: in

text: Section

text: Special

text: case:

text: glitch

text: at

text: zero

text: spin

text: For

text: the

text: special

text: case

text: where

text: Ω

text: the

text: velocity

text: ﬁeld

text: part

text: of

text: the

text: initial

text: data

text: is

text: zero.

text: For

text: the

text: displacement

text: ﬁeld,

text: B

text: although

text: there

text: is

text: no

text: change

text: in

text: angular

text: velocity

text: of

text: the

text: star

text: at

text: the

text: glitch,

text: we

text: still

text: have

text: a

text: ﬁnite

text: value

text: of

text: the

text: parameter

text: z

text: z

text: 2πGρ

text: A

text: and

text: the

text: displacement

text: ﬁeld

text: initial

text: data

text: takes

text: the

text: same

text: form

text: as

text: for

text: the

text: general,

text: rotating

text: case.

text: Our

text: only

text: free

text: parameter

text: in

text: this

text: Ω

text: case

text: is

text: z,

text: which

text: we

text: have

text: set

text: to

text: unity.

text: B

text: The

text: eigenfunctions

text: we

text: are

text: projecting

text: against

text: are

text: those

text: of

text: the

text: nonrotating

text: star

text: with

text: time-dependence

text: ξα

text: (x,

text: t)

text: ξα

text: (x)eiωαt.

text: The

text: initial

text: data

text: is

text: then

text: a

text: sum

text: over

text: these

text: modes,

text: with

text: each

text: eigenfunction

text: ξα

text: excited

text: by

text: some

text: amplitude

text: bα.

text: The

text: initial

text: data

text: is

text: real,

text: so

text: to

text: ensure

text: that

text: the

text: sum

text: over

text: the

text: is

text: also

text: real

text: it

text: as

text: ξID(x)

text: (x)

text: b∗αξ∗α(x)

text: α

text: ξ˙ID(x)

text: iωαbαξα

text: (x)

text: iωαb∗αξ∗α(x)

text: α

text: The

text: complex

text: conjugate

text: is

text: not

text: really

text: necessary

text: at

text: the

text: moment,

text: where

text: we

text: are

text: projecting

text: against

text: the

text: real

text: eigenfunctions

text: of

text: a

text: nonrotating

text: star.

text: However,

text: it

text: will

text: be

text: required

text: later

text: when

text: we

text: extend

text: to

text: the

text: rotating

text: case

text: and

text: the

text: eigenfunctions

text: have

text: an

text: imaginary

text: part,

text: so

text: for

text: consistency

text: we

text: keep

text: it

text: in

text: here.

text: We

text: will

text: scale

text: our

text: eigenfunctions

text: so

text: that

text: they

text: are

text: orthonormal

text: with

text: respect

text: to

text: the

text: inner

text: product

text: deﬁned

text: in

text: Section

text: ξβ

text: ξα

text: δαβ.

text: By

text: taking

text: the

text: inner

text: products

text: of

text: ξID

text: and

text: ξ˙ID

text: an

text: we

text: that

text: the

text: amplitudes

text: bα

text: satisfy

text: bα

text: i

text: ξα

text: ξ˙ID

text: iωα

text: ξα

text: ξID

text: ωα

text: Figure

text: shows

text: the

text: results

text: of

text: the

text: projection

text: for

text: diﬀerent

text: values

text: of

text: b,

text: ranging

text: from

text: the

text: high

text: value

text: of

text: b

text: down

text: to

text: the

text: physical

text: range

text: of

text: b

text: and

text: b

text: The

text: x-axis

text: shows

text: the

text: radial

text: number

text: of

text: the

text: mode,

text: while

text: the

text: y-axis

text: shows

text: the

text: amplitude

text: of

text: that

text: mode.

text: In

text: all

text: of

text: these

text: plots,

text: the

text: hybrid

text: ﬂuid-elastic

text: mode

text: discussed

text: in

text: Section

text: has

text: the

text: greatest

text: amplitude;

text: this

text: is

text: the

text: mode

text: with

text: the

text: frequency

text: closest

text: to

text: the

text: fundamental

text: Kelvin

text: mode

text: of

text: a

text: purely

text: ﬂuid

text: star.

text: For

text: the

text: higher

text: values

text: of

text: b,

text: the

text: lowest

text: order

text: modes

text: are

text: also

text: excited

text: to

text: signiﬁcant

text: amplitudes:

text: these

text: are

text: the

text: shear

text: modes

text: with

text: the

text: lowest

text: number

text: of

text: radial

text: nodes.

text: However,

text: for

text: physical

text: values

text: of

text: b

text: (b

text: only

text: the

text: ﬂuid-elastic

text: mode

text: is

text: appreciably

text: excited.

text: Finally,

text: as

text: a

text: check

text: we

text: can

text: also

text: show

text: that

text: we

text: are

text: reproducing

text: the

text: initial

text: data

text: correctly

text: by

text: reconstructing

text: the

text: sum

text: over

text: the

text: eigenfunctions

text: with

text: our

text: calculated

text: amplitudes

text: bα.

text: The

text: results

text: of

text: this

text: are

text: shown

text: in

text: Appendix

text: The

text: general

text: case:

text: glitch

text: at

text: arbitrary

text: spin

text: We

text: will

text: now

text: move

text: on

text: to

text: the

text: projection

text: of

text: the

text: initial

text: data

text: in

text: the

text: general

text: case,

text: where

text: Star

text: B

text: is

text: rotating

text: before

text: the

text: glitch.

text: Here

text: we

text: will

text: have

text: to

text: deal

text: with

text: the

text: zero

text: eigenvalue

text: l

text: toroidal

text: mode

text: that

text: our

text: velocity

text: initial

text: data

text: is

text: built

text: from.

text: As

text: this

text: mode

text: is

text: zero

text: frequency,

text: it

text: will

text: not

text: have

text: the

text: eiωt

text: time-dependence

text: of

text: the

text: other

text: modes

text: and

text: we

text: will

text: have

text: to

text: consider

text: it

text: c

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text: 22 L. Keer et al. The complex coeﬃcients bα contain the amplitude and phase of each mode. It will be convenient to write the zero frequency eigenfunction ξ1 in a similar way to the other eigenfunctio(cid:16)ns by (cid:0)deﬁ(cid:1)ni(cid:17)ng (0) i (cid:16) − (cid:0) (cid:1)∗(cid:17) C1 = b1 b1 , (131) 2 1 ∗ D1 = b1 + b1 , (132) 2 where b1 is a comp(cid:20)lex (cid:16)consta(cid:0)nt,(cid:1)so(cid:17)that ((cid:16)130) b(cid:0)eco(cid:1)m(cid:17)es(cid:21) (cid:88) (cid:104) (cid:105) ∞ ξ(x, t) = i b1 − b1 ∗ t + 1 b1 + b1 ∗ ξ1 (x) + 1 bαξα(x)eiωαt + b∗αξ∗α(x)e−iωαt . (133) 2 2 (0) 2 α=2 Diﬀerentiating this, the velocity ﬁeld is (cid:16) (cid:0) (cid:1) (cid:17) (cid:88) (cid:2) (cid:3) ∞ i − ∗ 1 − ∗ ∗ ξ˙(x, t) = b1 b1 ξ1 (x) + iωα bαξα(x, t) iωα b αξ α(x, t) , (134) 2 (0) 2 (0) (0) α=2 where we have used the fact that there are no rotational corrections to the zeroth order eigenvalues ω in the m = 0 (0) case we are interested in. We can combine ξ1 with the o(cid:40)ther eigenfunctions by making the deﬁnition (0) 1 α = 1, ω˜α = (135) (0) ωα otherwise. (0) Specialising to t = 0, we can then use this along with our deﬁnitions of C (131) and D (132) to write the initial data in the neat form (cid:88) ∞ ξID(x) ≡ ξ(x, 0) = 1 (cid:88) [(cid:2)bαξα(x) + c.c.] , (cid:3) (136) 2 α=1 ∞ ξ˙ID(x) ≡ ξ˙(x, 0) = 1 iω˜α bαξα(x) + c.c. . (137) 2 (0) α=1 Next we expand the eigenfunctions in the sum to ﬁrst order in rotation (102). Introducing the notation (cid:88) (cid:16) (cid:17) (cid:88) ∞ ∞ ≡ ξα = δαβ + λαβ ξβ Λαβξβ , (138) (1) (0) (0) β=1 β=1 we then have      (cid:88) (cid:88) (cid:88)      ∞ ∞ ∗ ∞ ξID(x) = bα Λαβξβ +(bα) (Λ∗)αβξβ ,  (139) (cid:88) 2 (cid:88) (0) 2 (cid:88) (0) α=1  β=1  β=1   ∞ ∞ ∗ ∞ ξ˙ID(x) = iω˜α bα Λαβξβ − iω˜α (bα) (Λ∗)αβξβ , (140) 2 (0) 2 (0) α=1 β=1 β=1 where the corrections to ξ1 are λ1β = 0. We can take an inner product with another zeroth order eigenfunction ξγ to (0) (0) obtain (cid:68) (cid:69) (cid:88) (cid:20) (cid:21) ∞ ∗ (cid:68)ξγ , ξID(x)(cid:69) = (cid:88) (cid:20)bα (Λαγ) + (bα) ((Λ∗)αγ) , (cid:21) (141) (0) 2 2 α=1 ∞ ∗ ξγ , ξ˙ID(x) = iω˜α bα (Λαγ) − iω˜α (bα) ((Λ∗)αγ) . (142) (0) 2 2 α=1 In practice, when carrying out a projection numerically we will only consider a ﬁnite sum, cutting oﬀ at some suﬃciently large β = N. We can then view (141) and (142) as vectors of N equations, so that we can rewrite them as (cid:13) c 0000 RAS, MNRAS 000, 000–000 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L.

text: Keer

text: et

text: al.

text: The

text: complex

text: coeﬃcients

text: bα

text: contain

text: the

text: amplitude

text: and

text: phase

text: of

text: each

text: mode.

text: It

text: will

text: be

text: convenient

text: to

text: write

text: the

text: zero

text: frequency

text: eigenfunction

text: ξ1

text: in

text: a

text: similar

text: way

text: to

text: the

text: other

text: by

text: i

text: where

text: is

text: a

text: ξ(x,

text: t)

text: i

text: t

text: ξ1

text: (x)

text: bαξα(x)eiωαt

text: b∗αξ∗α(x)e−iωαt

text: Diﬀerentiating

text: this,

text: the

text: velocity

text: ﬁeld

text: is

text: i

text: ξ˙(x,

text: t)

text: ξ1

text: (x)

text: iωα

text: bαξα(x,

text: t)

text: iωα

text: b

text: αξ

text: α(x,

text: t)

text: where

text: we

text: have

text: used

text: the

text: fact

text: that

text: there

text: are

text: no

text: rotational

text: corrections

text: to

text: the

text: zeroth

text: order

text: eigenvalues

text: ω

text: in

text: the

text: m

text: case

text: we

text: are

text: interested

text: in.

text: We

text: can

text: combine

text: ξ1

text: with

text: the

text: eigenfunctions

text: by

text: making

text: the

text: deﬁnition

text: α

text: ω˜α

text: ωα

text: otherwise.

text: Specialising

text: to

text: t

text: we

text: can

text: then

text: use

text: this

text: along

text: with

text: our

text: deﬁnitions

text: of

text: C

text: and

text: D

text: to

text: write

text: the

text: initial

text: data

text: in

text: the

text: neat

text: form

text: ξID(x)

text: ξ(x,

text: c.c.]

text: ξ˙ID(x)

text: ξ˙(x,

text: iω˜α

text: bαξα(x)

text: c.c.

text: Next

text: we

text: expand

text: the

text: eigenfunctions

text: in

text: the

text: sum

text: to

text: ﬁrst

text: order

text: in

text: rotation

text: Introducing

text: the

text: notation

text: ξα

text: δαβ

text: λαβ

text: ξβ

text: Λαβξβ

text: we

text: then

text: have

text: ξID(x)

text: bα

text: Λαβξβ

text: +(bα)

text: (Λ∗)αβξβ

text: β=1

text: ξ˙ID(x)

text: iω˜α

text: bα

text: Λαβξβ

text: iω˜α

text: (bα)

text: (Λ∗)αβξβ

text: where

text: the

text: corrections

text: to

text: ξ1

text: are

text: λ1β

text: We

text: can

text: take

text: an

text: inner

text: product

text: with

text: another

text: zeroth

text: order

text: eigenfunction

text: ξγ

text: to

text: obtain

text: (Λαγ)

text: (bα)

text: ((Λ∗)αγ)

text: ξγ

text: ξ˙ID(x)

text: iω˜α

text: bα

text: (Λαγ)

text: iω˜α

text: (bα)

text: ((Λ∗)αγ)

text: In

text: practice,

text: when

text: carrying

text: out

text: a

text: projection

text: numerically

text: we

text: will

text: only

text: consider

text: a

text: ﬁnite

text: sum,

text: cutting

text: oﬀ

text: at

text: some

text: suﬃciently

text: large

text: β

text: N.

text: We

text: can

text: then

text: view

text: and

text: as

text: vectors

text: of

text: N

text: equations,

text: so

text: that

text: we

text: can

text: rewrite

text: them

text: as

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

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text: 24 L. Keer et al. Zeroth order First order l=1 zero (no l=0 as frequency incompressible) l=1 l=2 l=1 l=2 l=3 l=2 l=3 (l=4 out of included range) = velocity ﬁeld initial data = displacement ﬁeld initial data Figure 8. Schematic showing how the zeroth order modes couple to other modes at ﬁrst order in rotation. The displacement ﬁeld has an l = 2 form and couples to l = 1 and l = 3; this is shown in blue. The velocity ﬁeld, shown in read, has the form of a zero frequency l = 1 mode; it has no rotational corrections. The other modes in the range we are considering are also shown. The l = 1 modes only couple to l = 2, while for the l = 3 modes, only the l = 2 coupling falls into the considered range.        I 0 0 ε[ST12] 0 0   N N   0 IN ε[TS12]N 0 0 0     0 ε[ST21]N IN 0 0 ε[ST23]N  . (154) ε[TS21] 0 0 I ε[TS23] 0 N N N 0 0 0 ε[ST32] I 0 N N 0 0 ε[TS32] 0 0 I N N × Each entry represents an N N matrix. Most oﬀ-diagonal entries are zero apart from a few blocks. For l = 2 these are the l = 1 and 3 toroidal corrections to the spheroidal eigenfunctions, [ST21] and [ST23] , and the l = 1 and 3 spheroidal N N corrections to the toroidal eigenfunctions, [TS21] and [TS23] . For l = 1 there are only l = 2 corrections; this is also true N N for l = 3 as we are not including l = 4. 6.3 Results of the projection − − − We carried out the projection using this numerical scheme for the cases b = 10 1, 10 2, 10 3. As with the calculation of the rotational corrections of the eigenfunctions, we cut oﬀ at N = 10, 15 and 40 respectively, with these values chosen so that the hybrid ﬂuid-elastic mode is within the range for each case. The increasingly high n value of this mode for small b, × and the fact that the calculation of the 6N 6N Λ matrix is computationally intensive, meant that we did not investigate any smaller values of b. For given b, we also need to consider what parameter range of t our scheme is valid for. We have used the slow rotation approximation to calculate the modes. Considering the form of the eigenvalue equation for the rotating star (9√8), we can (cid:28) ∼ expect this to be a good approximation when ω2A is large compared to εωB, i.e when ΩωB 1. We have ΩB t, but we also have to take into consideration√which mode ω we are looking at. The elastic mode√s scale with b, so that for the lowest (cid:28) modes with small n value we need t 1. The hybrid ﬂuid-like m√ode has a scaling Gρ – we have scaled our numerical b (cid:28) results so that this is of order unity, so for this mode we only need t 1. (cid:13) c 0000 RAS, MNRAS 000, 000–000 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L.

text: Keer

text: et

text: al.

text: Zeroth

text: order

text: First

text: order

text: zero

text: (no

text: as

text: frequency

text: incompressible)

text: out

text: of

text: included

text: range)

text: velocity

text: ﬁeld

text: initial

text: data

text: displacement

text: ﬁeld

text: initial

text: data

text: Figure

text: Schematic

text: showing

text: how

text: the

text: zeroth

text: order

text: modes

text: couple

text: to

text: other

text: modes

text: at

text: ﬁrst

text: order

text: in

text: rotation.

text: The

text: displacement

text: ﬁeld

text: has

text: an

text: l

text: form

text: and

text: couples

text: to

text: l

text: and

text: l

text: this

text: is

text: shown

text: in

text: blue.

text: The

text: velocity

text: ﬁeld,

text: shown

text: in

text: read,

text: has

text: the

text: form

text: of

text: a

text: zero

text: frequency

text: l

text: mode;

text: it

text: has

text: no

text: rotational

text: corrections.

text: The

text: other

text: modes

text: in

text: the

text: range

text: we

text: are

text: considering

text: are

text: also

text: shown.

text: The

text: l

text: modes

text: only

text: couple

text: to

text: l

text: while

text: for

text: the

text: l

text: modes,

text: only

text: the

text: l

text: coupling

text: falls

text: into

text: the

text: considered

text: range.

text: I

text: N

text: N

text: IN

text: IN

text: I

text: N

text: N

text: N

text: I

text: N

text: N

text: I

text: N

text: N

text: Each

text: entry

text: represents

text: an

text: N

text: N

text: matrix.

text: Most

text: oﬀ-diagonal

text: entries

text: are

text: zero

text: apart

text: from

text: a

text: few

text: blocks.

text: For

text: l

text: these

text: are

text: the

text: l

text: and

text: toroidal

text: corrections

text: to

text: the

text: spheroidal

text: eigenfunctions,

text: and

text: and

text: the

text: l

text: and

text: spheroidal

text: N

text: N

text: corrections

text: to

text: the

text: toroidal

text: eigenfunctions,

text: and

text: For

text: l

text: there

text: are

text: only

text: l

text: corrections;

text: this

text: is

text: also

text: true

text: N

text: N

text: for

text: l

text: as

text: we

text: are

text: not

text: including

text: l

text: Results

text: of

text: the

text: projection

text: We

text: carried

text: out

text: the

text: projection

text: using

text: this

text: numerical

text: scheme

text: for

text: the

text: cases

text: b

text: As

text: with

text: the

text: calculation

text: of

text: the

text: rotational

text: corrections

text: of

text: the

text: eigenfunctions,

text: we

text: cut

text: oﬀ

text: at

text: N

text: and

text: respectively,

text: with

text: these

text: values

text: chosen

text: so

text: that

text: the

text: hybrid

text: ﬂuid-elastic

text: mode

text: is

text: within

text: the

text: range

text: for

text: each

text: case.

text: The

text: increasingly

text: high

text: n

text: value

text: of

text: this

text: mode

text: for

text: small

text: b,

text: and

text: the

text: fact

text: that

text: the

text: calculation

text: of

text: the

text: Λ

text: matrix

text: is

text: computationally

text: intensive,

text: meant

text: that

text: we

text: did

text: not

text: investigate

text: any

text: smaller

text: values

text: of

text: b.

text: For

text: given

text: b,

text: we

text: also

text: need

text: to

text: consider

text: what

text: parameter

text: range

text: of

text: t

text: our

text: scheme

text: is

text: valid

text: for.

text: We

text: have

text: used

text: the

text: slow

text: rotation

text: approximation

text: to

text: calculate

text: the

text: modes.

text: Considering

text: the

text: form

text: of

text: the

text: eigenvalue

text: equation

text: for

text: the

text: rotating

text: star

text: we

text: can

text: expect

text: this

text: to

text: be

text: a

text: good

text: approximation

text: when

text: is

text: large

text: compared

text: to

text: εωB,

text: i.e

text: when

text: ΩωB

text: We

text: have

text: ΩB

text: t,

text: but

text: we

text: also

text: have

text: to

text: take

text: into

text: consideration√which

text: mode

text: ω

text: we

text: are

text: looking

text: at.

text: The

text: elastic

text: mode√s

text: scale

text: with

text: b,

text: so

text: that

text: for

text: the

text: lowest

text: modes

text: with

text: small

text: n

text: value

text: we

text: need

text: t

text: The

text: hybrid

text: ﬂuid-like

text: m√ode

text: has

text: a

text: scaling

text: Gρ

text: we

text: have

text: scaled

text: our

text: numerical

text: b

text: results

text: so

text: that

text: this

text: is

text: of

text: order

text: unity,

text: so

text: for

text: this

text: mode

text: we

text: only

text: need

text: t

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

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text: SCHUBERT CALCULUS AND SHIFTING OF INTERVAL POSITROID VARIETIES 15 ⊆ A ⊆ If X Gr ( n) is T-invariant and irreducible, its matroid is the collection B(X) := {S A ∈ k · [n] : S X}. A positroid is one arising from X = T V where V has all nonnegative real Plu¨ cker coordinates [Pos]. The matroids of the {Π } are positroids, whose connected ﬂats f are intervals (not just cyclic intervals), hence the term “interval positroid variety”. Under Grassmannian duality Gr (An) =∼ Gr (An), o→ne can check that the positroid k n−k variety Π is corresponded with the positroid variety Π (cid:55) ◦ . This does not preserve f (i i+n) f−1 the subclass of interval positroid varieties (other than the Richardson varieties). The ad- ditional power available from dualizing is exploited in [KnLed]. One interpretation of proposition 2.5 is that interval positroids are “dual transversal” matroids. Consider the dots in f as a set of n − k choosy brides b, each of whom will only marry a groom within a certain height range [i , j ]. Then each S is an acceptable set of (cid:12) (cid:12) b b groom(cid:12) s. Hall’s Mar(cid:12)riage Theorem (from which this terminology is derived) says that if ⊂ some groomset S [n], |S| = rank(f) is unmarriable, it is because there is a set B of brides ∩ ∪ with S ∈ [i, j] < |B|. Proposition 2.5 goes further in two ways: it says that if S is (i,j) B unmarriable, then (1) there is an interval [a, b] where B is the brides with ranges in that interval, and the grooms in that interval aren’t numerous enough, (2) even if one includes the grooms none of those brides wants. ⊆ ∈ N For S [n], r , let ∈ ≤ X ≤ := {rowspan(M) : M M × , rank M = k, rank (columns S of M) r}. S r k n If S is a cyclic interval, call this a basic positroid variety. Clearly every positroid variety is an intersection of basic ones, and one can show that no basic positroid variety is an intersection of other positroid varieties. (cid:0) (cid:1) (cid:48) Theorem 2.6. Let J be a bounded juggling pattern, and i, j = J(i) one of its dots. Let C = (j ) be the columns of those dots minimally Northwest(cid:91)of (i, J(i)), ordered Northeast/Southwest, and r = |[i, j − 1]| − #{dots in J that are weakly Southwest of (i, j − 1)}. Then ↔ ∩ Π X = Π (cid:48)◦ (cid:48) . J [i,j−1]<r J (j j ) (cid:48)∈ j C Proof. An intersection of positroid varieties is a reduced union of positroid varieties [KLS14, ∩ corollary 4.4], so we just need to det↔ermine which such occur in Π X . We alert the J [i,j−1]<r reader that this is perhaps the subtlest combinatorial argument in the paper. By the↔deﬁnition of C, each J ◦ (j j(cid:48)) (cid:109) J is a covering relation in afﬁne Bruhat order, so Π (cid:48)◦ (cid:48) ⊂ Π . Also, the dot in column j(cid:48) of ↔J moves down to row i, providing another J (j j ) J ⊆ ⊇ dot weakly Southwest of (i, j − 1), hence Π (cid:48)◦ (cid:48) X . Together these prove the J (j j ) [i,j−1]<r containment. ⊆ ∩ For the reverse containment, we↔need to show that for any Π (cid:48) Π X , there (cid:48) ∈ ⊆ J J (cid:48) [i≤,j−1]<r exists a j C such that Π (cid:48) Π ◦ (cid:48) . In rank matrix terms, we need r(J ) r(J), with J J (j j ) (cid:48) × (cid:48) strict inequality on some rectangle [J−1(j ) + 1, i] [j , j − 1], not just at (i, j − 1). (cid:108) (cid:108) (cid:108) (cid:48) Consider a saturated chain J = J J . . . J = J in strong afﬁne Bruhat order, 0 1 k ≥ ≥ ≥ so in particular we have entrywise inequalities r(J ) r(J ) . . . r(J ), and more 0 1 k ##LTLine##

text: SCHUBERT

font: AJRIYF+URWPalladioL-Roma

text: CALCULUS

text: AND

text: SHIFTING

text: OF

text: INTERVAL

text: POSITROID

text: VARIETIES

text: A

text: If

text: X

text: Gr

text: n)

text: is

text: T-invariant

text: and

text: irreducible,

text: its

text: matroid

font: PQWQTW+URWPalladioL-Bold

text: is

text: the

text: collection

text: B(X)

text: {S

text: A

text: k

text: [n]

text: S

text: X}.

text: A

text: positroid

text: is

text: one

text: arising

text: from

text: X

text: T

text: V

text: where

text: V

text: has

text: all

text: nonnegative

text: real

text: Plu¨

text: cker

text: coordinates

text: [Pos].

text: The

text: matroids

text: of

text: the

text: {Π

text: are

text: positroids,

text: whose

text: connected

text: ﬂats

text: f

text: are

text: intervals

text: (not

text: just

text: cyclic

text: intervals),

text: hence

text: the

text: term

text: “interval

text: positroid

text: variety”.

text: Under

text: Grassmannian

text: duality

text: Gr

text: (An)

text: Gr

text: (An),

text: o→ne

text: can

text: check

text: that

text: the

text: positroid

text: k

text: n−k

text: variety

text: Π

text: is

text: corresponded

text: with

text: the

text: positroid

text: variety

text: Π

text: This

text: does

text: not

text: preserve

text: f

text: (i

text: i+n)

text: the

text: subclass

text: of

text: interval

text: positroid

text: varieties

text: (other

text: than

text: the

text: Richardson

text: varieties).

text: The

text: ad-

text: ditional

text: power

text: available

text: from

text: dualizing

text: is

text: exploited

text: in

text: [KnLed].

text: One

text: interpretation

text: of

text: proposition

text: is

text: that

text: interval

text: positroids

text: are

text: “dual

text: transversal”

text: matroids.

text: Consider

text: the

text: dots

text: in

text: f

text: as

text: a

text: set

text: of

text: n

text: k

text: choosy

text: brides

text: b,

text: each

text: of

text: whom

text: will

text: only

text: marry

text: a

text: groom

text: within

text: a

text: certain

text: height

text: range

text: [i

text: j

text: Then

text: each

text: S

text: is

text: an

text: acceptable

text: set

text: of

text: b

text: b

text: s.

text: Hall’s

text: Theorem

text: (from

text: which

text: this

text: terminology

text: is

text: derived)

text: says

text: that

text: if

text: some

text: groomset

text: S

text: [n],

text: |S|

text: rank(f)

text: is

text: unmarriable,

text: it

text: is

text: because

text: there

text: is

text: a

text: set

text: B

text: of

text: brides

text: with

text: S

text: [i,

text: j]

text: |B|.

text: Proposition

text: goes

text: further

text: in

text: two

text: ways:

text: it

text: says

text: that

text: if

text: S

text: is

text: (i,j)

text: B

text: unmarriable,

text: then

text: there

text: is

text: an

text: interval

text: [a,

text: b]

text: where

text: B

text: is

text: the

text: brides

text: with

text: ranges

text: in

text: that

text: interval,

text: and

text: the

text: grooms

text: in

text: that

text: interval

text: aren’t

text: numerous

text: enough,

text: even

text: if

text: one

text: includes

text: the

text: grooms

text: of

text: those

text: brides

text: wants.

text: N

text: For

text: S

text: [n],

text: r

text: let

text: X

text: {rowspan(M)

text: M

text: M

text: rank

text: M

text: k,

text: rank

text: (columns

text: S

text: of

text: M)

text: r}.

text: S

text: r

text: k

text: n

text: If

text: S

text: is

text: a

text: cyclic

text: interval,

text: call

text: this

text: a

text: basic

text: positroid

text: variety.

text: Clearly

text: every

text: positroid

text: variety

text: is

text: an

text: intersection

text: of

text: basic

text: ones,

text: and

text: one

text: can

text: show

text: that

text: no

text: basic

text: positroid

text: variety

text: is

text: an

text: intersection

text: of

text: other

text: positroid

text: varieties.

text: Theorem

text: Let

text: J

text: be

text: a

text: bounded

text: juggling

text: pattern,

text: and

text: i,

text: j

text: J(i)

text: one

text: of

text: its

text: dots.

text: Let

text: C

text: (j

text: be

text: the

text: columns

text: of

text: those

text: dots

text: minimally

text: (i,

text: J(i)),

text: ordered

text: Northeast/Southwest,

text: and

text: r

text: |[i,

text: j

text: #{dots

text: in

text: J

text: that

text: are

text: weakly

text: Southwest

text: of

text: (i,

text: j

text: Then

text: Π

text: X

text: Π

text: J

text: J

text: (j

text: j

text: j

text: C

text: Proof.

text: An

text: intersection

text: of

text: positroid

text: varieties

text: is

text: a

text: reduced

text: union

text: of

text: positroid

text: varieties

text: corollary

text: so

text: we

text: just

text: need

text: to

text: det↔ermine

text: which

text: such

text: occur

text: in

text: Π

text: X

text: We

text: alert

text: the

text: J

text: reader

text: that

text: this

text: is

text: perhaps

text: the

text: subtlest

text: combinatorial

text: argument

text: in

text: the

text: paper.

text: By

text: the↔deﬁnition

text: of

text: C,

text: each

text: J

text: (j

text: J

text: is

text: a

text: covering

text: relation

text: in

text: afﬁne

text: Bruhat

text: order,

text: so

text: Π

text: Π

text: Also,

text: the

text: dot

text: in

text: column

text: of

text: ↔J

text: moves

text: down

text: to

text: row

text: i,

text: providing

text: another

text: J

text: (j

text: j

text: J

text: dot

text: weakly

text: Southwest

text: of

text: (i,

text: j

text: hence

text: Π

text: X

text: Together

text: these

text: prove

text: the

text: J

text: (j

text: j

text: containment.

text: For

text: the

text: reverse

text: containment,

text: we↔need

text: to

text: show

text: that

text: for

text: any

text: Π

text: Π

text: X

text: there

text: J

text: J

text: exists

text: a

text: j

text: C

text: such

text: that

text: Π

text: Π

text: In

text: rank

text: matrix

text: terms,

text: we

text: need

text: r(J

text: r(J),

text: with

text: J

text: J

text: (j

text: j

text: strict

text: inequality

text: on

text: some

text: rectangle

text: i]

text: [j

text: j

text: not

text: just

text: at

text: (i,

text: j

text: Consider

text: a

text: saturated

text: chain

text: J

text: J

text: J

text: J

text: J

text: in

text: strong

text: afﬁne

text: Bruhat

text: order,

text: k

text: so

text: in

text: particular

text: we

text: have

text: entrywise

text: inequalities

text: r(J

text: r(J

text: r(J

text: and

text: more

text: k

text: ##LTLine##


text: 16 ALLEN KNUTSON speciﬁcally each r(J ) − r(J ) is a matrix of 0s and 1s with the 1s in a rectangle. Then m m+1 there exists a smallest m such that r(J ) > r(J ) , and that Π is therefore contained 0 i,j−1 m i,j−1 Jm in X . With this we can reduce to the case k = m. [i,j−1]<r To describe our goal another way, each covering relation moves the dots at the NW and SE corners of an otherwise empty rectangle to the NE and SW corners. We want to show that the union of these rectangles from the covering relations in the chain (J , . . . , J ) 0 k contains one of the maximal rectangles in the staircase of J (above (i, j)), the set of boxes weakly Southeast of some box of C and weakly Northwest of (i, j). We will prove this by induction on k. The k = 1 case is easy – the only rectangle must include (i, j), so must have it as the Southeast corner, and hence the Northwest corner column must be in C. Consider the corresponding staircases for J , . . . , J with Northwest corner sets C = (cid:108) 0 k C , . . . , C . If the covering relation J J gives an increase in the staircase, or leaves 0 k−1 0 1 it the same, then we can use induction. Otherwise, one checks that one of the dots d in (cid:48) C must move South or East inside the staircase to d , as pictured (these being Southern moves): By induction, the remaining covering relations from J to J give rectangles that cover a (cid:38) 1 k rectangle R connecting (i, j) to one of the NW corners of J ’s staircase. If that corner is not (cid:48) 1 (cid:48) d , then it is one of the corners of J and we’re done. If that corner is d , then R union the 0 (cid:48) (cid:3) rectangle acquired during the d d move covers the rectangle connecting (i, j) to d. (cid:0) (cid:1)3. COMBINATORIAL AND GEOMETRIC SHIFTING The classic combinatorial shift operations deﬁned in [EKR61] concern the sets [n] := ⊆ {1, 2, . . . , n} and [n] := {S [n] : |S| = k}. Before getting into them, we establish a k basic correspondence between collections of subsets (the combinatorial side) and certain subschemes of the Grassmannian (the geometrical side). (cid:18) (cid:19) → → The connect(cid:102)ion to geometry begins with the correspondence [n] A (cid:55) ∈ A coord : Gr ( n)T, S coord(S) := {(cid:126)v n that use only coordinates from S} k k between k-subsets and the T-ﬁxed points, the coordinate subspaces. ⊆ A (cid:12) (cid:13) If X Gr ( n) is a closed T-invariant sub(cid:18)sche(cid:19)me, not just a point, we ca(cid:18)n no(cid:19)netheless k look at its ﬁxed points XT, and write ∈ [n] ∈ ⊆ [n] coord−1(X) := coord−1(XT) = S : coord(S) XT . k k

text: ALLEN

text: KNUTSON

text: speciﬁcally

text: each

text: r(J

text: r(J

text: is

text: a

text: matrix

text: of

text: and

text: with

text: the

text: in

text: a

text: rectangle.

text: Then

text: m

text: there

text: exists

text: a

text: smallest

text: m

text: such

text: that

text: r(J

text: r(J

text: and

text: that

text: Π

text: is

text: therefore

text: contained

text: m

text: Jm

text: in

text: X

text: With

text: this

text: we

text: can

text: reduce

text: to

text: the

text: case

text: k

text: m.

text: To

text: describe

text: our

text: goal

text: another

text: way,

text: each

text: covering

text: relation

text: moves

text: the

text: dots

text: at

text: the

text: NW

text: and

text: SE

text: corners

text: of

text: an

text: otherwise

text: empty

text: rectangle

text: to

text: the

text: NE

text: and

text: SW

text: corners.

text: We

text: want

text: to

text: show

text: that

text: the

text: union

text: of

text: these

text: rectangles

text: from

text: the

text: covering

text: relations

text: in

text: the

text: chain

text: (J

text: J

text: k

text: contains

text: one

text: of

text: the

text: maximal

text: rectangles

text: in

text: the

text: staircase

text: of

text: J

text: (above

text: (i,

text: j)),

text: the

text: set

text: of

text: boxes

text: weakly

text: Southeast

text: of

text: some

text: box

text: of

text: C

text: and

text: weakly

text: Northwest

text: of

text: (i,

text: j).

text: We

text: will

text: prove

text: this

text: by

text: induction

text: on

text: k.

text: The

text: k

text: case

text: is

text: easy

text: the

text: only

text: rectangle

text: must

text: include

text: (i,

text: j),

text: so

text: must

text: have

text: it

text: as

text: the

text: Southeast

text: corner,

text: and

text: hence

text: the

text: Northwest

text: corner

text: column

text: must

text: be

text: in

text: C.

text: Consider

text: the

text: corresponding

text: staircases

text: for

text: J

text: J

text: with

text: Northwest

text: corner

text: sets

text: C

text: k

text: C

text: C

text: If

text: the

text: covering

text: relation

text: J

text: J

text: gives

text: an

text: increase

text: in

text: the

text: staircase,

text: or

text: leaves

text: it

text: the

text: same,

text: then

text: we

text: can

text: use

text: induction.

text: Otherwise,

text: one

text: checks

text: that

text: one

text: of

text: the

text: dots

text: d

text: in

text: C

text: must

text: move

text: South

text: or

text: East

text: inside

text: the

text: staircase

text: to

text: d

text: as

text: pictured

text: (these

text: being

text: Southern

text: moves):

text: By

text: induction,

text: the

text: remaining

text: covering

text: relations

text: from

text: J

text: to

text: J

text: give

text: rectangles

text: that

text: cover

text: a

text: k

text: rectangle

text: R

text: connecting

text: (i,

text: j)

text: to

text: one

text: of

text: the

text: NW

text: corners

text: of

text: J

text: ’s

text: staircase.

text: If

text: that

text: corner

text: is

text: not

text: d

text: then

text: it

text: is

text: one

text: of

text: the

text: corners

text: of

text: J

text: and

text: we’re

text: done.

text: If

text: that

text: corner

text: is

text: d

text: then

text: R

text: union

text: the

text: rectangle

text: acquired

text: during

text: the

text: d

text: d

text: move

text: covers

text: the

text: rectangle

text: connecting

text: (i,

text: j)

text: to

text: d.

text: COMBINATORIAL

text: AND

text: GEOMETRIC

text: SHIFTING

text: The

text: classic

text: combinatorial

text: shift

text: operations

text: deﬁned

text: in

text: concern

text: the

text: sets

text: [n]

text: n}

text: and

text: [n]

text: {S

text: [n]

text: |S|

text: k}.

text: Before

text: getting

text: into

text: them,

text: we

text: establish

text: a

text: k

text: basic

text: correspondence

text: between

text: collections

text: of

text: subsets

text: (the

text: combinatorial

text: side)

text: and

text: certain

text: subschemes

text: of

text: the

text: Grassmannian

text: (the

text: geometrical

text: side).

text: The

text: to

text: geometry

text: begins

text: with

text: the

text: correspondence

text: [n]

text: A

text: A

text: coord

text: Gr

text: n)T,

text: S

text: coord(S)

text: n

text: that

text: use

text: only

text: coordinates

text: from

text: S}

text: k

text: k

text: between

text: k-subsets

text: and

text: the

text: T-ﬁxed

text: points,

text: the

text: coordinate

text: subspaces.

text: A

text: If

text: X

text: Gr

text: n)

text: is

text: a

text: closed

text: T-invariant

text: not

text: just

text: a

text: point,

text: we

text: k

text: look

text: at

text: its

text: ﬁxed

text: points

text: XT,

text: and

text: write

text: [n]

text: [n]

text: S

text: coord(S)

text: XT

text: k

text: k


text: SCHUBERT CALCULUS AND SHIFTING OF INTERVAL POSITROID VARIETIES 17 (cid:0) (cid:1) ⊆ To forestall confusion when talking(cid:92)about sets of sets, we will call any S [n] a subset C ⊆ and any [n] a collection. Extend coord beyond subsets to collections, as follows: k C ∈ A coord( ) := {V Gr ( n) : p (V) = 0} k S ∈C S/ where p is the Plu¨ cker coordinate. This is the “bracket ring” construction of [W75], in C S C which is somewhat needlessly assumed to be a matroid, presumably because coord( ) is reducible otherwise.7 To study these operations, we ﬁrst need a basic result about Plu¨ cker coordinates: (cid:0) (cid:1) ⊆ A ∈ ∈ Lemma 3.1. Let X Gr ( n) be T-invariant and reduced. If S [n] , and coord(S) / X, then k k the Plu¨cker coordinate p vanishes on X. S (cid:14) Proof. Consider th→e one-pa→rameter subgroup ∈ G (cid:55) 0 if k S d : T, t diag(td1, . . . , tdn) where d = m k 1 otherwise A The sink of d’s Białynicki-Birula decomposition of Gr ( n) is the point coord(S), and its (cid:54) k basin of attraction is the big cell p = 0. If X meets this cell, then since X is d-invariant (cid:51)S (being T-invariant) and closed, X coord(S), contradiction. Hence X is set-theoretically contained in the divisor {p = 0}, and since it was assumed reduced X is contained in that S (cid:3) divisor scheme-theoretically as well. (cid:0) (cid:1) ⊆ Proposition 3.2. For C [n] , k coord−1(co(cid:0)ord(C)T) = C(cid:1) . ⊆ A For X Gr ( n), closed, reduced and T-invariant but possibly reducible, k ⊆ X coord coord−1(X) . Proof. The ﬁrst is ta(cid:92)utological: (cid:92) ∈ A ∈ A (cid:54) coord(C)T = {V Gr ( n)T : p (V) = 0} = {V Gr ( n)T : V = coord(S)} k S k ∈ ∈ S/C S/C ∈ A ∈ ∈ = {V Gr ( n)T : coord−1(V) / {S : S / C}} k ∈ A ∈ = {V Gr ( n)T : coord−1(V) C} k C 7 It is well-known that if X is irreducible and T-invariant, then := coord−1(XT) is the bases of a matroid, ∈ · C meaning that for each π S , the collection π has a unique Bruhat minimum. Proof: Let ρˇ be a regular n A dominant coweight, so its Białynicki-Birula decomposition of Gr ( n) is the Bruhat decomposition. If X is ∈ · k irreducible, then for each π S , π X will have a unique open Białynicki-Birula stratum, whose center is n this unique Bruhat minimum. See e.g. [BGW03]. ##LTLine##

text: SCHUBERT

text: CALCULUS

text: AND

text: SHIFTING

text: OF

text: INTERVAL

text: POSITROID

text: VARIETIES

text: To

text: forestall

text: confusion

text: when

text: sets

text: of

text: sets,

text: we

text: will

text: call

text: any

text: S

text: [n]

text: a

text: subset

text: C

text: and

text: any

text: [n]

text: a

text: collection.

text: Extend

text: coord

text: beyond

text: subsets

text: to

text: collections,

text: as

text: follows:

text: k

text: C

text: A

text: coord(

text: {V

text: Gr

text: n)

text: p

text: (V)

text: k

text: S

text: ∈C

text: S/

text: where

text: p

text: is

text: the

text: Plu¨

text: cker

text: coordinate.

text: This

text: is

text: the

text: “bracket

text: ring”

text: construction

text: of

text: in

text: C

text: S

text: C

text: which

text: is

text: somewhat

text: needlessly

text: assumed

text: to

text: be

text: a

text: matroid,

text: presumably

text: because

text: coord(

text: is

text: reducible

text: To

text: study

text: these

text: operations,

text: we

text: ﬁrst

text: need

text: a

text: basic

text: result

text: about

text: Plu¨

text: cker

text: coordinates:

text: A

text: Lemma

text: Let

text: X

text: Gr

text: n)

text: be

text: T-invariant

text: and

text: reduced.

text: If

text: S

text: [n]

text: and

text: coord(S)

text: X,

text: then

text: k

text: k

text: the

text: Plu¨cker

text: coordinate

text: p

text: vanishes

text: on

text: X.

text: S

text: Proof.

text: Consider

text: th→e

text: one-pa→rameter

text: subgroup

text: G

text: if

text: k

text: S

text: d

text: T,

text: t

text: tdn)

text: where

text: d

text: m

text: k

text: otherwise

text: A

text: The

text: sink

text: of

text: d’s

text: Białynicki-Birula

text: decomposition

text: of

text: Gr

text: n)

text: is

text: the

text: point

text: coord(S),

text: and

text: its

text: k

text: basin

text: of

text: attraction

text: is

text: the

text: big

text: cell

text: p

text: If

text: X

text: meets

text: this

text: cell,

text: then

text: since

text: X

text: is

text: d-invariant

text: (being

text: T-invariant)

text: and

text: closed,

text: X

text: coord(S),

text: contradiction.

text: Hence

text: X

text: is

text: set-theoretically

text: contained

text: in

text: the

text: divisor

text: {p

text: and

text: since

text: it

text: was

text: assumed

text: reduced

text: X

text: is

text: contained

text: in

text: that

text: S

text: divisor

text: scheme-theoretically

text: as

text: well.

text: Proposition

text: For

text: C

text: [n]

text: k

text: A

text: For

text: X

text: Gr

text: n),

text: closed,

text: reduced

text: and

text: T-invariant

text: but

text: possibly

text: reducible,

text: k

text: X

text: coord

text: Proof.

text: The

text: ﬁrst

text: is

text: A

text: A

text: coord(C)T

text: {V

text: Gr

text: n)T

text: p

text: (V)

text: {V

text: Gr

text: n)T

text: V

text: coord(S)}

text: k

text: S

text: k

text: S/C

text: S/C

text: A

text: {V

text: Gr

text: n)T

text: {S

text: S

text: C}}

text: k

text: A

text: {V

text: Gr

text: n)T

text: C}

text: k

text: C

text: It

text: is

text: well-known

text: that

text: if

text: X

text: is

text: irreducible

text: and

text: T-invariant,

text: then

text: is

text: the

text: bases

text: of

text: a

text: matroid,

text: C

text: meaning

text: that

text: for

text: each

text: π

text: S

text: the

text: collection

text: π

text: has

text: a

text: unique

text: Bruhat

text: minimum.

text: Proof:

text: Let

text: ρˇ

text: be

text: a

text: regular

text: n

text: A

text: dominant

text: coweight,

text: so

text: its

text: Białynicki-Birula

text: decomposition

text: of

text: Gr

text: n)

text: is

text: the

text: Bruhat

text: decomposition.

text: If

text: X

text: is

text: k

text: irreducible,

text: then

text: for

text: each

text: π

text: S

text: π

text: X

text: will

text: have

text: a

text: unique

text: open

text: Białynicki-Birula

text: stratum,

text: whose

text: center

text: is

text: n

text: this

text: unique

text: Bruhat

text: minimum.

text: See

text: e.g.

text: ##LTLine##


text: SCHUBERT CALCULUS AND SHIFTING OF INTERVAL POSITROID VARIETIES 3 ∪ Consider the label set {0, 1} {A, B, . . .}, where only the latter group are called letters, and consider the following tile schema, with pipes connecting the edges of a square: Call these the crossing and elbows tiles, and the a = b = 0 elbows the equivariant tile2. We will often want to determine a tile from its South and East labels, and this can be done uniquely unless both are 0. We will tile these together, such that the boundary labels of adjoining tiles match up, making continuous “pipes” from boundary to boundary bearing well-deﬁned labels. De- ﬁne an IP pipe dream (the IP for “interval positroid”) to be a ﬁlling of the upper triangle × of an n n matrix, such that • on the East edges (of each (i, n) square), there are no 1 labels, • on the South edges (below each (i, i) square), there are no 0 labels, • on the West edges (West of each (i, i) square), there are only 0 labels (we will derive this from other conditions, in proposition 4.2), • on the North edges (above each (i, 1) square), there are only 0s and 1s, • no two pipes of the same label cross, and ﬁnally, • no two lettered pipes cross twice. This is the only nonlocal condition. § In fact the 1 acts more like a special letter than like the 0 (especially in ); for example the “no two lettered pipes cross twice” rule applies even if 1 is considered to be a letter, because of the second condition on crossing tiles. (cid:12) (cid:12) Each lettered pipe connects a horizontal edge below the diagonal to a vertical edge on (cid:54) the East side. Since a = b in crossing tiles, the ith −B− from the left must connect to the ith B from the top. By the nonlocal condition, the ith A pipe will cross the jth B pipe either once or not at all, and can be predicted from the boundary and the Jordan curve theorem. 2The 0s and 1s on these tiles are not quite the same as those on the puzzle pieces from [KnTao03]; see theorem 5.1 for the connection. ##LTLine##

text: SCHUBERT

text: CALCULUS

text: AND

text: SHIFTING

text: OF

text: INTERVAL

text: POSITROID

text: VARIETIES

text: Consider

text: the

text: label

text: set

text: {A,

text: B,

text: where

text: only

text: the

text: latter

text: group

text: are

text: called

text: letters,

text: and

text: consider

text: the

text: following

text: tile

text: schema,

text: with

text: pipes

text: connecting

text: the

text: edges

text: of

text: a

text: square:

text: Call

text: these

text: the

text: crossing

text: and

text: elbows

text: tiles,

text: and

text: the

text: a

text: b

text: elbows

text: the

text: equivariant

text: We

text: will

text: often

text: want

text: to

text: determine

text: a

text: tile

text: from

text: its

text: South

text: and

text: East

text: labels,

text: and

text: this

text: can

text: be

text: done

text: uniquely

text: unless

text: both

text: are

text: We

text: will

text: tile

text: these

text: together,

text: such

text: that

text: the

text: boundary

text: labels

text: of

text: adjoining

text: tiles

text: match

text: up,

text: making

text: continuous

text: “pipes”

text: from

text: boundary

text: to

text: boundary

text: bearing

text: well-deﬁned

text: labels.

text: De-

text: ﬁne

text: an

text: IP

text: pipe

text: dream

text: (the

text: IP

text: for

text: “interval

text: positroid”)

text: to

text: be

text: a

text: ﬁlling

text: of

text: the

text: upper

text: triangle

text: of

text: an

text: n

text: n

text: matrix,

text: such

text: that

text: on

text: the

text: East

text: edges

text: (of

text: each

text: (i,

text: n)

text: square),

text: there

text: are

text: no

text: labels,

text: on

text: the

text: South

text: edges

text: (below

text: each

text: (i,

text: i)

text: square),

text: there

text: are

text: no

text: labels,

text: on

text: the

text: West

text: edges

text: (West

text: of

text: each

text: (i,

text: i)

text: square),

text: there

text: are

text: only

text: labels

text: (we

text: will

text: derive

text: this

text: from

text: other

text: conditions,

text: in

text: proposition

text: on

text: the

text: North

text: edges

text: (above

text: each

text: (i,

text: square),

text: there

text: are

text: only

text: and

text: no

text: two

text: pipes

text: of

text: the

text: same

text: label

text: cross,

text: and

text: ﬁnally,

text: no

text: two

text: lettered

text: pipes

text: cross

text: twice.

text: This

text: is

text: the

text: only

text: nonlocal

text: condition.

text: In

text: fact

text: the

text: acts

text: more

text: like

text: a

text: special

text: letter

text: than

text: like

text: the

text: (especially

text: in

text: for

text: example

text: the

text: “no

text: two

text: lettered

text: pipes

text: cross

text: twice”

text: rule

text: applies

text: even

text: if

text: is

text: considered

text: to

text: be

text: a

text: letter,

text: because

text: of

text: the

text: second

text: condition

text: on

text: crossing

text: tiles.

text: Each

text: lettered

text: pipe

text: connects

text: a

text: horizontal

text: edge

text: below

text: the

text: diagonal

text: to

text: a

text: vertical

text: edge

text: on

text: the

text: East

text: side.

text: Since

text: a

text: b

text: in

text: crossing

text: tiles,

text: the

text: ith

text: −B−

text: from

text: the

text: left

text: must

text: connect

text: to

text: the

text: ith

text: B

text: from

text: the

text: top.

text: By

text: the

text: nonlocal

text: condition,

text: the

text: ith

text: A

text: pipe

text: will

text: cross

text: the

text: jth

text: B

text: pipe

text: either

text: once

text: or

text: not

text: at

text: all,

text: and

text: can

text: be

text: predicted

text: from

text: the

text: boundary

text: and

text: the

text: Jordan

text: curve

text: theorem.

text: and

text: on

text: these

text: tiles

text: are

text: not

text: quite

text: the

text: same

text: as

text: those

text: on

text: the

text: puzzle

text: pieces

text: from

text: see

text: theorem

text: for

text: the

text: connection.

text: ##LTLine##


text: 24 ALLEN KNUTSON → ∩ ∈ A If there is such an S, let M = S \ {i} = S X (S), and L = {V Gr ( n) : ∪ i j ⊆ k coord(M) < V < coord(M {i, j}). So far we’ve determined that LT X. Since X ⊆ is assumed T-convex, L X. → → → ⊇ (cid:51) The key fact is that L is X -invariant. Hence X (X) X (L) = L i j i j i j coord(S), cont(cid:103)radicting the choice of S. · ∈ A ∈ A × (4) The space {(t, ex→p(te ) x) : t 1, x X} is isomorphi→c to 1 X, hence irreducible. ij So its closure Ψ (X) is irreducible, and the image Ψ (X) of that is irre→ducible. i j i j To show the projection is degree 1, it sufﬁces to ﬁnd a point in Ψ (X) over i j w(cid:103)hich the map is an isomorphism. By the non-invariance assumption, there exists ∪ ∈ ∪ ∈ a s→ubset M su→ch that coord(M {i}) X, coord(M {j}) / X. Hence the map → (cid:16) Ψ (X) Ψ (X) takes i j i j ∪ (cid:55) ∪ (0, coord(M {i}) coord(M {i}). (cid:103) ⊇ Let D be the Plu¨ cker diviso→r {p ∪ = 0}. By lemma 3.1, D X. Since D is (cid:8) A M {j} P × (cid:9)A codimension 1 in Gr ( n), Ψ D is codimension 1 in 1 Gr ( n), and is easily k i j k seen to lie in the hypersurface ∈ P × A H := ([t, u], V) 1 Gr ( n) : tp ∪ + up ∪ = 0 . k M {i} M {j} → → → (cid:103) (cid:103) So far we have the maps → → ∪ (cid:16) A {(0, coord(M {i}))} (cid:44) Ψ (X) (cid:44) Ψ (D) (cid:44) H Gr ( n). i j i j k ∪ (cid:16) A Now we claim that the ﬁber over coord(M {i}) of this last projection H Gr ( n) ∪ k is already a reduced point (namely, (0, coord(M {i}))); it is ∪ {([t, u], coord(M {i})) : tp ∪ = 0} →M {i} (cid:103) and the remaining projective coordinat→e p ∪ does not vanish. Hence the ﬁber coord(M ∪ {i}) of Ψ (XM) {i} Gr (An) is a reduced point. (cid:3) i j k Finally we are ready to give a theorem that can, in certain cases, calculate shifts. (While we hoped to use it with proposition 2.5 to compute the shifts in theorem 1.3, we will instead use proposition 3.14.) (cid:83) (cid:83) (cid:83) (cid:48) (cid:48) ⊆ A Theorem 3.11. Let X, X→, . . . , X Gr ( n) be T-invariant subvarieties of the same dimension. 1 m k (cid:48) ∀ (cid:48) (cid:48) (cid:54) ∅ Assume t→hat coord(X coord−1(XT)) = m X as schemes, and i, (X )T \ (cid:54) (X )T = . i (cid:48)j i=1 m i j=i j Then X (X) = m X . i j i=1 m (cid:84) Proof. Since X is reduced, ⊆ (cid:84) ∈ A X ∈C {V Gr ( n) : p (V) = 0} by lemma 3.1, and → S k S ⊆ → ∈ A (cid:83) X (X) ∈ C {V Gr ( n) : p (V) = 0} by proposition 3.9. i j S Xj i k S → ⊆ (cid:48) We know the right side by assumption, with the result that X (X) m X . i j i=1 m

text: ALLEN

text: KNUTSON

label: list

text: A

text: If

text: there

text: is

text: such

text: an

text: S,

text: let

text: M

text: S

text: {i}

text: S

text: X

font: unknown

text: (S),

text: and

text: L

text: {V

text: Gr

text: n)

text: i

text: j

text: k

text: coord(M)

text: V

text: coord(M

text: {i,

text: j}).

text: So

text: far

text: we’ve

text: determined

text: that

text: LT

text: X.

text: Since

text: X

text: is

text: assumed

text: T-convex,

text: L

text: X.

text: The

text: key

text: fact

text: is

text: that

text: L

text: is

text: X

text: -invariant.

text: Hence

text: X

text: (X)

text: X

text: (L)

text: L

text: i

text: j

text: i

text: j

text: i

text: j

text: coord(S),

text: the

text: choice

text: of

text: S.

text: A

text: A

text: The

text: space

text: {(t,

text: ex→p(te

text: x)

text: t

text: x

text: X}

text: is

text: isomorphi→c

text: to

text: X,

text: hence

text: irreducible.

text: ij

text: So

text: its

text: closure

text: Ψ

text: (X)

text: is

text: irreducible,

text: and

text: the

text: image

text: Ψ

text: (X)

text: of

text: that

text: is

text: irre→ducible.

text: i

text: j

text: i

text: j

text: To

text: show

text: the

text: projection

text: is

text: degree

text: it

text: sufﬁces

text: to

text: ﬁnd

text: a

text: point

text: in

text: Ψ

text: (X)

text: over

text: i

text: j

text: the

text: map

text: is

text: an

text: isomorphism.

text: By

text: the

text: non-invariance

text: assumption,

text: there

text: exists

text: a

text: s→ubset

text: M

text: su→ch

text: that

text: coord(M

text: {i})

text: X,

text: coord(M

text: {j})

text: X.

text: Hence

text: the

text: map

text: Ψ

text: (X)

text: Ψ

text: (X)

text: takes

text: i

text: j

text: i

text: j

text: coord(M

text: {i})

text: coord(M

text: {i}).

text: Let

text: D

text: be

text: the

text: Plu¨

text: cker

text: diviso→r

text: {p

text: By

text: lemma

text: D

text: X.

text: Since

text: D

text: is

text: A

text: M

text: {j}

text: P

text: codimension

text: in

text: Gr

text: n),

text: Ψ

text: D

text: is

text: codimension

text: in

text: Gr

text: n),

text: and

text: is

text: easily

text: k

text: i

text: j

text: k

text: seen

text: to

text: lie

text: in

text: the

text: hypersurface

text: P

text: A

text: H

text: ([t,

text: u],

text: V)

text: Gr

text: n)

text: tp

text: up

text: k

text: M

text: {i}

text: M

text: {j}

text: So

text: far

text: we

text: have

text: the

text: maps

text: A

text: coord(M

text: {i}))}

text: Ψ

text: (X)

text: Ψ

text: (D)

text: H

text: Gr

text: n).

text: i

text: j

text: i

text: j

text: k

text: A

text: Now

text: we

text: claim

text: that

text: the

text: ﬁber

text: over

text: coord(M

text: {i})

text: of

text: this

text: last

text: projection

text: H

text: Gr

text: n)

text: k

text: is

text: already

text: a

text: reduced

text: point

text: (namely,

text: coord(M

text: {i})));

text: it

text: is

text: {([t,

text: u],

text: coord(M

text: {i}))

text: tp

text: →M

text: {i}

text: and

text: the

text: remaining

text: projective

text: coordinat→e

text: p

text: does

text: not

text: vanish.

text: Hence

text: the

text: ﬁber

text: coord(M

text: {i})

text: of

text: Ψ

text: (XM)

text: {i}

text: Gr

text: (An)

text: is

text: a

text: reduced

text: point.

text: i

text: j

text: k

text: Finally

text: we

text: are

text: ready

text: to

text: give

text: a

text: theorem

text: that

text: can,

text: in

text: certain

text: cases,

text: calculate

text: shifts.

text: (While

text: we

text: hoped

text: to

text: use

text: it

text: with

text: proposition

text: to

text: compute

text: the

text: shifts

text: in

text: theorem

text: we

text: will

text: instead

text: use

text: proposition

text: A

text: Theorem

text: Let

text: X,

text: X→,

text: X

text: Gr

text: n)

text: be

text: T-invariant

text: subvarieties

text: of

text: the

text: same

text: dimension.

text: m

text: k

text: Assume

text: t→hat

text: coord(X

text: m

text: X

text: as

text: schemes,

text: and

text: i,

text: (X

text: )T

text: (X

text: )T

text: i

text: m

text: i

text: j=i

text: j

text: Then

text: X

text: (X)

text: m

text: X

text: i

text: j

text: m

text: Proof.

text: Since

text: X

text: is

text: reduced,

text: A

text: X

text: ∈C

text: {V

text: Gr

text: n)

text: p

text: (V)

text: by

text: lemma

text: and

text: S

text: k

text: S

text: A

text: X

text: (X)

text: C

text: {V

text: Gr

text: n)

text: p

text: (V)

text: by

text: proposition

text: i

text: j

text: S

text: Xj

text: i

text: k

text: S

text: We

text: know

text: the

text: right

text: side

text: by

text: assumption,

text: with

text: the

text: result

text: that

text: X

text: (X)

text: m

text: X

text: i

text: j

text: m