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text: WEBS OF TYPE Q 23 Proof. To show Π is well-deﬁned it suﬃces to verify the deﬁning relations of U˙ (q(m))≥ hold in q-Web↑. m 0 This follows by direct calculations using the deﬁning relations of q-Web↑ along with the identities proven in ?? . That the image of Π lies in the given subcategory is immediate. That the functor is full amounts to the m fact that the images of the generating morphisms of U˙ (q(m))≥ also generate the morphisms of q-Web↑. 0 Since every merge and split is a special case of a ladder, it is straightforward to verify that one can obtain every possible merge, split, and dot (possibly tensored by identity strands on the left and/or right) which lies in this subcategory. Since every web which is a morphism in this subcategory is a composition of such (cid:3) diagrams, it follows that Π is full. m Remark 5.1.2. Recall the functors Θ (cid:48) from Remark 3.6.5. The Π functors are compatible in the sense ◦ m ,m (cid:48) ≥ that Π (cid:48) Θ (cid:48) and Π are isomorphic functors for all positive integers m m. m m ,m m → As shown below, there is also a functor Ψ : q-Web↑ q(n)-ModS. n ≥ Proposition 5.2.1. For every n 1 there exists an essentially surjective functor of monoidal supercate- gories, → Ψ : q-Web↑ q(n)-ModS, n given on objects by ↑ ⊗ · · · ⊗ Ψ ( ) = S = Sλ(V ) = Sλ1(V ) Sλt(V ), n λ λ n n n where λ = (λ , . . . , λ ). On morphisms the functor is deﬁned by sending the dot, merge, and split, respectively, 1 t (cid:88) to the following maps: → · · · (cid:55)→ k − | | ··· | | · · · · · · Sk(V ) Sk(V ), v v ( 1) v1 + + vt−1 v c(v ) v ; n n 1 k 1 t k (cid:88)t=1 ⊗ → · · · ⊗ · · · (cid:55)→ · · · · · · Sk(V ) Sl(V ) Sk+l(V ), v v w w v v w w ; n n n 1 k 1 l 1 k 1 l → ⊗ · · · (cid:55)→ − · · · ⊗ · · · Sk+l(Vn) Sk(Vn) Sl(Vn), v1 vk+l ( 1)εI,Jvi1 vik vj1 vjl. { · · · } { · · · } ∪ { · · · } The last sum is over all I = i < < i and J = j < < j for which I J = 1, , k + l . The ∈ Z 1 k · · · 1· · · −l · · · element εI,J 2 is deﬁned by the formula vi1 vikvj1 vjl = ( 1)εI,Jv1 vk+l. Proof. The interested reader can verify that these maps satisfy the deﬁning relations of q-Web↑. That the (cid:3) functor is essentially surjective is clear. ⊗ → ⊗ One can verify β↑ ↑ is sent to the graded ﬂip map Sk(V ) Sl(V ) Sl(V ) Sk(V ) given by · · · ⊗ · · · (cid:55)→k, l− | | ··· | | | | ··· | | · · · ⊗n · · · n n n v1 vk w1 wl ( 1)( v1 + + vk )( w1 + + wl )w1 wl v1 vk. Also, we could have deﬁned the functor Ψ using the functors Φ . n m,n       Remark 5.2.2. Recall the functors Φ from Proposition 3.6.4. Then,  k  m,nk+l   k l  Ψ = Φ (h 1 ), Ψ = Φ (e(l)1 ), Ψ = Φ (f(l)1 ). n 1,n 1 (k) n 2,n 1 (k,l) n 2,n 1 (k+l,0) k k l k+l ≥ ◦ In particular, for any m, n 1 we have Ψ Π = Φ . The well-deﬁnedness of Ψ can also be deduced n m m,n n from the well-deﬁnedness of the functors Φ and the compatibility given in Remark 3.6.5. m,n ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WEBS

label: paragraph

text: OF

text: TYPE

text: Q

text: Proof.

text: To

text: show

text: Π

text: is

text: well-deﬁned

text: it

text: suﬃces

text: to

text: verify

text: the

text: deﬁning

text: relations

text: of

text: U˙

text: (q(m))≥

text: hold

text: in

text: q-Web↑.

text: m

text: This

text: follows

text: by

text: direct

text: calculations

text: using

text: the

text: deﬁning

text: relations

text: of

text: q-Web↑

text: along

text: with

text: the

text: identities

text: proven

text: in

text: That

text: the

text: image

text: of

text: Π

text: lies

text: in

text: the

text: given

text: subcategory

text: is

text: immediate.

text: That

text: the

text: functor

text: is

text: full

text: amounts

text: to

text: the

text: m

text: fact

text: that

text: the

text: images

text: of

text: the

text: generating

text: morphisms

text: of

text: U˙

text: (q(m))≥

text: also

text: generate

text: the

text: morphisms

text: of

text: q-Web↑.

text: Since

text: every

text: merge

text: and

text: split

text: is

text: a

text: special

text: case

text: of

text: a

text: ladder,

text: it

text: is

text: straightforward

text: to

text: verify

text: that

text: one

text: can

text: obtain

text: every

text: possible

text: merge,

text: split,

text: and

text: dot

text: (possibly

text: tensored

text: by

text: identity

text: strands

text: on

text: the

text: left

text: and/or

text: right)

text: which

text: lies

text: in

text: this

text: subcategory.

text: Since

text: every

text: web

text: which

text: is

text: a

text: morphism

text: in

text: this

text: subcategory

text: is

text: a

text: composition

text: of

text: such

text: diagrams,

text: it

text: follows

text: that

text: Π

text: is

text: full.

text: m

text: Remark

text: Recall

text: the

text: functors

text: Θ

text: from

text: Remark

text: The

text: Π

text: functors

text: are

text: compatible

text: in

text: the

text: sense

text: m

text: ,m

text: that

text: Π

text: Θ

text: and

text: Π

text: are

text: isomorphic

text: functors

text: for

text: all

text: positive

text: integers

text: m

text: m.

text: m

text: m

text: ,m

text: m

text: As

text: shown

text: below,

text: there

text: is

text: also

text: a

text: functor

text: Ψ

text: q-Web↑

text: q(n)-ModS.

text: n

text: Proposition

text: For

text: every

text: n

text: there

text: exists

text: an

text: essentially

text: surjective

text: functor

text: of

text: monoidal

text: supercate-

text: gories,

text: Ψ

text: q-Web↑

text: q(n)-ModS,

text: n

text: given

text: on

text: objects

text: by

text: Ψ

text: S

text: Sλ(V

text: Sλt(V

text: n

text: λ

text: λ

text: n

text: n

text: n

text: where

text: λ

text: (λ

text: λ

text: On

text: morphisms

text: the

text: functor

text: is

text: deﬁned

text: by

text: sending

text: the

text: dot,

text: merge,

text: and

text: split,

text: respectively,

text: t

text: to

text: the

text: following

text: maps:

text: k

text: Sk(V

text: Sk(V

text: v

text: v

text: v

text: c(v

text: v

text: n

text: n

text: k

text: t

text: k

text: Sk(V

text: Sl(V

text: Sk+l(V

text: v

text: v

text: w

text: w

text: v

text: v

text: w

text: w

text: n

text: n

text: n

text: k

text: l

text: k

text: l

text: Sk+l(Vn)

text: Sk(Vn)

text: Sl(Vn),

text: vk+l

text: vik

text: vjl.

text: The

text: last

text: sum

text: is

text: over

text: all

text: I

text: i

text: i

text: and

text: J

text: j

text: j

text: for

text: which

text: I

text: J

text: k

text: l

text: The

text: Z

text: k

text: 1·

text: −l

text: element

text: εI,J

text: is

text: deﬁned

text: by

text: the

text: formula

text: vjl

text: vk+l.

text: Proof.

text: The

text: interested

text: reader

text: can

text: verify

text: that

text: these

text: maps

text: satisfy

text: the

text: deﬁning

text: relations

text: of

text: q-Web↑.

text: That

text: the

text: functor

text: is

text: essentially

text: surjective

text: is

text: clear.

text: One

text: can

text: verify

text: β↑

text: is

text: sent

text: to

text: the

text: graded

text: ﬂip

text: map

text: Sk(V

text: Sl(V

text: Sl(V

text: Sk(V

text: given

text: by

text: l−

text: ⊗n

text: n

text: n

text: n

text: vk

text: wl

text: vk

text: wl

text: wl

text: vk.

text: Also,

text: we

text: could

text: have

text: deﬁned

text: the

text: functor

text: Ψ

text: using

text: the

text: functors

text: Φ

text: n

text: m,n

text: Remark

text: Recall

text: the

text: functors

text: Φ

text: from

text: Proposition

text: Then,

text: k

text: m,nk+l

text: k

text: l

text: Ψ

text: Φ

text: (h

text: Ψ

text: Φ

text: Ψ

text: Φ

text: n

text: (k)

text: n

text: (k,l)

text: n

text: k

text: k

text: l

text: k+l

text: In

text: particular,

text: for

text: any

text: m,

text: n

text: we

text: have

text: Ψ

text: Π

text: Φ

text: The

text: well-deﬁnedness

text: of

text: Ψ

text: can

text: also

text: be

text: deduced

text: n

text: m

text: m,n

text: n

text: from

text: the

text: well-deﬁnedness

text: of

text: the

text: functors

text: Φ

text: and

text: the

text: compatibility

text: given

text: in

text: Remark

text: m,n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: WEBS OF TYPE Q 29 ↑ · · · ↑ ↓ · · · ↓ ↓ · · · ↓ ↑ · · · ↑ In addition, given objects of the form a = and b = , then there a1 (cid:0) ar ar+1 at b1 bs bs+1 (cid:1)bu is a superspace map, −∼→ ↑ · · · ↑ ↑ · · · ↑ ↑ · · · ↑ ↑ · · · ↑ Homq-Web↑↓(a, b) = Homq-Web↑↓ bs b1 a1 ar, bs+1 bu at ar+1 , (6.12) given on diagrams by bs+·1· ·bu at· ·a·r+1 b1· · ·bs bs+·1· ·bu · · · · · · (cid:55)→ w w . · · · · · · · · · · · · · · · · · · a1 ar ar+1 at bs b1 a1 ar The inverse is given by a similar map (thanks to relation (6.1)). As in the proof of Theorem 5.4.1, whenever a morphism between Hom-spaces in q-Web↑↓ is deﬁned by applying some combination of compositions and tensor products of morphisms we can apply the functor Ψ and obtain a corresponding morphism in the category of q(n)-modules. In particular, commuting diagrams go to commuting diagrams and isomorphisms go to isomorphisms. Theorem 6.5.1. There is a fully faithful functor of symmetric monoidal supercategories → q-Web↑ q-Web↑↓ which takes objects and morphisms in q-Web↑ to objects and morphisms of the same name in q-Web↑↓. Proof. By construction the deﬁning relations on morphisms of q-Web↑ hold in q-Web↑↓. Consequently, there is a well-deﬁned functor of monoidal supercategories. As discussed in ?? , the relations listed in Lemma 4.5.2 hold for all possible orientations of the strands in a diagram. Now if a, b are objects of q-Web↑ and if d is a web diagram which lies in Homq-Web↑↓(a, b), then considering the end behavior of strands we see that we can use these relations, planar isotopies, and (6.1) to eliminate all cups, caps, and rightward crossings. Furthermore, by (6.4) and Lemma 6.2.1 there are no bubbles in d. Taken together, this implies d (cid:3) is equal to a web diagram which lies in Homq-Web↑(a, b). Fullness and faithfulness follows. We will identify q-Web↑ as a full su(cid:8)bcategory of q-Web↑↓ via(cid:9) this functor. Let q(n)-ModS S∗ denote , the full monoidal subsupercategory of q(n)-supermodules generated by the objects ∗ | ≥ Sp (V ) , Sp (V ) p 1 . n n → → We next show the functor Ψ : q-Web↑ q(n)-ModS can be extended to Ψ : q-Web↑↓ q(n)-ModS S∗. n n , Set the notation ∗ ⊗ → C ev : Sk(V ) Sk(V ) , (6.13) k n n C → ⊗ ∗ coev : Sk(V ) Sk(V ) , (6.14) k n n for the evaluation and coevaluation maps deﬁned in ?? . Theorem 6.5.2. There is an essentially surjective, full functor of symmetric monoidal supercategories → Ψ : q-Web↑↓ q(n)-ModS S∗ n , given on generating objects by ↑ Ψ ( ) = Sk(V ), n k n ↓ ∗ Ψ ( ) = Sk(V ) n k n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WEBS

text: OF

text: TYPE

text: Q

text: In

text: addition,

text: given

text: objects

text: of

text: the

text: form

text: a

text: and

text: b

text: then

text: there

label: equation

text: ar

text: at

text: bs

text: is

text: a

text: superspace

text: map,

text: Homq-Web↑↓(a,

text: b)

text: Homq-Web↑↓

text: bs

text: ar,

text: bu

text: at

text: given

text: on

text: diagrams

text: by

text: ·bu

text: at·

text: ·bs

text: ·bu

text: w

text: w

text: ar

text: at

text: bs

text: ar

text: The

text: inverse

text: is

text: given

text: by

text: a

text: similar

text: map

text: (thanks

text: to

text: relation

text: As

text: in

text: the

text: proof

text: of

text: Theorem

text: whenever

text: a

text: morphism

text: between

text: Hom-spaces

text: in

text: q-Web↑↓

text: is

text: deﬁned

text: by

text: applying

text: some

text: combination

text: of

text: compositions

text: and

text: tensor

text: products

text: of

text: morphisms

text: we

text: can

text: apply

text: the

text: functor

text: Ψ

text: and

text: obtain

text: a

text: corresponding

text: morphism

text: in

text: the

text: category

text: of

text: q(n)-modules.

text: In

text: particular,

text: commuting

text: diagrams

text: go

text: to

text: commuting

text: diagrams

text: and

text: isomorphisms

text: go

text: to

text: isomorphisms.

text: Theorem

text: There

text: is

text: a

text: fully

text: faithful

text: functor

text: of

text: symmetric

text: monoidal

text: supercategories

text: q-Web↑

text: q-Web↑↓

text: which

text: takes

text: objects

text: and

text: morphisms

text: in

text: q-Web↑

text: to

text: objects

text: and

text: morphisms

text: of

text: the

text: same

text: name

text: in

text: q-Web↑↓.

text: Proof.

text: By

text: construction

text: the

text: deﬁning

text: relations

text: on

text: morphisms

text: of

text: q-Web↑

text: hold

text: in

text: q-Web↑↓.

text: Consequently,

text: there

text: is

text: a

text: well-deﬁned

text: functor

text: of

text: monoidal

text: supercategories.

text: As

text: discussed

text: in

text: the

text: relations

text: listed

text: in

text: Lemma

text: hold

text: for

text: all

text: possible

text: orientations

text: of

text: the

text: strands

text: in

text: a

text: diagram.

text: Now

text: if

text: a,

text: b

text: are

text: objects

text: of

text: q-Web↑

text: and

text: if

text: d

text: is

text: a

text: web

text: diagram

text: which

text: lies

text: in

text: Homq-Web↑↓(a,

text: b),

text: then

text: considering

text: the

text: end

text: behavior

text: of

text: strands

text: we

text: see

text: that

text: we

text: can

text: use

text: these

text: relations,

text: planar

text: isotopies,

text: and

text: to

text: eliminate

text: all

text: cups,

text: caps,

text: and

text: rightward

text: crossings.

text: Furthermore,

text: by

text: and

text: Lemma

text: there

text: are

text: no

text: bubbles

text: in

text: d.

text: Taken

text: together,

text: this

text: implies

text: d

text: is

text: equal

text: to

text: a

text: web

text: diagram

text: which

text: lies

text: in

text: Homq-Web↑(a,

text: b).

text: Fullness

text: and

text: faithfulness

text: follows.

text: We

text: will

text: identify

text: q-Web↑

text: as

text: a

text: full

text: of

text: q-Web↑↓

text: this

text: functor.

text: Let

text: q(n)-ModS

text: S∗

text: denote

text: the

text: full

text: monoidal

text: subsupercategory

text: of

text: q(n)-supermodules

text: generated

text: by

text: the

text: objects

text: Sp

text: (V

text: Sp

text: (V

text: p

text: n

text: n

text: We

text: next

text: show

text: the

text: functor

text: Ψ

text: q-Web↑

text: q(n)-ModS

text: can

text: be

text: extended

text: to

text: Ψ

text: q-Web↑↓

text: q(n)-ModS

text: S∗.

text: n

text: n

text: Set

text: the

text: notation

text: C

text: ev

text: Sk(V

text: Sk(V

text: k

text: n

text: n

text: C

text: coev

text: Sk(V

text: Sk(V

text: k

text: n

text: n

text: for

text: the

text: evaluation

text: and

text: coevaluation

text: maps

text: deﬁned

text: in

text: Theorem

text: There

text: is

text: an

text: essentially

text: surjective,

text: full

text: functor

text: of

text: symmetric

text: monoidal

text: supercategories

text: Ψ

text: q-Web↑↓

text: q(n)-ModS

text: S∗

text: n

text: given

text: on

text: generating

text: objects

text: by

text: Ψ

text: Sk(V

text: n

text: k

text: n

text: Ψ

text: Sk(V

text: n

text: k

text: n

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: WEBS OF TYPE Q 33 First, we deﬁne a hook to be a left-justiﬁed array of boxes in which only the ﬁrst row may have more than one box: · · : Given the shape of λ(n), the skew shape [µ/λ(n)] is an ordinary partition and, hence, can be thought of as consisting of a series of hooks wedged inside each other with the corner of each hook lying on the diagonal of [µ/λ(n)]. We number the hooks of [µ/λ(n)] from the upper left to the lower right. We deﬁne T to be the shifted tableau of shape [µ/λ(n)] whose i-th hook has the form µ,n · · (cid:48) i i i i . (cid:48) i (cid:48) i : (cid:48) i i (cid:48) The unmarked i at the very bottom takes priority over all of the i , so that if the i-th hook has only one row then every entry will be i. For example, if µ = (8, 5, 4, 2) and n = 2, then we have (cid:48) T = 1 1 1 1 1 µ,2 (cid:48) (cid:48) 1 2 2 (cid:48) (cid:48) 1 2 3 1 2 (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) whose corresponding word is w(T ) = 121 2 31 2 21 1111. µ,2 Clearly T satisﬁes the conditions of Deﬁnition 8.1.1 and is a shifted tableau. Since ν is just the number µ,n i of boxes in the i-th hook, the content ν = (ν , ν , . . . ) of T is a strict partition. Indeed, at its largest, the 1 2 µ,n (i + 1)-st hook extends as far to the right and one box farther down than the i-th hook, which is precisely − − when ν = ν 1; otherwise ν < ν 1. And, as was previously observed, property (b) of the shifted i+1 i i+1 i Littlewood-Richardson rule is satisﬁed. ≥ It remains to show that T has the lattice property. Let i 2 and let l denote the length of the µ,n ≤ ∈ word w = w(T ). Suppose m (j) = m − (j) for 0 j < l. If m (j) = m − (j) = 0, then w − { (cid:48) (cid:48) µ−,n − (cid:48)} i (cid:54) i(cid:48) 1 i i 1 l j 1, 1 , 2, 2 , . . . , i 1, (i 1) so w − = i, i . The case of m (j) = m − (j) > 0 occurs precisely if the width − l j i i 1 of the (i 1)-st hook exceeds that of the i-th hook by one box (e.g. the 2nd and 3rd hooks in the example). − ∈ { − − (cid:48)} (cid:54) (cid:48) In that case w − lies in the (i 1)-st hook, so w − i 1, (i 1) and w − = i, i . Thus T satisﬁes l j ≤l j − l j (cid:48) µ,n condition (1) of the lattice property. Since m (l) m − (l) 1, and between every pair of i in w is at least − (cid:48) (cid:54) ≤ i ≤ i 1 one (i 1) , we have m (j) = m − (j) for l j 2l. Thus T satisﬁes condition (2) of the lattice property, i i 1 µ,n completing the proof. (cid:3) Recall the character of the simple q(m)-supermodule labelled by the strict partition λ, L (λ), has char- m acter given by a power of two multiple of the Schur P-function P (x , . . . , x ) (e.g. see [CW2, Theorem λ 1 m 3.48]). Furthermore, tensor products of q(m)-supermodules corresponds to the multiplication of functions. (cid:54) ⊗ Consequently, fµ = 0 if and only L (µ) is a direct summand of L (λ) L (ν). λ,ν m m m ≥ Corollary 8.1.5. Fix a strict partition µ with n < (cid:96)(µ) and ﬁx a m (cid:96)(µ). Then there is a strict partition ⊗ ν such that the simple q(m)-supermodule L (µ) is a direct summand of the tensor product L (λ(n)) L (ν) m ⊗ | | m µ and, hence, is a direct summand of L (λ(n)) V ν . m m | | | |−| | Proof. By Proposition 8.1.4 there exists a strict partition ν with ν = µ λ(n) , of shape µ/λ(n), and with (cid:54) ⊗ fµ = 0. This implies L (µ) is a direct summand of L (λ(n)) L (ν), as asserted. The ﬁnal statement λ(n),ν m m| | m (cid:3) follows from the fact that L (ν) is a direct summand of V ν by Theorem 3.4.1. m m ##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##

text: WEBS

text: OF

text: TYPE

text: Q

text: First,

text: we

text: deﬁne

text: a

text: hook

text: to

text: be

text: a

text: left-justiﬁed

text: array

text: of

text: boxes

text: in

text: which

text: only

text: the

text: ﬁrst

text: row

text: may

text: have

text: more

text: than

text: one

text: box:

text: Given

text: the

text: shape

text: of

text: λ(n),

text: the

text: skew

text: shape

text: [µ/λ(n)]

text: is

text: an

text: ordinary

text: partition

text: and,

text: hence,

text: can

text: be

text: thought

text: of

text: as

text: consisting

text: of

text: a

text: series

text: of

text: hooks

text: wedged

text: inside

text: each

text: other

text: with

text: the

text: corner

text: of

text: each

text: hook

text: lying

text: on

text: the

text: diagonal

text: of

text: [µ/λ(n)].

text: We

text: number

text: the

text: hooks

text: of

text: [µ/λ(n)]

text: from

text: the

text: upper

text: left

text: to

text: the

text: lower

text: right.

text: We

text: deﬁne

text: T

text: to

text: be

text: the

text: shifted

text: tableau

text: of

text: shape

text: [µ/λ(n)]

text: whose

text: i-th

text: hook

text: has

text: the

text: form

text: µ,n

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: The

text: unmarked

text: i

text: at

text: the

text: very

text: bottom

text: takes

text: priority

text: over

text: all

text: of

text: the

text: i

text: so

text: that

text: if

text: the

text: i-th

text: hook

text: has

text: only

text: one

text: row

text: then

text: every

text: entry

text: will

text: be

text: i.

text: For

text: example,

text: if

text: µ

text: and

text: n

text: then

text: we

text: have

text: T

text: whose

text: corresponding

text: word

text: is

text: w(T

text: Clearly

text: T

text: satisﬁes

text: the

text: conditions

text: of

text: Deﬁnition

text: and

text: is

text: a

text: shifted

text: tableau.

text: Since

text: ν

text: is

text: just

text: the

text: number

text: µ,n

text: i

text: of

text: boxes

text: in

text: the

text: i-th

text: hook,

text: the

text: content

text: ν

text: (ν

text: ν

text: of

text: T

text: is

text: a

text: strict

text: partition.

text: Indeed,

text: at

text: its

text: largest,

text: the

text: µ,n

text: (i

text: hook

text: extends

text: as

text: far

text: to

text: the

text: right

text: and

text: one

text: box

text: farther

text: down

text: than

text: the

text: i-th

text: hook,

text: which

text: is

text: precisely

text: when

text: ν

text: ν

text: otherwise

text: ν

text: ν

text: And,

text: as

text: was

text: previously

text: observed,

text: property

text: (b)

text: of

text: the

text: shifted

text: i

text: i

text: Littlewood-Richardson

text: rule

text: is

text: satisﬁed.

text: It

text: remains

text: to

text: show

text: that

text: T

text: has

text: the

text: lattice

text: property.

text: Let

text: i

text: and

text: let

text: l

text: denote

text: the

text: length

text: of

text: the

text: µ,n

text: word

text: w

text: w(T

text: Suppose

text: m

text: (j)

text: m

text: (j)

text: for

text: j

text: l.

text: If

text: m

text: (j)

text: m

text: (j)

text: then

text: w

text: µ−,n

text: i

text: i

text: i

text: l

text: j

text: i

text: (i

text: so

text: w

text: i,

text: i

text: The

text: case

text: of

text: m

text: (j)

text: m

text: (j)

text: occurs

text: precisely

text: if

text: the

text: width

text: l

text: j

text: i

text: i

text: of

text: the

text: (i

text: hook

text: exceeds

text: that

text: of

text: the

text: i-th

text: hook

text: by

text: one

text: box

text: (e.g.

text: the

text: and

text: hooks

text: in

text: the

text: example).

text: In

text: that

text: case

text: w

text: lies

text: in

text: the

text: (i

text: hook,

text: so

text: w

text: i

text: (i

text: and

text: w

text: i,

text: i

text: Thus

text: T

text: satisﬁes

text: l

text: j

text: ≤l

text: j

text: l

text: j

text: µ,n

text: condition

text: of

text: the

text: lattice

text: property.

text: Since

text: m

text: (l)

text: m

text: (l)

text: and

text: between

text: every

text: pair

text: of

text: i

text: in

text: w

text: is

text: at

text: least

text: i

text: i

text: one

text: (i

text: we

text: have

text: m

text: (j)

text: m

text: (j)

text: for

text: l

text: j

text: Thus

text: T

text: satisﬁes

text: condition

text: of

text: the

text: lattice

text: property,

text: i

text: i

text: µ,n

text: completing

text: the

text: proof.

text: Recall

text: the

text: character

text: of

text: the

text: simple

text: q(m)-supermodule

text: labelled

text: by

text: the

text: strict

text: partition

text: λ,

text: L

text: (λ),

text: has

text: char-

text: m

text: acter

text: given

text: by

text: a

text: power

text: of

text: two

text: multiple

text: of

text: the

text: Schur

text: P-function

text: P

text: (x

text: x

text: (e.g.

text: see

text: Theorem

text: λ

text: m

text: Furthermore,

text: tensor

text: products

text: of

text: q(m)-supermodules

text: corresponds

text: to

text: the

text: multiplication

text: of

text: functions.

text: Consequently,

text: fµ

text: if

text: and

text: only

text: L

text: (µ)

text: is

text: a

text: direct

text: summand

text: of

text: L

text: (λ)

text: L

text: (ν).

text: λ,ν

text: m

text: m

text: m

text: Corollary

text: Fix

text: a

text: strict

text: partition

text: µ

text: with

text: n

text: and

text: ﬁx

text: a

text: m

text: Then

text: there

text: is

text: a

text: strict

text: partition

text: ν

text: such

text: that

text: the

text: simple

text: q(m)-supermodule

text: L

text: (µ)

text: is

text: a

text: direct

text: summand

text: of

text: the

text: tensor

text: product

text: L

text: (λ(n))

text: L

text: (ν)

text: m

text: m

text: µ

text: and,

text: hence,

text: is

text: a

text: direct

text: summand

text: of

text: L

text: (λ(n))

text: V

text: ν

text: m

text: m

text: Proof.

text: By

text: Proposition

text: there

text: exists

text: a

text: strict

text: partition

text: ν

text: with

text: ν

text: µ

text: λ(n)

text: of

text: shape

text: µ/λ(n),

text: and

text: with

text: fµ

text: This

text: implies

text: L

text: (µ)

text: is

text: a

text: direct

text: summand

text: of

text: L

text: (λ(n))

text: L

text: (ν),

text: as

text: asserted.

text: The

text: ﬁnal

text: statement

text: λ(n),ν

text: m

text: m|

text: m

text: follows

text: from

text: the

text: fact

text: that

text: L

text: (ν)

text: is

text: a

text: direct

text: summand

text: of

text: V

text: ν

text: by

text: Theorem

text: m

text: m

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text: 34 GORDON C. BROWN AND JONATHAN R. KUJAWA References [Br] G. Brown, Webs for permutation supermodules of type Q, preprint. [BDK] G. Brown, N. Davidson, J. Kujawa, Quantum webs of type Q, preprint. [Bru] J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q(n), Adv. Math., 182 (1) (2004), 28-77. O [BD2] J. Brundan and N. Davidson, Type A blocks of super category , J. Algebra, 473 (2017), 447-480. O [BD3] , Type C blocks of super category , Math. Z., to appear. [BE] J. Brundan and A. Ellis, Monoidal supercategories, Commun. Math. Phys. 351, 1045-1089. [BK] J. Brundan and A. Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z. 239 (2002), 27-68. [CKM] S. Cautis, J. Kamnitzer, and S. Morrison, Webs and quantum skew Howe duality, Math. Ann. 360-1-2 (2014), 351-390. [Chen] C.-W. Chen, Reduction method for representations of queer Lie superalgebras, J. Math. Phys., 57 (5), (2016). [CKW] S.-J. Cheng, J.-H. Kwon, and W. Wang. Character formulae for queer Lie superalgebras and canonical bases of types A/C, Comm. Math. Phys., 352(3) (2017), 1091–1119. [CW1] S.-J. Cheng and W. Wang, Remarks on Schur-Howe-Sergeev duality, Lett. Math. Phys. 52 (2000), 143-153. [CW2] S.-J. Cheng and W. Wang, Dualities and representations of Lie superalgebras, Amer. Math. Soc., 2012. [CK] J. Comes and J. Kujawa, A basis theorem for the degenerate aﬃne oriented Brauer-Cliﬀord supercategory, submitted. [CK2] , Higher level twisted Heisenberg supercategories, preprint. [DW] J. Du and J. Wan, Presenting queer Schur superalgebras, Int. Math. Res. Not. 8 (2015), 2210-2272. [EGNO] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Amer. Math. Soc., 2015. [Ki] D. Kim, Graphical calculus on representations of quantum lie algebras, 2003. PhD thesis, University of California, Davis. [Kl] A. Kleshchev, Linear and Projective Representations of Symmetric Groups, Cam. Tr. Math. 163 (2005), CUP. [Ku] G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151. [Mo] S. Morrison, A Diagrammatic Category for the Representation Theory of Uq (sln), PhD thesis, University of California, Berkeley, 2007. [N] M. Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), 190-257. [PS] I. Penkov and V. Serganova, Characters of irreducible G-modules and cohomology of G/P for the Lie supergroup G = Q(N), J. Math. Sci. (New York) 84 (5) (1997), 1382-1412. [QS] H. Queﬀelec and A. Sartori, Mixed skew Howe duality and link invariants, arXiv:1504.01225. [RT] D.E.V. Rose and D. Tubbenhauer, Symmetric webs, Jones-Wenzl recursions and q-Howe duality, Int. Math. Res. Not. IMRN 2016, no. 17, 5249-5290. [RTW] G. Rumer, E. Teller, and H. Weyl, Eine fu¨r die Valenztheorie geeignete Basis der bina¨ren Vektorinvarianten, Nachrichten von der Ges. der Wiss. Zu G¨ottingen, Math.-Phys. Klasse (1932), 498-504 (German). [Se1] A.N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras GL(n,m) and Q(n), Math. USSR Sbornik 51 (1985), 419-427. [Se2] A.N. Sergeev, The Howe duality and the projective representations of symmetric groups, Rep. Th. 3 (1999), 416-434. [ST] A. Sartori and D. Tubbenhauer, Webs and q-Howe dualities in types BCD, arXiv:1701.02932. [St] J. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), 87-134. [TVW] D. Tubbenhauer, P. Vaz, and P. Wedrich, Super q-Howe duality and web categories, Algebr. Geom. Topol. 17 (2017), no. 6, 3703-3749. [W] H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 91 (1987), 5-9. Fort Worth, TX Email address: gcbrown@utexas.edu Department of Mathematics, University of Oklahoma, Norman, OK 73019 Email address: kujawa@math.ou.edu ##LTLine## ##LTLine##

text: GORDON

text: C.

text: BROWN

text: AND

text: JONATHAN

text: R.

text: KUJAWA

text: References

text: [Br]

text: G.

text: Brown,

text: Webs

text: for

text: permutation

text: supermodules

text: of

text: type

text: Q,

text: preprint.

text: [BDK]

text: G.

text: Brown,

text: N.

text: Davidson,

text: J.

text: Kujawa,

text: Quantum

text: webs

text: of

text: type

text: Q,

text: preprint.

text: [Bru]

text: J.

text: Brundan,

text: Kazhdan-Lusztig

text: polynomials

text: and

text: character

text: formulae

text: for

text: the

text: Lie

text: superalgebra

text: q(n),

text: Adv.

text: Math.,

text: O

text: J.

text: Brundan

text: and

text: N.

text: Davidson,

text: Type

text: A

text: blocks

text: of

text: super

text: category

text: J.

text: Algebra,

text: O

text: Type

text: C

text: blocks

text: of

text: super

text: category

text: Math.

text: Z.,

text: to

text: appear.

text: [BE]

text: J.

text: Brundan

text: and

text: A.

text: Ellis,

text: Monoidal

text: supercategories,

text: Commun.

text: Math.

text: Phys.

text: [BK]

text: J.

text: Brundan

text: and

text: A.

text: Kleshchev,

text: Projective

text: representations

text: of

text: symmetric

text: groups

text: via

text: Sergeev

text: duality,

text: Math.

text: Z.

text: [CKM]

text: S.

text: Cautis,

text: J.

text: Kamnitzer,

text: and

text: S.

text: Morrison,

text: Webs

text: and

text: quantum

text: skew

text: Howe

text: duality,

text: Math.

text: Ann.

text: [Chen]

text: C.-W.

text: Chen,

text: Reduction

text: method

text: for

text: representations

text: of

text: queer

text: Lie

text: superalgebras,

text: J.

text: Math.

text: Phys.,

text: [CKW]

text: S.-J.

text: Cheng,

text: J.-H.

text: Kwon,

text: and

text: W.

text: Wang.

text: Character

text: formulae

text: for

text: queer

text: Lie

text: superalgebras

text: and

text: canonical

text: bases

text: of

text: types

text: A/C,

text: Comm.

text: Math.

text: Phys.,

text: S.-J.

text: Cheng

text: and

text: W.

text: Wang,

text: Remarks

text: on

text: Schur-Howe-Sergeev

text: duality,

text: Lett.

text: Math.

text: Phys.

text: S.-J.

text: Cheng

text: and

text: W.

text: Wang,

text: Dualities

text: and

text: representations

text: of

text: Lie

text: superalgebras,

text: Amer.

text: Math.

text: Soc.,

text: [CK]

text: J.

text: Comes

text: and

text: J.

text: Kujawa,

text: A

text: basis

text: theorem

text: for

text: the

text: degenerate

text: aﬃne

text: oriented

text: Brauer-Cliﬀord

text: supercategory,

text: submitted.

text: Higher

text: level

text: twisted

text: Heisenberg

text: supercategories,

text: preprint.

text: [DW]

text: J.

text: Du

text: and

text: J.

text: Wan,

text: Presenting

text: queer

text: Schur

text: superalgebras,

text: Int.

text: Math.

text: Res.

text: Not.

text: [EGNO]

text: P.

text: Etingof,

text: S.

text: Gelaki,

text: D.

text: Nikshych,

text: and

text: V.

text: Ostrik,

text: Tensor

text: Categories,

text: Amer.

text: Math.

text: Soc.,

text: [Ki]

text: D.

text: Kim,

text: Graphical

text: calculus

text: on

text: representations

text: of

text: quantum

text: lie

text: algebras,

text: PhD

text: thesis,

text: University

text: of

text: California,

text: Davis.

text: [Kl]

text: A.

text: Kleshchev,

text: Linear

text: and

text: Projective

text: Representations

text: of

text: Symmetric

text: Groups,

text: Cam.

text: Tr.

text: Math.

text: CUP.

text: [Ku]

text: G.

text: Kuperberg,

text: Spiders

text: for

text: rank

text: Lie

text: algebras,

text: Comm.

text: Math.

text: Phys.

text: [Mo]

text: S.

text: Morrison,

text: A

text: Diagrammatic

text: Category

text: for

text: the

text: Representation

text: Theory

text: of

text: Uq

text: (sln),

text: PhD

text: thesis,

text: University

text: of

text: California,

text: Berkeley,

text: [N]

text: M.

text: Nazarov,

text: Young’s

text: symmetrizers

text: for

text: projective

text: representations

text: of

text: the

text: symmetric

text: group,

text: Adv.

text: Math.

text: [PS]

text: I.

text: Penkov

text: and

text: V.

text: Serganova,

text: Characters

text: of

text: irreducible

text: G-modules

text: and

text: cohomology

text: of

text: G/P

text: for

text: the

text: Lie

text: supergroup

text: G

text: Q(N),

text: J.

text: Math.

text: Sci.

text: (New

text: York)

text: [QS]

text: H.

text: Queﬀelec

text: and

text: A.

text: Sartori,

text: Mixed

text: skew

text: Howe

text: duality

text: and

text: link

text: invariants,

text: [RT]

text: D.E.V.

text: Rose

text: and

text: D.

text: Tubbenhauer,

text: Symmetric

text: webs,

text: Jones-Wenzl

text: recursions

text: and

text: q-Howe

text: duality,

text: Int.

text: Math.

text: Res.

text: Not.

text: IMRN

text: no.

text: [RTW]

text: G.

text: Rumer,

text: E.

text: Teller,

text: and

text: H.

text: Weyl,

text: Eine

text: fu¨r

text: die

text: Valenztheorie

text: geeignete

text: Basis

text: der

text: bina¨ren

text: Vektorinvarianten,

text: Nachrichten

text: von

text: der

text: Ges.

text: der

text: Wiss.

text: Zu

text: G¨ottingen,

text: Math.-Phys.

text: Klasse

text: (German).

text: A.N.

text: Sergeev,

text: Tensor

text: algebra

text: of

text: the

text: identity

text: representation

text: as

text: a

text: module

text: over

text: the

text: Lie

text: superalgebras

text: GL(n,m)

text: and

text: Q(n),

text: Math.

text: USSR

text: Sbornik

text: A.N.

text: Sergeev,

text: The

text: Howe

text: duality

text: and

text: the

text: projective

text: representations

text: of

text: symmetric

text: groups,

text: Rep.

text: Th.

text: [ST]

text: A.

text: Sartori

text: and

text: D.

text: Tubbenhauer,

text: Webs

text: and

text: q-Howe

text: dualities

text: in

text: types

text: BCD,

text: [St]

text: J.

text: Stembridge,

text: Shifted

text: tableaux

text: and

text: the

text: projective

text: representations

text: of

text: symmetric

text: groups,

text: Adv.

text: Math.

text: [TVW]

text: D.

text: Tubbenhauer,

text: P.

text: Vaz,

text: and

text: P.

text: Wedrich,

text: Super

text: q-Howe

text: duality

text: and

text: web

text: categories,

text: Algebr.

text: Geom.

text: Topol.

text: no.

text: [W]

text: H.

text: Wenzl,

text: On

text: sequences

text: of

text: projections,

text: C.

text: R.

text: Math.

text: Rep.

text: Acad.

text: Sci.

text: Canada

text: Fort

text: Worth,

text: TX

text: Email

text: address:

text: gcbrown@utexas.edu

text: Department

text: of

text: Mathematics,

text: University

text: of

text: Oklahoma,

text: Norman,

text: OK

text: Email

text: address:

text: kujawa@math.ou.edu

text: ##LTLine##

text: ##LTLine##


text: Moment Approximations for Shallow Flow J. Kowalski and M. Torrilhon is traditionally known as the Saint-Venant equations. The idea itself, however, namely deriving a shallow ﬂow model by means of depth-averaging, is not restricted to a constitutive equation representing water and has been applied to many other ﬂuid and granular ﬂow rheologies [27, 11, 2]. Robust and eﬃcient numerical methods have been developed to solve them [23, 32]. Quite naturally any shallow ﬂow formulation comes at the price of loosing vertical information, such as information on the velocity proﬁle. This is not critical if the velocity proﬁle is constant throughout the ﬂow depth and vertical acceleration can be neglected. In the geophysical ﬂow context such a situation is often referred to as ’plug-ﬂow’. In many realistic situations, however, velocity proﬁles deviate signiﬁcantly from a vertically constant value. This can be observed both in large scale ﬁeld experiments [17] as well as in small scale laboratory experiments [28, 26]. The computational shallow ﬂow community encounters this obvious modeling error by intro- ducing the concept of a shape factor. The shape factor is determined based on an assumed parametrization of the velocity proﬁle. Simple polynomial and exponential parametrizations re- sult in shape factors that are independent of ﬁeld variables, and enter the system as additional constants [14, 18]. Though this serves as a ﬁrst order correction, it is also restrictive, as the chosen parametrization, say a linear velocity proﬁle, is assumed to be an appropriate choice throughout the duration of the ﬂow. Data acquired during transient granular chute ﬂow exper- iments again indicate that this is typically not the case and a strong regime dependency of the velocity proﬁle can be observed [28]. The capability to model a regime dependent velocity (and shear) proﬁle would be highly relevant in the geophysical context, as it allows to assess erosion rates and to determine coupling forces, both of which are of special interest from an application perspective. In our work we address this problem and propose a shallow ﬂow formulation based on vertical moments. Moment methods proved to be a powerful tool and have been successfully applied in various scientiﬁc ﬁelds, such as in rareﬁed gas dynamics [10, 33]. In the shallow ﬂow context, moment methods allow to preserve information on the vertical ﬂow structure, while we can still make use of the simplifying and complexity reducing framework of depth-averaging. This idea is not entirely new and we follow the spirit of previously published articles on secondary ﬂow eﬀects in curved open channel ﬂow and meandering rivers [9, 31, 34], sediment transport and mixing processes in rivers [1, 8] as well as vertical stratiﬁcation in shallow geophysical mixtures [19]. While these works focus on pragmatic model derivations that are strongly motivated by speciﬁc applications, the focus of our paper is a fundamental analysis and validation of a generic model cascade shallow ﬂows based on moment equations. We complement derivation and analysis of the shallow moment hierarchy with a tailored ver- tically resolved reference model that will allow us to assess the moment model’s accuracy. The vertically resolved system uses the nonlinear mapping of the shallow ﬂow system onto its scaled counterpart with a constant depth of unity. Such a transformation has been used, e.g., in [22] to study thin melting ﬁlms in phase-change processes. Since then similar mappings have been applied to a variety of processes, e.g. contact phase-change and ablation processes [21, 30]. Com- paring results of the reference system with the shallow ﬂow moment models allows us for the ﬁrst time to analyze and discuss the model error introduced during depth-averaging. The generic formulation of the shallow ﬂow moment hierarchy and its comparison to the vertically resolved reference solutions are the major innovations of our work. In Sec. 2 we give a quick overview of the technical approach of the paper. In Sec. 3, we introduce the coordinate mapping and derive the reference shallow ﬂow system. It consists of a ﬁrst contribution similar to the shallow ﬂow equation, and a second contribution that accounts for a kinematically consistent vertical coupling. We close this section by particularizing the reference shallow ﬂow system for a Newtonian constitutive relation. A moment expansion based on the reference shallow ﬂow system is the content of Sec. 4. In Sec. 5, we present numerical results both for the vertically resolved reference model, and for the shallow moment systems. A discussion of these results includes the role of the vertical coupling factor in the vertically resolved model and the performance of the shallow moment hierarchy with respect to the reference results. We 2 ##LTLine##

text: Moment

text: Approximations

text: for

text: Shallow

text: Flow

text: J.

text: Kowalski

text: and

text: M.

text: Torrilhon

text: is

text: traditionally

text: known

text: as

text: the

text: Saint-Venant

text: equations.

text: The

text: idea

text: itself,

text: however,

text: namely

text: deriving

text: a

text: shallow

text: ﬂow

text: model

text: by

text: means

text: of

text: depth-averaging,

text: is

text: not

text: restricted

text: to

text: a

text: constitutive

text: equation

text: representing

text: water

text: and

text: has

text: been

text: applied

text: to

text: many

text: other

text: ﬂuid

text: and

text: granular

text: ﬂow

text: rheologies

text: Robust

text: and

text: eﬃcient

text: numerical

text: methods

text: have

text: been

text: developed

text: to

text: solve

text: them

text: Quite

text: naturally

text: any

text: shallow

text: ﬂow

text: formulation

text: comes

text: at

text: the

text: price

text: of

text: loosing

text: vertical

text: information,

text: such

text: as

text: information

text: on

text: the

text: velocity

text: proﬁle.

text: This

text: is

text: not

text: critical

text: if

text: the

text: velocity

text: proﬁle

text: is

text: constant

text: throughout

text: the

text: ﬂow

text: depth

text: and

text: vertical

text: acceleration

text: can

text: be

text: neglected.

text: In

text: the

text: geophysical

text: ﬂow

text: context

text: such

text: a

text: situation

text: is

text: often

text: referred

text: to

text: as

text: ’plug-ﬂow’.

text: In

text: many

text: realistic

text: situations,

text: however,

text: velocity

text: proﬁles

text: deviate

text: signiﬁcantly

text: from

text: a

text: vertically

text: constant

text: value.

text: This

text: can

text: be

text: observed

text: both

text: in

text: large

text: scale

text: ﬁeld

text: experiments

text: as

text: well

text: as

text: in

text: small

text: scale

text: laboratory

text: experiments

text: The

text: computational

text: shallow

text: ﬂow

text: community

text: encounters

text: this

text: obvious

text: modeling

text: error

text: by

text: intro-

text: ducing

text: the

text: concept

text: of

text: a

text: shape

text: factor.

text: The

text: shape

text: factor

text: is

text: determined

text: based

text: on

text: an

text: assumed

text: parametrization

text: of

text: the

text: velocity

text: proﬁle.

text: Simple

text: polynomial

text: and

text: exponential

text: parametrizations

text: re-

text: sult

text: in

text: shape

text: factors

text: that

text: are

text: independent

text: of

text: ﬁeld

text: variables,

text: and

text: enter

text: the

text: system

text: as

text: additional

text: constants

text: Though

text: this

text: serves

text: as

text: a

text: ﬁrst

text: order

text: correction,

text: it

text: is

text: also

text: restrictive,

text: as

text: the

text: chosen

text: parametrization,

text: say

text: a

text: linear

text: velocity

text: proﬁle,

text: is

text: assumed

text: to

text: be

text: an

text: appropriate

text: choice

text: throughout

text: the

text: duration

text: of

text: the

text: ﬂow.

text: Data

text: acquired

text: during

text: transient

text: granular

text: chute

text: ﬂow

text: exper-

text: iments

text: again

text: indicate

text: that

text: this

text: is

text: typically

text: not

text: the

text: case

text: and

text: a

text: strong

text: regime

text: dependency

text: of

text: the

text: velocity

text: proﬁle

text: can

text: be

text: observed

text: The

text: capability

text: to

text: model

text: a

text: regime

text: dependent

text: velocity

text: (and

text: shear)

text: proﬁle

text: would

text: be

text: highly

text: relevant

text: in

text: the

text: geophysical

text: context,

text: as

text: it

text: allows

text: to

text: assess

text: erosion

text: rates

text: and

text: to

text: determine

text: coupling

text: forces,

text: both

text: of

text: which

text: are

text: of

text: special

text: interest

text: from

text: an

text: application

text: perspective.

text: In

text: our

text: work

text: we

text: address

text: this

text: problem

text: and

text: propose

text: a

text: shallow

text: ﬂow

text: formulation

text: based

text: on

text: vertical

text: moments.

text: Moment

text: methods

text: proved

text: to

text: be

text: a

text: powerful

text: tool

text: and

text: have

text: been

text: successfully

text: applied

text: in

text: various

text: scientiﬁc

text: ﬁelds,

text: such

text: as

text: in

text: rareﬁed

text: gas

text: dynamics

text: In

text: the

text: shallow

text: ﬂow

text: context,

text: moment

text: methods

text: allow

text: to

text: preserve

text: information

text: on

text: the

text: vertical

text: ﬂow

text: structure,

text: while

text: we

text: can

text: still

text: make

text: use

text: of

text: the

text: simplifying

text: and

text: complexity

text: reducing

text: framework

text: of

text: depth-averaging.

text: This

text: idea

text: is

text: not

text: entirely

text: new

text: and

text: we

text: follow

text: the

text: spirit

text: of

text: previously

text: published

text: articles

text: on

text: secondary

text: ﬂow

text: eﬀects

text: in

text: curved

text: open

text: channel

text: ﬂow

text: and

text: meandering

text: rivers

text: sediment

text: transport

text: and

text: mixing

text: processes

text: in

text: rivers

text: as

text: well

text: as

text: vertical

text: stratiﬁcation

text: in

text: shallow

text: geophysical

text: mixtures

text: While

text: these

text: works

text: focus

text: on

text: pragmatic

text: model

text: derivations

text: that

text: are

text: strongly

text: motivated

text: by

text: speciﬁc

text: applications,

text: the

text: focus

text: of

text: our

text: paper

text: is

text: a

text: fundamental

text: analysis

text: and

text: validation

text: of

text: a

text: generic

text: model

text: cascade

text: shallow

text: ﬂows

text: based

text: on

text: moment

text: equations.

text: We

text: complement

text: derivation

text: and

text: analysis

text: of

text: the

text: shallow

text: moment

text: hierarchy

text: with

text: a

text: tailored

text: ver-

text: tically

text: resolved

text: reference

text: model

text: that

text: will

text: allow

text: us

text: to

text: assess

text: the

text: moment

text: model’s

text: accuracy.

text: The

text: vertically

text: resolved

text: system

text: uses

text: the

text: nonlinear

text: mapping

text: of

text: the

text: shallow

text: ﬂow

text: system

text: onto

text: its

text: scaled

text: counterpart

text: with

text: a

text: constant

text: depth

text: of

text: unity.

text: Such

text: a

text: transformation

text: has

text: been

text: used,

text: e.g.,

text: in

text: to

text: study

text: thin

text: melting

text: ﬁlms

text: in

text: phase-change

text: processes.

text: Since

text: then

text: similar

text: mappings

text: have

text: been

text: applied

text: to

text: a

text: variety

text: of

text: processes,

text: e.g.

text: contact

text: phase-change

text: and

text: ablation

text: processes

text: Com-

text: paring

text: results

text: of

text: the

text: reference

text: system

text: with

text: the

text: shallow

text: ﬂow

text: moment

text: models

text: allows

text: us

text: for

text: the

text: ﬁrst

text: time

text: to

text: analyze

text: and

text: discuss

text: the

text: model

text: error

text: introduced

text: during

text: depth-averaging.

text: The

text: generic

text: formulation

text: of

text: the

text: shallow

text: ﬂow

text: moment

text: hierarchy

text: and

text: its

text: comparison

text: to

text: the

text: vertically

text: resolved

text: reference

text: solutions

text: are

text: the

text: major

text: innovations

text: of

text: our

text: work.

text: In

text: Sec.

text: we

text: give

text: a

text: quick

text: overview

text: of

text: the

text: technical

text: approach

text: of

text: the

text: paper.

text: In

text: Sec.

text: we

text: introduce

text: the

text: coordinate

text: mapping

text: and

text: derive

text: the

text: reference

text: shallow

text: ﬂow

text: system.

text: It

text: consists

text: of

text: a

text: ﬁrst

text: contribution

text: similar

text: to

text: the

text: shallow

text: ﬂow

text: equation,

text: and

text: a

text: second

text: contribution

text: that

text: accounts

text: for

text: a

text: kinematically

text: consistent

text: vertical

text: coupling.

text: We

text: close

text: this

text: section

text: by

text: particularizing

text: the

text: reference

text: shallow

text: ﬂow

text: system

text: for

text: a

text: Newtonian

text: constitutive

text: relation.

text: A

text: moment

text: expansion

text: based

text: on

text: the

text: reference

text: shallow

text: ﬂow

text: system

text: is

text: the

text: content

text: of

text: Sec.

text: In

text: Sec.

text: we

text: present

text: numerical

text: results

text: both

text: for

text: the

text: vertically

text: resolved

text: reference

text: model,

text: and

text: for

text: the

text: shallow

text: moment

text: systems.

text: A

text: discussion

text: of

text: these

text: results

text: includes

text: the

text: role

text: of

text: the

text: vertical

text: coupling

text: factor

text: in

text: the

text: vertically

text: resolved

text: model

text: and

text: the

text: performance

text: of

text: the

text: shallow

text: moment

text: hierarchy

text: with

text: respect

text: to

text: the

text: reference

text: results.

text: We

text: ##LTLine##


text: Moment Approximations for Shallow Flow J. Kowalski and M. Torrilhon z hs(t;x;y) ³ ³ = 1 u(t;x;y;z) u~(t;x; y;³) hb(t;x;y) ³ = 0 x;y x;y Figure 1: In the physical space (left plot) the ﬂow is constrained in z-direction between the basal topography at z = h (t, x, y) and the free surface at z = h (t, x, y). The mapping (7) leads to a projected b s ∈ depth coordinate ζ (right plot) and to a ﬂow that remains in the interval ζ [0, 1]. − p(t, x, y, z) = (h (t, x, y) z) ρge , (6) b z The kinematic boundary conditions (61) and (62) are not aﬀected by back-transformation to dimensional formulation and stay unaltered. 3.1 Mapping The system (3)-(6) already represents our reference system, however, in order to render it more accessible we formulate the system in terms of the scaled and normalized vertical variable ζ deﬁned by − z h (t, x, y) ζ = b (7) h(t, x, y) − with the height of the ﬂow as h := h h . This kind of mapping immediately implies z = s b h(t, x, y)ζ + h (t, x, y) and eﬀectively transforms the free surface ﬂow onto a domain with a b constant height of unity, see Fig. 1. In order to transform the governing equations, we initially consider an arbitrary function depending on space and time ψ(t, x, y, z). Its mapped counterpart ψ˜(t, x, y, ζ) is given by ψ˜ (t, x, y, ζ) = ψ (t, x, y, h(t, x, y)ζ + h (t, x, y)) , (8) b which implies (cid:0) (cid:1) − − ψ (t, x, y, z) = ψ˜ t, x, y, h(t, x, y) 1(z h (t, x, y)) . (9) b Later, we want to transform the set of reduced balance laws (3)-(5). This requires diﬀerential operators of the mapping ψ, wh(cid:16)ich (cid:17)read (cid:16) (cid:17) − ∈ { } h∂ ψ = ∂ hψ˜ ∂ ∂ (ζh + h ) ψ˜ s t, x, y (10) s s ζ s b h∂ ψ = ∂ ψ˜ (11) z ζ 4 ##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##

text: Moment

text: Approximations

text: for

text: Shallow

text: Flow

text: J.

text: Kowalski

text: and

text: M.

text: Torrilhon

text: z

font: default

text: hs(t;x;y)

text: ³

text: ³

text: u(t;x;y;z)

text: u~(t;x;

text: hb(t;x;y)

text: ³

text: x;y

text: x;y

text: Figure

text: In

label: caption

text: the

text: physical

text: space

text: (left

text: plot)

text: the

text: ﬂow

text: is

text: constrained

text: in

text: z-direction

text: between

text: the

text: basal

text: topography

text: at

text: z

text: h

text: (t,

text: x,

text: y)

text: and

text: the

text: free

text: surface

text: at

text: z

text: h

text: (t,

text: x,

text: y).

text: The

text: mapping

text: leads

text: to

text: a

text: projected

text: b

text: s

text: depth

text: coordinate

text: ζ

text: (right

text: plot)

text: and

text: to

text: a

text: ﬂow

text: that

text: remains

text: in

text: the

text: interval

text: ζ

text: p(t,

text: x,

text: y,

text: z)

text: (h

text: (t,

text: x,

text: y)

text: z)

text: ρge

text: b

text: z

text: The

text: kinematic

text: boundary

text: conditions

text: and

text: are

text: not

text: aﬀected

text: by

text: back-transformation

text: to

text: dimensional

text: formulation

text: and

text: stay

text: unaltered.

text: Mapping

text: The

text: system

text: already

text: represents

text: our

text: reference

text: system,

text: however,

text: in

text: order

text: to

text: render

text: it

text: more

text: accessible

text: we

text: formulate

text: the

text: system

text: in

text: terms

text: of

text: the

text: scaled

text: and

text: normalized

text: vertical

text: variable

text: ζ

text: deﬁned

text: by

text: z

text: h

text: (t,

text: x,

text: y)

text: ζ

text: b

text: h(t,

text: x,

text: y)

text: with

text: the

text: height

text: of

text: the

text: ﬂow

text: as

text: h

text: h

text: h

text: This

text: kind

text: of

text: mapping

text: immediately

text: implies

text: z

text: s

text: b

text: h(t,

text: x,

text: y)ζ

text: h

text: (t,

text: x,

text: y)

text: and

text: eﬀectively

text: transforms

text: the

text: free

text: surface

text: ﬂow

text: onto

text: a

text: domain

text: with

text: a

text: b

text: constant

text: height

text: of

text: unity,

text: see

text: Fig.

text: In

text: order

text: to

text: transform

text: the

text: governing

text: equations,

text: we

text: initially

text: consider

text: an

text: arbitrary

text: function

text: depending

text: on

text: space

text: and

text: time

text: ψ(t,

text: x,

text: y,

text: z).

text: Its

text: mapped

text: counterpart

text: ψ˜(t,

text: x,

text: y,

text: ζ)

text: is

text: given

text: by

text: ψ˜

text: (t,

text: x,

text: y,

text: ζ)

text: ψ

text: (t,

text: x,

text: y,

text: h(t,

text: x,

text: y)ζ

text: h

text: (t,

text: x,

text: y))

text: b

text: which

text: implies

text: ψ

text: (t,

text: x,

text: y,

text: z)

text: ψ˜

text: t,

text: x,

text: y,

text: h(t,

text: x,

text: y)

text: h

text: (t,

text: x,

text: y))

text: b

text: Later,

text: we

text: want

text: to

text: transform

text: the

text: set

text: of

text: reduced

text: balance

text: laws

text: This

text: requires

text: diﬀerential

text: operators

text: of

text: the

text: mapping

text: ψ,

text: h∂

text: ψ

text: hψ˜

text: (ζh

text: h

text: ψ˜

text: s

text: t,

text: x,

text: y

text: s

text: s

text: ζ

text: s

text: b

text: h∂

text: ψ

text: ψ˜

text: z

text: ζ

text: ##LTFigure##

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: Supplementary Information 7 Free energy, the Fokker-Planck equation, and variational optimization A brief, self-contained, introduction to the properties of the Fokker-Planck equation in the context of variational optimization is provided. 7.1 Stochastic processes and the Fokker-Planck equation A general class of stochastic processes is given by the Itô stochastic differential equation: √ dx(t) = µ(x(t), t) dt + 2σ(x(t), t) dw(t) (15) ∈ This equation describes the evolution of a random vector x X under a drift µ(x(t), t) dt and a diffusion σ(x(t), t) dw(t). The diffusion coefﬁcient σ(x(t), t) scales a Wiener process w(t) which − ∼ N describes the accumulation of gaussian noise w(t + dt) w(t) (0, dt). In general x and t µ(x(t), t) are n-dimensional random vectors, σ(x(t), t) an (n, m)-dimensional matrix, and w(t) an m-dimensional vector. The Fokker-Planck equation describes the time-evolution of the probability density ρ of the random vector x: −∇ ρ˙ = (µρ) + ∆(σ2ρ) (16) ∞ × → ∞ · where ρ : [0, ) M [0, ) is a probability density ﬂow(cid:112)(i.e. at each timepoint t, ρ(t, ) deﬁnes a density), M is the underlying manifold, and σ2 := σσT is known as the diffusion tensor. The simplest case of the FP equation is Brownian motion with stationary (i.e. time-independent) and − invariant (i.e. position-independent) diffusion σ(x(t), t) = β 1 and zero drift µ(x(t), t) = 0. In this case, the FP equation (16) is the heat equation with inverse temperature β controlling the “speed” of the diffusion: − ρ˙ = β 1∆ρ (17) 7.2 Langevin dynamics: drift potential The special case in which the drift µ is a conservative time-homogeneous vector ﬁeld corresponds to Langevin dynamics. Here the vector ﬁeld µ is generated by a scalar potential µ(x(t), t) = −∇ Ψ(x(t)). Further assuming a “white noise” diffusion with temperature β results in the stochastic dynamics: (cid:112) ∇ − dx(t) = Ψ(x(t)) dt + 2β 1 dw(t) (18) In this case, the FP equation (Eqn. 16) becomes ∇ · ∇ − ρ˙ = [( Ψ)ρ] + β 1∆ρ (19) and the corresponding velocity ﬁeld v is −∇ − − ∇ v = Ψ β 1 (log ρ) (20) 7 ##LTLine## ##LTLine## ##LTLine##

text: Supplementary

text: Information

label: section

text: Free

text: energy,

text: the

text: Fokker-Planck

text: equation,

text: and

text: variational

text: optimization

text: A

text: brief,

text: self-contained,

text: introduction

text: to

text: the

text: properties

text: of

text: the

text: Fokker-Planck

text: equation

text: in

text: the

text: context

text: of

text: variational

text: optimization

text: is

text: provided.

text: Stochastic

text: processes

text: and

text: the

text: Fokker-Planck

text: equation

text: A

text: general

text: class

text: of

text: stochastic

text: processes

text: is

text: given

text: by

text: the

text: Itô

text: stochastic

text: differential

text: equation:

text: dx(t)

text: µ(x(t),

text: t)

text: dt

text: t)

text: dw(t)

text: This

text: equation

text: describes

text: the

text: evolution

text: of

text: a

text: random

text: vector

text: x

text: X

text: under

text: a

text: drift

text: µ(x(t),

text: t)

text: dt

text: and

text: a

text: diffusion

text: σ(x(t),

text: t)

text: dw(t).

text: The

text: diffusion

text: coefﬁcient

text: σ(x(t),

text: t)

text: scales

text: a

text: Wiener

text: process

text: w(t)

text: which

text: N

text: describes

text: the

text: accumulation

text: of

text: gaussian

text: noise

text: w(t

text: dt)

text: w(t)

text: dt).

text: In

text: general

text: x

text: and

text: t

text: µ(x(t),

text: t)

text: are

text: n-dimensional

text: random

text: vectors,

text: σ(x(t),

text: t)

text: an

text: (n,

text: m)-dimensional

text: matrix,

text: and

text: w(t)

text: an

text: m-dimensional

text: vector.

text: The

text: Fokker-Planck

text: equation

text: describes

text: the

text: time-evolution

text: of

text: the

text: probability

text: density

text: ρ

text: of

text: the

text: random

text: vector

text: x:

text: ρ˙

text: (µρ)

text: ∆(σ2ρ)

text: where

text: ρ

text: M

text: is

text: a

text: probability

text: density

text: at

text: each

text: timepoint

text: t,

text: ρ(t,

text: deﬁnes

text: a

text: density),

text: M

text: is

text: the

text: underlying

text: manifold,

text: and

text: σ2

text: σσT

text: is

text: known

text: as

text: the

text: diffusion

text: tensor.

text: The

text: simplest

text: case

text: of

text: the

text: FP

text: equation

text: is

text: Brownian

text: motion

text: with

text: stationary

text: (i.e.

text: time-independent)

text: and

text: invariant

text: (i.e.

text: position-independent)

text: diffusion

text: σ(x(t),

text: t)

text: β

text: and

text: zero

text: drift

text: µ(x(t),

text: t)

text: In

text: this

text: case,

text: the

text: FP

text: equation

text: is

text: the

text: heat

text: equation

text: with

text: inverse

text: temperature

text: β

text: controlling

text: the

text: “speed”

text: of

text: the

text: diffusion:

text: ρ˙

text: β

text: 1∆ρ

text: Langevin

text: dynamics:

text: drift

text: potential

text: The

text: special

text: case

text: in

text: which

text: the

text: drift

text: µ

text: is

text: a

text: conservative

text: time-homogeneous

text: vector

text: ﬁeld

text: corresponds

text: to

text: Langevin

text: dynamics.

text: Here

text: the

text: vector

text: ﬁeld

text: µ

text: is

text: generated

text: by

text: a

text: scalar

text: potential

text: µ(x(t),

text: t)

text: Ψ(x(t)).

text: Further

text: assuming

text: a

text: “white

text: noise”

text: diffusion

text: with

text: temperature

text: β

text: results

text: in

text: the

text: stochastic

text: dynamics:

text: dx(t)

text: Ψ(x(t))

text: dt

text: 2β

text: dw(t)

text: In

text: this

text: case,

text: the

text: FP

text: equation

text: (Eqn.

text: becomes

text: ρ˙

text: Ψ)ρ]

text: β

text: 1∆ρ

text: and

text: the

text: corresponding

text: velocity

text: ﬁeld

text: v

text: is

text: v

text: Ψ

text: β

text: (log

text: ρ)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: −∇ such that ρ˙ = (ρv). The expression j := ρv is referred to as the current density which reﬂects ∈ the rate of probability density ﬂow at a particular point x X. 7.3 Free energy minimization J The FP equation is intimately related to an associ(cid:2)ated free energy(cid:3) functional of the density ρ: J E − [ρ] = Ψ + β 1 log ρ ρ − = E [ρ] + β 1S [ρ] (21) E −H where we have the energy E := [Ψ] and entropy S := [ρ] terms respectively. The free energy J ρ is a Lyapunov function for the FP dynamics (Eqn. 19). Conversely, and remarkably, the FP equation is a gradient ﬂow of the free energy functional [4]. Speciﬁcally, the FP equation deﬁnes the J trajectory of steepest descent on the free energy with respect to the 2-Wasserstein metric d on the space P(X) of densities ρ, ρ(cid:48) ∈ P(X) on X: (cid:90) W2 (cid:48) (cid:48) d (ρ, ρ ) := inf d(x, x )2dλ (22) W2 λ∈M(ρ,ρ(cid:48)) X×X (cid:48) × where M(ρ, ρ ) is the set of probability densities λ on X X whose marginals are equal to ρ and (cid:48) ρ respectively. Although no derivations in this work will rely on the deﬁnition of the Wasserstein metric, it may be interesting to point out that the appearance of this metric at this juncture is related to its theoretical foundations in optimal transport theory [15]. Discrete Fokker-Planck theory Here, our primary interest lies in discrete Markov decision problems thus the translation of continuous (cid:80) FP theory to the discrete domain is reviewed [2, 8]. X is a discrete state-space (e.g. a graph) with ∈ → states x X. A density on this state-space is a discrete distribution ρ : X [0, 1] such that ∈ ρ(x) = 1. A discrete potential Ψ is deﬁned as a mapping from states to real numbers x X→ R Ψ : X . The discrete form of the Fokker-Planck equation (Eqn. 23) is based on a Markov chain approximation of Langevin stochastic dynamics and retains the key properties of the continuous case [2, 8]: (cid:88) (cid:48) (cid:48) − (cid:48) − ρ˙(x) = (cid:88) K(x, x ) (cid:2)[Ψ(x ) Ψ(x) + log(cid:3)ρ(x ) log ρ(x)] (cid:48)∈N x (x) (cid:48) (cid:48) − = K(x, x ) Ψ¯[ρ](x ) Ψ¯[ρ](x) x(cid:26)(cid:48)∈N(x) = ∆Ψ¯[ρ](x) (cid:48) (cid:48) (cid:48) ρ(x ) if Ψ¯(x ) > Ψ¯(x) K(x, x ) = (cid:48) (23) ρ(x) if Ψ¯(x ) < Ψ¯(x) where Ψ¯ is the generalized potential: − Ψ¯(x) := Ψ(x) + β 1 log ρ(x) (24) The function K can be interpreted as the kernel of a discrete Laplacian operator ∆ applied to the generalized potential Ψ¯ [16]. 8 From trajectories to transitions: expectations of linear functions Results are derived which will be used in the derivations in the main paper. An absorbing3 Markov ∈ X chain with stochastic matrix π is considered. Let f(x) be a function of state trajectories x 3Analogous results can be derived for regular Markov chains. 8 ##LTLine##

text: such

text: that

text: ρ˙

text: (ρv).

text: The

text: expression

text: j

text: ρv

text: is

text: referred

text: to

text: as

text: the

text: current

text: density

text: which

text: reﬂects

text: the

text: rate

text: of

text: probability

text: density

text: ﬂow

text: at

text: a

text: particular

text: point

text: x

text: X.

text: Free

text: energy

text: minimization

text: J

text: The

text: FP

text: equation

text: is

text: intimately

text: related

text: to

text: an

text: free

text: functional

text: of

text: the

text: density

text: ρ:

text: J

text: E

text: [ρ]

text: Ψ

text: β

text: log

text: ρ

text: ρ

text: E

text: [ρ]

text: β

text: [ρ]

text: E

text: −H

text: where

text: we

text: have

text: the

text: energy

text: E

text: [Ψ]

text: and

text: entropy

text: S

text: [ρ]

text: terms

text: respectively.

text: The

text: free

text: energy

text: J

text: ρ

text: is

text: a

text: Lyapunov

text: function

text: for

text: the

text: FP

text: dynamics

text: (Eqn.

text: Conversely,

text: and

text: remarkably,

text: the

text: FP

text: equation

text: is

text: a

text: gradient

text: ﬂow

text: of

text: the

text: free

text: energy

text: functional

text: Speciﬁcally,

text: the

text: FP

text: equation

text: deﬁnes

text: the

text: J

text: trajectory

text: of

text: steepest

text: descent

text: on

text: the

text: free

text: energy

text: with

text: respect

text: to

text: the

text: metric

text: d

text: on

text: the

text: space

text: P(X)

text: of

text: densities

text: ρ,

text: P(X)

text: on

text: X:

text: d

text: (ρ,

text: ρ

text: inf

text: d(x,

text: x

text: X×X

text: where

text: M(ρ,

text: ρ

text: is

text: the

text: set

text: of

text: probability

text: densities

text: λ

text: on

text: X

text: X

text: whose

text: marginals

text: are

text: equal

text: to

text: ρ

text: and

text: ρ

text: respectively.

text: Although

text: no

text: derivations

text: in

text: this

text: work

text: will

text: rely

text: on

text: the

text: deﬁnition

text: of

text: the

text: Wasserstein

text: metric,

text: it

text: may

text: be

text: interesting

text: to

text: point

text: out

text: that

text: the

text: appearance

text: of

text: this

text: metric

text: at

text: this

text: juncture

text: is

text: related

text: to

text: its

text: theoretical

text: foundations

text: in

text: optimal

text: transport

text: theory

text: Discrete

text: Fokker-Planck

text: theory

text: Here,

text: our

text: primary

text: interest

text: lies

text: in

text: discrete

text: Markov

text: decision

text: problems

text: thus

text: the

text: translation

text: of

text: continuous

text: FP

text: theory

text: to

text: the

text: discrete

text: domain

text: is

text: reviewed

text: X

text: is

text: a

text: discrete

text: state-space

text: (e.g.

text: a

text: graph)

text: with

text: states

text: x

text: X.

text: A

text: density

text: on

text: this

text: state-space

text: is

text: a

text: discrete

text: distribution

text: ρ

text: X

text: such

text: that

text: ρ(x)

text: A

text: discrete

text: potential

text: Ψ

text: is

text: deﬁned

text: as

text: a

text: mapping

text: from

text: states

text: to

text: real

text: numbers

text: x

text: X→

text: R

text: Ψ

text: X

text: The

text: discrete

text: form

text: of

text: the

text: Fokker-Planck

text: equation

text: (Eqn.

text: is

text: based

text: on

text: a

text: Markov

text: chain

text: approximation

text: of

text: Langevin

text: stochastic

text: dynamics

text: and

text: retains

text: the

text: key

text: properties

text: of

text: the

text: continuous

text: case

text: ρ˙(x)

text: K(x,

text: x

text: Ψ(x)

text: log

text: ρ(x)]

text: x

text: (x)

text: K(x,

text: x

text: Ψ¯[ρ](x

text: Ψ¯[ρ](x)

text: ∆Ψ¯[ρ](x)

text: ρ(x

text: if

text: Ψ¯(x

text: Ψ¯(x)

text: K(x,

text: x

text: ρ(x)

text: if

text: Ψ¯(x

text: Ψ¯(x)

text: where

text: Ψ¯

text: is

text: the

text: generalized

text: potential:

text: Ψ¯(x)

text: Ψ(x)

text: β

text: log

text: ρ(x)

text: The

text: function

text: K

text: can

text: be

text: interpreted

text: as

text: the

text: kernel

text: of

text: a

text: discrete

text: Laplacian

text: operator

text: applied

text: to

text: the

text: generalized

text: potential

text: Ψ¯

text: From

text: trajectories

text: to

text: transitions:

text: expectations

text: of

text: linear

text: functions

text: Results

text: are

text: derived

text: which

text: will

text: be

text: used

text: in

text: the

text: derivations

text: in

text: the

text: main

text: paper.

text: An

text: Markov

text: X

text: chain

text: with

text: stochastic

text: matrix

text: π

text: is

text: considered.

text: Let

text: f(x)

text: be

text: a

text: function

text: of

text: state

text: trajectories

text: x

text: results

text: can

text: be

text: derived

text: for

text: regular

text: Markov

text: chains.

text: ##LTLine##


text: function f(x) := 1x implies that Eπ[f] returns an entry of the fundamental matrix which equals the expected number of times state x is encountered: E π[1x] = Dx0x . (31) Applying discrete Fokker-Planck dynamics requires us to develop analogous arguments for pairs (cid:48) (cid:48) ∈ X (cid:48) ∈ X of trajectories. Let f(x, x ) be a linear sum of states x, x in trajectories x, x . The (cid:48) (cid:48) (cid:48) contribution f(x, x ) to f(x, x ) for a speciﬁc state pair (x, x ) is weighted by the multiplicity of (cid:48) X (cid:48) the state combination (x, x ) in the set of trajectories according to the distribution π(x, x ) = (cid:48) π(x)π(x ). Analogous to previous arguments, this is equivalent to taking the expectation with (cid:48) respect to the probability of generating a trajectory according to π with x and then x appearing on arbitrary timesteps: (cid:88) E (cid:48) (cid:48) [f] = π(x)π(x )f(x, x ) π (cid:88) (cid:88) (cid:88) (cid:88) (cid:48)∈X x,x − (cid:48)− = (cid:88) T 1 T 1 pt (x|x )pt(cid:48)(x(cid:48)|x)f(x, x(cid:48)) π 0 π (cid:48)∈X (cid:48) (cid:48) x,x T,T >0 t=0 t =0 (cid:48) = D D (cid:48)f(x, x ) (32) (cid:48)∈X x0x xx x,x (cid:48) Finally, we consider the case where f(x, x ) decomposes as a sum of transition pairs within each trajectory: (cid:88) (cid:48) (cid:48) (cid:48) f(x, x ) = f(x, y) + f(x , y ) (33) (cid:48) (cid:48)∈X x,y,x ,y Then (cid:88) (cid:8) (cid:9) E E E (cid:48) (cid:48) [f] = D D (cid:48) | [f(x, y)] + (cid:48)| (cid:48) [f(x , y )] (34) π (cid:48)∈X x0x xx π(y x) π(y x ) x,x 10

text: function

text: f(x)

font: unknown

text: implies

text: that

text: Eπ[f]

text: returns

text: an

text: entry

text: of

text: the

text: fundamental

text: matrix

text: which

text: equals

text: the

text: expected

text: number

text: of

text: times

text: state

text: x

text: is

text: encountered:

text: E

text: Applying

text: discrete

text: Fokker-Planck

text: dynamics

text: requires

text: us

text: to

text: develop

text: analogous

text: arguments

text: for

text: pairs

text: X

text: X

text: of

text: trajectories.

text: Let

text: f(x,

text: x

text: be

text: a

text: linear

text: sum

text: of

text: states

text: x,

text: x

text: in

text: trajectories

text: x,

text: x

text: The

text: contribution

text: f(x,

text: x

text: to

text: f(x,

text: x

text: for

text: a

text: speciﬁc

text: state

text: pair

text: (x,

text: x

text: is

text: weighted

text: by

text: the

text: multiplicity

text: of

text: X

text: the

text: state

text: combination

text: (x,

text: x

text: in

text: the

text: set

text: of

text: trajectories

text: according

text: to

text: the

text: distribution

text: π(x,

text: x

text: π(x)π(x

text: Analogous

text: to

text: previous

text: arguments,

text: this

text: is

text: equivalent

text: to

text: taking

text: the

text: expectation

text: with

text: respect

text: to

text: the

text: probability

text: of

text: generating

text: a

text: trajectory

text: according

text: to

text: π

text: with

text: x

text: and

text: then

text: x

text: appearing

text: on

text: arbitrary

text: timesteps:

text: E

text: [f]

text: π(x)π(x

text: )f(x,

text: x

text: π

text: x,x

text: T

text: T

text: pt

text: (x|x

text: π

text: π

text: x,x

text: T,T

text: t

text: D

text: D

text: x

text: xx

text: x,x

text: Finally,

text: we

text: consider

text: the

text: case

text: where

text: f(x,

text: x

text: decomposes

text: as

text: a

text: sum

text: of

text: transition

text: pairs

text: within

text: each

text: trajectory:

text: f(x,

text: x

text: f(x,

text: y)

text: f(x

text: y

text: x,y,x

text: ,y

text: Then

text: E

text: E

text: E

text: [f]

text: D

text: D

text: [f(x,

text: y)]

text: [f(x

text: y

text: π

text: xx

text: π(y

text: x)

text: π(y

text: x

text: x,x