TITLE: Hurwitz numbers and $t$-cores
QUESTION [7 upvotes]: For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$
be the Hurwitz number which, for the purposes
of this posting, will be defined by:
\begin{equation}
 H(k,d)
\, := \ d! \,  \sum_{\lambda \, \vdash d}
\, \nu_{\scriptscriptstyle T}^k(\lambda)
\ \ \text{where} \ \nu_{\scriptscriptstyle T}(\lambda) := \binom{d}{2} \cdot
{{\chi^\lambda_{\scriptscriptstyle T}} \over {\dim(\lambda)} } 
\end{equation}
and where $\chi^\lambda$ is the
character value of the irreducible representation $V_\lambda$ of the symmetric group $S_d$ corresponding to the partition $\lambda \vdash d$ evaluated at any representative transposition (taken from the conjugacy class $T$ of all transpositions) and where $\dim(\lambda)$ is the dimension of $V_\lambda$. The Hurwitz number $H(k,d)$ can be interpreted, using the Verlinde formula, as counting the
the number of homomorphisms (up to conjugation in $S_d$)
\begin{equation}
\rho : \pi_1 \Big( \Bbb{T}^2_k , \, \mathrm{base \, point}\Big)
\longrightarrow S_d
\end{equation}
where $\Bbb{T}^2_k$ is the 2-torus with $k$ punctures.
We can assemble these Hurwitz numbers into the following
bivariate generating function
\begin{equation}
\begin{array}{ll}
H(x;q) 
&\displaystyle = \
1 \ + \ \sum_{d \geq 1} \, \sum_{k \geq 0} \, H(k,d) \, {x^k \over {k!}} \, q^d \\ 
&\displaystyle = \ 1 \ + \ \sum_{\lambda \ne \emptyset} \, q^{|\lambda|} \, \exp \big\{ x \, \nu_{\scriptscriptstyle T}(\lambda) \big\}
\end{array}
\end{equation}
whose logarithm has a "genus" expansion
\begin{equation}
\log H(x;\tau) =  F_1(\tau) \ + \ \sum_{g \geq 2} \, F_g(\tau) \,
{x^{2g-2} \over {(2g-2)!}} 
\end{equation}
where we set $q = e^{2\pi i \tau}$ and each $\tau$-series $F_g(\tau)$ is known to be a quasi-modular form.
Now let $t \geq 2$ be an integer and let us consider the following
$t$-core analogues:
\begin{equation}
\begin{array}{l}
\displaystyle H_t(x;q) 
\, := \ 1 \ + \ \sum_{\stackrel{\scriptstyle \text{$t$-cores}}{\lambda \,\ne \, \emptyset}} \, q^{|\lambda|} \, \exp \
\big\{ x \, \nu_{\scriptscriptstyle T}(\lambda) \big\} \\
\displaystyle F_{g; \, t}(\tau) \, := \ \text{
the coefficient of} \ {x^{2g-2} \over {(2g-2)!}} \ \text{in}
\ \log H_t(x;\tau)
\end{array}
\end{equation}
Question 1: Does the generating function $H_t(x;q)$ have
a nice closed expression, e.g. some sort of product formula?
Question 2: Does the $\tau$-series $F_{g; \, t}(\tau)$ have any
kind of modular property?
thanks, ines.
Post Script:
As a kind of stupid example, consider the case of $2$-cores, which are precisely the stair-case partitions. The Murnaghan-Nakayama rule tells us that $\chi^\lambda_{\scriptscriptstyle T}$ can be evaluated recursively as the (signed) sum of dimensions $\dim(\mu)$ of partitions $\mu \vdash |\lambda| -2$ obtained by removing skew-hooks of size $2$ from the border of $\lambda$, i.e.
\begin{equation}
\chi^\lambda_{\scriptscriptstyle T} \ = \
\sum_{\stackrel{\scriptstyle \lambda \, = \, \mu + \sigma}{\sigma \, \vdash \, 2}} \, \big(-1 \big)^{\#(\sigma)-1} \, \dim(\mu)
\end{equation}
where $\#(\sigma)$ is the number of parts of $\sigma$. Of course there are no skew-hooks of size $2$ which can be excised from a stair-case partition, so $\chi^\lambda_{\scriptscriptstyle T} = 0$
for any $2$-core partition $\lambda$ and consequently
\begin{equation}
\begin{array}{ll}
H_2(x \, ;q) 
&\displaystyle = \ 1  \ + \ \sum_{\stackrel{\scriptstyle \text{$2$-cores}}{\lambda \, \ne \, \emptyset}} \, q^{|\lambda|} \\
&\displaystyle = \ 1 \ + \ \sum_{d \geq 1} \, q^{{1 \over 2}d(d+1)} \\
&\displaystyle = \ \prod_{d \geq 1} \big(1 - q^{2d}\big) \cdot \big(1 + q^d \big)
\end{array}
\end{equation}
Furthermore
\begin{equation}
\begin{array}{ll}
\log H_2(x \, ; q) 
&\displaystyle = \ 
\sum_{d \geq 1} \, 
\log(1 - q^{2d}) \, + \, \log(1 + q^d) \\
&\displaystyle = \ F_{1 ; 2}(\tau)
\end{array}
\end{equation}
The interesting computation begins with $3$-cores.
Post-Post Script: One possible approach to the problem may be to
take advantage of the Garvan-Kim-Stanton correspondence (GKS for short) which is a bijection
\begin{equation}
\lambda \stackrel{\phi}{\Longleftrightarrow} \vec{n}
\end{equation}
between $t$-cores $\lambda$ and integer vectors $\vec{n} = \big(n_0, n_1, \dots, n_{t-1} \big)$ with
zero coordinate sum $n_0 + \cdots + n_{t-1} = 0$ such that
\begin{equation}
|\lambda| \ = \ {t \over 2} \| \vec{n} \|^2 \, + \, \vec{b} \cdot \vec{n}
\end{equation}
where $\vec{b} = \big(0 ,1 , \dots, t-1 \big)$. The trick might be to
express the quantity $\nu_{\scriptscriptstyle T}(\lambda)$ in terms of the coordinates of the corresponding GKS-vector $\vec{n}$.
Consider the case of $3$-cores: If my understanding of O. Brunat and R. Nath's pointed abacus construction is correct (see https://arxiv.org/pdf/2101.01512.pdf) a $3$-core partition $\lambda$
with GKS-vector $\vec{n}= \big(n_0, n_1, n_2 \big)$ has an arm
of length $3p + r$ with residue $0 \leq r \leq 2$
if and only if $n_r$ is positive and $0 \leq p \leq n_r - 1$. Likewise $\lambda$ will have a leg of length $3p + 2 - r$ with $0 \leq r \leq 2$ if and only if $n_r$ is negative and $0 \leq p \leq | n_r | - 1 $.
As mentioned in the comments
\begin{equation}
\begin{array}{ll}
\nu_{\scriptscriptstyle T}(\lambda)
&\displaystyle = \ {1 \over 2} \, \sum_{j=1}^k \, \Big(a_j + {1 \over 2} \Big)^2 - \Big(b_j + {1 \over 2} \Big)^2 \\
&\displaystyle = \ \sum_{j=1}^k \,  a_j + {1 \over 2} a_j^2  
\ - \ \sum_{j=1}^k b_j + {1 \over 2} b_j^2 
\end{array}
\end{equation}
where $a_j$ and $b_j$ are the respective $j$-th arm and length lengths
of the partition $\lambda$.
So it should be possible to write $H_3(x \, ; q)$ as a piecewise polynomial function of the GKS-coordinates $n_0$, $n_1$, $n_2$. As illustration consider the situation where $n_0 < 0$ and $n_1 \geq -n_0$ and $n_2 = -n_0 - n_1 \leq 0$ which is one of the of six possible sign configurations of the three GKS-coordinates $n_0$, $n_1$, and $n_2$.
By the Brunat-Nath recipe only $n_1$ will contribute arm lengths
while $n_0$ and $n_2$ will contribute leg lengths. The arm contribution
to $\nu_{\scriptscriptstyle T}(\lambda)$ will be
\begin{equation}
\begin{array}{ll}
\displaystyle \sum_{j=1}^k a_j + {1 \over 2}a_j^2
&\displaystyle = \ \sum_{p=0}^{n_1 - 1} \, (3p+1) + {1 \over 2}(3p+1)^2 \\
&\displaystyle = \ {1 \over 2} 3n_1 \, + \, 3n_1(n_1-1) \, + \, {3 \over 4}n_1(n_1-1)(2n_1-1)
\end{array}
\end{equation}
while the leg contribution to $\nu_{\scriptscriptstyle T}(\lambda)$ will be
\begin{equation}
\begin{array}{l}
\displaystyle 
\sum_{j=1}^k b_j + {1 \over 2}b_j^2 \
\displaystyle 
= \ \left\{
\begin{array}{c}
\displaystyle \sum_{p=0}^{|n_0| -1} \, (3p + 2) + {1 \over 2}(3p+ 2)^2 \\
+ \\
\displaystyle \sum_{p=0}^{|n_2| -1} \, (3p) + {1 \over 2}(3p)^2
\end{array}
\right. \\
= \, \left\{
\begin{array}{c}
\displaystyle -4n_0 \, + \, {9 \over 2}n_0(n_0+1) \, - \, {3 \over 4}n_0(n_0+1)(2n_0+1)  \\
+ \\
\displaystyle {3 \over 2}n_2(n_2+1) \, - \, {3 \over 4}n_2(n_2+1)(2n_2+1)
\end{array}
\right. \\
= \, \left\{
\begin{array}{c}
\displaystyle -4n_0 \, + \, {9 \over 2}n_0(n_0+1) \, - \, {3 \over 4}n_0(n_0+1)(2n_0+1)  \\
+ \\
\displaystyle -{3 \over 2}(n_0+n_1)(1-n_0-n_1) \, + \, {3 \over 4}(n_0+n_1)(1-n_0 -n_1)(1-2n_0 - 2n_1)
\end{array}
\right. \\
\end{array}
\end{equation}
Taking the difference of the arm and leg contributions we get
the value of $\nu_{\scriptscriptstyle T}(\lambda)$ namely
\begin{equation}
\nu_{\scriptscriptstyle T}(\lambda)
\ = \
{1 \over 2}\Big(3n_1^2 - 3n_0^2\big(1 + 3n_1\big) + n_0\big(2 + 3n_1 - 9n_1^2 \big) \Big)
\end{equation}
So the lattice points of the cone in $\Bbb{Z}^2$ cut out by the inequalities $n_0 < 0$ and $n_1 \geq -n_0$ and $n_2 = -n_0 - n_1 \leq 0$
make the following contribution to $H_3(x \, ; q)$
\begin{equation}
\displaystyle
\sum_{n_0 < 0} \sum_{n_1 \geq -n_0} \, q^{3n_0^2 + 3n_1^2 + 3n_0n_1 - 2n_0 - n_1} 
\exp \Big\{ {x \over 2}\Big(3n_1^2 - 3n_0^2\big(1 + 3n_1\big) + n_0\big(2 + 3n_1 - 9n_1^2 \big) \Big) \Big\} 
\end{equation}
A similiar calculation can be undertaken for the remaining five cones
in $\Bbb{Z}^2$. Does anyone recognize this kind of sum?

REPLY [2 votes]: This posting shows how to handle the case of $3$-cores
and compute $H_3(x \, ; q)$.
Let's begin with the straight forward observation that $\lambda$ is a $t$-core if and only if its conjugate
partition $\lambda'$ is a $t$-core. Furthermore, taking
advantage of R. Stanley's comment
\begin{equation}
\begin{array}{ll}
\nu_{\scriptscriptstyle T}(\lambda) 
&\displaystyle = \ \eta( \lambda) \ - \ \eta(\lambda') \\
&\displaystyle = \ \sum_{i \geq 1} \, 
\binom{\lambda_i}{2} 
\ - \ \sum_{i \geq 1} \, 
\binom{\lambda_i'}{2}
\end{array}
\end{equation}
we see that $\nu_{\scriptscriptstyle T}(\lambda') = -\nu_{\scriptscriptstyle T}(\lambda)$. In particular
$\nu_{\scriptscriptstyle T}(\lambda)$ vanishes when
$\lambda$ is a self-conjugate partition. This means that
\begin{equation}
H_t( x \, ; q) \ = \ 1 + 
\displaystyle  \sum_{\ \lambda \, = \, \lambda'}
\, q^{|\lambda|} 
\ +  \displaystyle \sum_{\nu_{\scriptscriptstyle T}(\lambda) \, > \, 0}
2 \cosh \big\{ \nu_{\scriptscriptstyle T}(\lambda) \, x \big\} \, q^{|\lambda|}
\end{equation}
where the sums are taken over non-empty $t$-core partitions. The self-conjugate $3$-cores
are precisely those $3$-cores whose GKS-vectors
are of the form $(a,0,-a)$ with $a \in \Bbb{Z}$.
For GKS-vectors $(a,0,-a)$ with $a > 0$ the corresponding $3$-core partition $\lambda$ will have size $|\lambda| = a(3a-2)$ and
first part $\lambda_1 = 3a-2$. For GKS-vectors $(-a,0,a)$ with
$a \geq 0$ the corresponding $3$-core partition $\lambda$
will have size $|\lambda| = a(3a+2)$ and first part $\lambda_1 = 3a$. These calculation are made using the Brunat-Nath set-up mentioned
in second post-script of my original post.
The set of $3$-cores can be arranged into the following
triangular hierarchy as depicted on page 142 of
these notes (https://qcpages.qc.cuny.edu/~chanusa/courses/636/14/notes/636fa14ch50.pdf) by Christopher Hanusa. After
staring at the alcove pattern a bit, I'll
guess that (1) a $3$-core partition is
self-conjugate if and only if $\nu_{\scriptscriptstyle T}(\lambda) = 0$
and (2) any $3$-core partition with with positive
$\nu_{\scriptscriptstyle T}$-value can be uniquely expressed as
$\rho^k \cdot \lambda$ where $\lambda = \big(\lambda_1, \dots, \lambda_\ell \big)$ is a self-conjugate $3$-core
partition, $k \geq 1$ is an integer, and
$\rho \cdot \lambda := \big(\lambda_1 + 2, \lambda_1, \dots, \lambda_\ell  \big)$. Here $\rho^k \cdot \lambda$ denotes the $k$-fold iteration of $\rho$.
Note that for a general partition $\lambda$ we have
\begin{equation}
\begin{array}{rl}
\displaystyle
\big| \rho \cdot \lambda \big| 
&\displaystyle = \ |\lambda| \, + \, \lambda_1 +2  \\
\displaystyle 
\nu_{\scriptscriptstyle T}\big( \rho \cdot \lambda \big)
&\displaystyle = \ \nu_{\scriptscriptstyle T}(\lambda) 
\, - \, |\lambda| \, + \, 
\binom{\lambda_1 +2}{2} 
\end{array}
\end{equation}
and using the Faulhaber formulae it's not too hard to see that
\begin{equation}
\begin{array}{rl}
\big| \rho^k \cdot \lambda \big|
&\displaystyle = \
|\lambda| \ + \ k\lambda_1 \ + \ k(k+1) \\
\displaystyle \nu_{\scriptscriptstyle T}\big(\rho^k \cdot \lambda \big) 
&\displaystyle = \ 
\left\{
\begin{array}{l}
\displaystyle \ \ \ \, 
\nu_{\scriptscriptstyle T}(\lambda) \, - \, k|\lambda|
\\
\displaystyle
+ \ {1 \over 2} \, k\lambda_1^2 \, + \, {1 \over 2} \, k(k+2)\lambda_1 \\  
\displaystyle + \ {1 \over 6} \, k(k+1)(2k+1)
\end{array}
\right.
\end{array}
\end{equation}
If we iteratively apply $\rho$ to a self-conjugate $3$-core $\lambda$ with GKS-vector $(a,0,-a)$
with $a > 0$ and tally the total contribution to $H_3(x \, ; q)$ made by $\rho^k \cdot \lambda$ and its conjugate partition as $k \geq 1$ varies we get:
\begin{equation}
\begin{array}{ll}
G^+_a(x \, ; q) 
&\displaystyle := \
\sum_{k \geq 1} 2 \cosh \big\{ U_+(a,k) \, x \big\}
\, q^{D_+(a,k)} \ \ \text{where} \\
U_+(a,k) 
&\displaystyle := \ 
\left\{
\begin{array}{l}
\displaystyle \ \ \ \, {1 \over 2} \, (a-2)(3a-2)k \\
\displaystyle + \ {1 \over 2} \, (3a-2)k(k+2) \\
\displaystyle + \ {1 \over 6} \, k(k+1)(2k+1)  
\end{array}
\right. \\
D_+(a,k)
&\displaystyle = \ a(3a-2) \ + \ (3a-1)k \ + \ k^2
\end{array}
\end{equation}
Similarly the total contribution to $H_3(x \, ; q)$ made by $\rho^k \cdot \lambda$ and its conjugate partition as $k \geq 1$ varies and where $\lambda$ is
a self-conjugate $3$-core with GKS-vector
$(-a,0,a)$ and $a \geq 0$ is:
\begin{equation}
\begin{array}{ll}
G^{-}_a(x \, ; q) 
&\displaystyle := \
\sum_{k \geq 1} 2 \cosh \big\{ U_{-}(a,k) \, x \big\}
\, q^{D_{-}(a,k)} \ \ \text{where} \\
U_{-}(a,k) 
&\displaystyle := \ 
\left\{
\begin{array}{l}
\displaystyle \ \ \ \, {1 \over 2} \, a(3a-4)k \\
\displaystyle + \ {3 \over 2} \, ak(k+2) \\
\displaystyle + \ {1 \over 6} \, k(k+1)(2k+1)  
\end{array}
\right. \\
D_{-}(a,k)
&\displaystyle = \ a(3a+2) \ + \ (3a+1)k \ + \ k^2
\end{array}
\end{equation}
The self-conjugate $3$-cores on the own
make a contribution of
\begin{equation}
G(q) \ = \ 1 + \ \sum_{a \, > \, 0} \ q^{a(3a+2)} \ + \ q^{a(3a-2)}
\end{equation}
and thus, when taken altogether, we get
\begin{equation}
H_3(x \, ; q) \ = \
G(q) \ + \ \sum_{a > 0} G^+_a(x \, ; q ) 
\ + \sum_{a \geq 0} G^{-}_a(x \, ; q) 
\end{equation}
Perhaps the right-hand sum will be familiar to
the readership of this posting.