ag.algebraic geometry - complement of "good reduction" points in p-adic shimura varieties

co.combinatorics – extensions of the Nekrasov-Okounkov system Answer

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co.combinatorics – extensions of the Nekrasov-Okounkov system

In what follows $lambda$ will breathe an integer partition (of dimension $|lambda| = n$),
the notation $b in lambda$ will explain that $b$ is a cell or field within the Young diagram of $lambda$, and $mathrm{h}(b)$ will denote the
swipe size of the swipe clear by $b$ in $lambda$‘s Young diagram.
Furthermore $dim(lambda)$ will denote the variety of saturated chains
$lambda^{(0)} subset cdots subset lambda^{(n)}$ within the Young lattice $Bbb{Y}$ birth on the vacant partition $lambda^{(0)} = emptyset$ and
ending at $lambda^{(n)} = lambda$. Equivalently $dim(lambda)$ is the dimension of the corresponding irreducible illustration $V_lambda$ of the symmetric group $S_n$.

The Nekrasov-Okounkov system is the next $t$-deformation of the classical id for the partition duty, particularly:

commence{equation}
sum_{n geq 0} , {z^n over {n !}} , sum_lambda ,
{dim^2(lambda) over {n!}} , prod_{b in lambda} , huge(
mathrm{h}^2(b) – t huge) =
prod_{okay geq 1} , huge(1 – z^okay huge)^{t-1}
aim{equation}

with the specialization $t=0$ equivalent to the well-known id for the producing duty of $p(n)$, the variety of integer partitions of $n geq 0$:

commence{equation}
sum_{n geq 0} , p(n) , z^n = prod_{okay geq 1} , huge( 1 – z^okay huge)^{-1}
aim{equation}

My first query issues the interior sum

commence{equation}
sum_lambda ,
{dim^2(lambda) over {n!}} , prod_{b in lambda} , huge(
mathrm{h}^2(b) – t huge)
aim{equation}

which we will perceive as an expectation worth for the $t$-statistic $H_t(lambda) = prod_{b in lambda} , huge(mathrm{h}^2(b) – t huge) $ with respect to the
Plancherel touchstone $mu^{(n)}_mathrm{P}(lambda) = {1 over {n!}} dim^2(lambda)$ on the clique of partitions of dimension $|lambda| =n$. Let’s denote this expectation worth as $langle H_t rangle_{mathrm{P},n}$. Clearly it’s a polynomial in $t$.

Question 1: Is there a pleasant (e.g. combinatorial) system
describing how $langle H_t rangle_{mathrm{P},n}$ factorizes
as a polynomial in $t$?

Instead of utilizing the (household of) Plancherel touchstone(s) $mu^{(n)}_mathrm{P}$
we might employ a coherent, ergodic household of measures $mu^{(n)}_varphi$
related to a different altenative of normalised, minimal, harmonic duty $varphi: Bbb{Y} longrightarrow Bbb{R}_{>0}$. Recall that the
touchstone $mu_varphi^{(n)}$
on the clique of partitions $lambda$ of dimension $|lambda| = n$ is outlined
by

commence{equation}
mu_varphi^{(n)} (lambda) := dim(lambda) , varphi(lambda)
aim{equation}

Among the provision of minimal, normalised, harmonic capabilities
are the Schur-states $varphi_{bf x}$ outlined by $varphi_{bf x}(lambda) = {scriptstyle frak{S}}_lambda(x_1, x_2, x_3, dots)$ the place
${scriptstyle frak{S}}_lambda$ is the Schur duty related to
$lambda in Bbb{Y}$ and
the place ${bf x} = (x_1, x_2, x_3, dots)$ is any sequence of constructive actual numbers with $0 leq x_k leq 1$
and $x_{okay+1} leq x_k$ for all $okay geq 1$ and with convergent sum
$sum_{okay geq 1} x_k = 1$. Accordingly outline the corresponding Schur touchstone(s) by
$mu^{(n)}_{bf x}(lambda) = dim(lambda) , {scriptstyle frak{S}}_lambda(x_1, x_2, x_3, dots)$.

Question 2: Is a model of the Nekrasov-Okounkov system
identified the place the position of the Plancherel touchstone $mu^{(n)}_mathrm{P}$ is changed by a Schur touchstone $mu^{(n)}_{bf x}$? Specifically
an id for

commence{equation}
sum_{n geq 0} , {z^n over {n !}} , sum_lambda ,
mu^{(n)}_mathrm{x} , prod_{b in lambda} , huge(
mathrm{h}^2(b) – t huge) = ?
aim{equation}

Thanks, ines.

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