16 problems

Judge–Zanello conjecture. There is a choice of $\epsilon^t_{a,d,j}\in\{0,1\}$ such that $\epsilon^t_{a,1,0}=1$, $\epsilon^t_{a,d,j}=0$ if $at/d-24j<0$, and

Judge–Keith–Zanello conjecture. For every odd positive integer $t$, $\delta_t=1/2$.

Let $\mathcal H_n$ be a cyclotomic Hecke algebra of type $G(m,1,n)$, let $S^{\underline\lambda}$ be its Specht module indexed by a multipartition $\underline\lambda$, and define ……

Let $\underline{\lambda}$ be a $d$-partition, let $\underline{\emptyset}$ denote the empty $d$-partition, and let $f_i^{(a)}$ be the divided-power operator associated with a residu…

Let $\boldsymbol{\kappa}=(\kappa_1,\kappa_2)$ be a bicharge, let $\boldsymbol{\lambda}\in\mathcal{MP}^{\boldsymbol{\kappa}}$, and let $\boldsymbol{\lambda}_{\mathrm K}$ denote its…

Let $\Pi$ be the set of pairs of partitions, let $i\in\mathbb{Z}_2$, and let $\mathcal{X}_i:\Pi\rightarrow\mathcal{UL}$ be the associated map to upper ledge diagrams. If…

Let $\mathfrak{g}=A^{(2)}_{2n}$ and let $\Lambda=a_0\Lambda_0+\dots+a_n\Lambda_n$ be a positive integral weight. A spin multipartition for $\Lambda$ is a set of $a_0+a_1+\dots+a_n$…

Let $e$ be the quantum characteristic, and let a face have initial residue set $I_0=\{1,2,\dots,t\}$. An $e$-regular multipartition is a multipartition satisfying the relevant $e$-…

Let $\mathbf{s}$ be a multicharge of the form … Let $\lambda$ be a cylindric $l$-partition, and let $m(\lambda)$ denote the unique $l$-partition with minimal $b$-invariant occurrin…

For $t\geq1$, let $p_t(n)$ be the $t$-multipartition function, and let $\delta_t$ denote the density of those $n$ for which $p_t(n)$ is odd, when this limit exists. Density-transfe…

Let $s$ be a positive integer and let $a,b$ satisfy $0\leq a,b<s$. For parameters $r$ and $c$, write $\mathcal{C}_{(r\,|\,0,c)}$ for the corresponding set of bipartitions, and let…

Let ${\boldsymbol\lambda},{\boldsymbol\mu}\in\mathscr{P}_{n}$. For the polynomials $d_{{\boldsymbol\lambda}{\boldsymbol\mu}}(q)$ and the defect…

Multipartition equidistribution conjecture. The density $\delta_t$ exists and equals $1/2$ for every odd positive integer $t$. Equivalently, if $t=2^k t_0$ with $t_0\geq 1$ odd, th…

Let $M^{\mathbf{s}}$ be the irreducible highest-weight module with canonical basis indexed by regular multipartitions, and let $d^{\mathbf{s}}_{\nu\mu}$ denote the coefficient of t…

Let $\mathcal{P}_r(n)$ be the set of $r$-partitions of $n$, let $\mathcal{K}_r(n)$ be the set of Kleshchev $r$-partitions with respect to $(e;\mathbf{Q})$, and let an $r$-partition…

Let $\mathcal{H}_n$ be an Ariki–Koike algebra of $G(r,1,n)$, and let $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ be multipartitions of $n$ with $r$ components. Let…