25 problems

Let $\min\mathbb{N}$ and $lin\{0,1,\dotsc,m\}$. For partitions $u$ and $\tau$, let $\mathsf{A}{l,m}( u,\tau)$ be the sequence of plethysm coefficients defined from the associated skew…

Let $u$ and $\tau$ be arbitrary partitions, and let $minmathbb{N}$. The notation $\mathsf{A}{m-1,m}( u,\tau)$ denotes the associated sequence of plethysm coefficients. **Wildon's sta…

Let $Pi{(b^a)}$ and $Pi{(a^b)}$ denote the loci of unordered set partitions whose block-size types are $(b^a)$ and $(a^b)$, respectively, and let ${rm gr}{bf I}( Pi)$ be the asso…

Mulmuley's conjecture. The generating function $\tilde{\mathsf{A}}_{\mu[\nu]}^\lambda(t)$ is the Ehrhart series of a polytope.

Denominator conjecture. The expression

The bivariate generating function is … A weighted lattice-point cone conjecture. The generating function $A_\mu(z,q)$ is a weighted generating function of lattice points of a disjo…

Let $a,b,c,d\in\mathbb{N}$ satisfy $ab=cd$ and $a=\min(a,b,c,d)$. Write $U\leqslant V$ when $U$ is isomorphic to a subrepresentation of $V$. Generalised Foulkes conjecture. … This…

Let $N$ be a positive integer, and let $m,n$ be integers with $m\leqslant n$. For representations of $\operatorname{SL}_N(\mathbb{C})$, write $U\leqslant V$ when $U$ is isomorphic…

Let $M(n,m)$ be the set of $n\times m$ nonnegative integer matrices with every row and column sum equal to $m$ and $n$, respectively, as in the paper, and let…

Seed-existence conjecture. For all $n\ge 0$, there exists a seed $S(n)$ which is a solution to Problems 1, 2, and 3, and therefore the crystal definition depending on $S(n)$ define…

Let $M,N\in\mathbb{N}$, let $d\in\mathbb{N}_0$, let $\mathbb{F}$ be the underlying field, and let $E$ be the natural $2$-dimensional $\mathbb{F}$-representation of…

Let $a_\lambda(d[n])$ denote the plethysm coefficient indexed by a partition $\lambda\vdash nd$. Foulkes' conjecture. If $d>n$, then … for all $\lambda\vdash nd$. This is a fundame…

Let $\rho^{(a)}_b$ and $\chi^{(r)}$ denote the characters used in the source, let $\boxtimes$ denote the indicated outer product, and let $\operatorname{Char}(S_N)$ be the cone of…

Let $Lie^{(2)}$ be the symmetric-function species used in the plethysm below, and let $V\!h_k(n)$ be the $S_n$-module whose Frobenius characteristic is the degree-$n$ term of…

Let $\nu$, $\lambda$, and $\lambda$ be partitions, with $\nu$ and $\lambda$ arbitrary. Write $\nu\sqcup(1)$ for the partition obtained by adjoining a part $1$ to $\nu$, and similar…

Let $a$ and $b$ be positive integers with $a<b$, and let $M$ be a multiset of size $ab$. A multiset partition divides $M$ into an unordered collection of submultisets. Weak Foulkes…

Let $a$ and $b$ be positive integers with $a<b$. For a polynomial representation $W$ of $GL_n$ of degree $ab$, consider the symmetrized weight spaces corresponding to the weights ……

Let $\alpha,\beta,\gamma,\delta$ be partitions such that … Write $\prec_s$ for strict Schur order and let $S_\rho$ denote the corresponding $q$-analogue used in the paper. The diag…

Let $a$ and $b$ be positive integers, set $n=ab$, let $c$ be a divisor of $n$ with $a\leq c\leq b$, and set $d=n/c$. Write $\preceq_s$ for Schur positivity. The q-Doran conjecture.…

For $0<a\leq b$, let $F_{a,b}(\mathbf{x};q)$ be the divided difference from the $q$-Foulkes conjecture, and let the overline operator remove the largest part from each Schur index.…

For each positive integer $n$, let $H_n(\mathbf{x};q)$ be the Hall–Littlewood polynomial used in the paper, and for $0<a\leq b$ define … Let $f_{a,b}=h_b[h_a]-h_a[h_b]$. The q-Foul…

Let $m,n$ be positive integers and let $\nu$ be a partition of $n$. For a partition $\tau$ of $mn$, write $p^{\tau}_{\nu,(m)}$ for the corresponding plethysm coefficient. Two-step…

Let $m,n$ be positive integers, let $\nu$ be a partition of $n$, and let $\nu'$ denote its conjugate partition. For a partition $\tau$ of $mn$, write $p^{\tau}_{\nu,(m)}$ for the c…

Let $m,n$ be positive integers, let $u$ be a partition of $n$, and let $u'$ denote its conjugate partition. For a partition $\tau$ of $mn$, write $p^{\tau}{nu,(m)}$ for the corresp…

Sign-reversing cancellation conjecture. There are a subset $S_0\subset\operatorname{SSTab}(\lambda[\mu])$ and a bijective map