41 problems

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 $m,n\in\mathbb{N}$, with $m$ even, and let $\lambda\vdash mn$ be an even partition with $l(\lambda)\le n$. The plethysm coefficient $a^\lambda_{(n),(m)}$ is defined by the mult…

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 $a\leq b$, and let $U$ be a finite-dimensional vector space. Define the natural $\mathsf{GL}(U)$-equivariant map … by embedding symmetric tensors into tensor powers, reordering…

Foulkes–Howe conjecture. The map

Let $V$ be a finite-dimensional complex vector space, and let $S^mV$ denote its $m$-th symmetric power. For integers $n \geq m$, consider the $GL(V)$-modules $S^n(S^mV)$ and…

GOSSZ's conjecture. There exist positive expressions for these coefficients.

Let $a\leq b<c\leq d$ be integers with $n=ad=bc$. For complete homogeneous symmetric functions, write $h_n\circ h_k$ for their plethysm, and let $\{s_\mu\}_{\mu\vdash n}$ be the Sc…

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…

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 $m,n$ be integers with $m\leqslant n$. The source refers to a composition of maps introduced in the surrounding discussion, with dualities distinguished according to its conven…

Let $a,d$ be positive integers and let $k\in\mathbb{Q}$ satisfy $k\geqslant1$ and $ka,kd\in\mathbb{Z}$. The source considers the $k$-fold plethystic map and the corresponding itera…

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…

Monotonicity conjecture. The sequences of plethysm coefficients in Theorems and are non-decreasing with respect to $M$.

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 $s_\lambda[s_\mu]=\sum_\nu a_{\lambda,\mu}^\nu s_\nu$ be the plethysm expansion, and let $\nu$ be the lexicographically largest partition $\kappa$ such that…

Let $a,b,c,d$ be positive integers with $ab<c d$ and $n=ad=bc$. For complete homogeneous symmetric functions, write $hncirc hk$ for their plethysm. Vessenes' conjecture. The symmet…

Let $1\le a\le b$ be integers, and let $\rho_a^{(b)}$ and $\rho_b^{(a)}$ be the characters in the source, viewed as virtual characters of $S_{ab}$. Foulkes' conjecture. … In the so…

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…

The paper introduces a relation on integer partitions through Schur positivity of differences of plethysms of homogeneous symmetric functions. Its restriction to one-part partition…

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 $m,n,p,q\in\mathbb{N}$ with $m\leqslant n$, $mn=pq$, and $p,q\geqslant m$. For a partition $\lambda$, let $\lambda_{[r]}$ denote the partition of $r$ obtained by appending a fi…