55 problems

Let $\mathfrak{so}(N)$ be the orthogonal Lie algebra, and let $\operatorname{ad}$ denote its adjoint representation. For a Young diagram $\lambda$, write $\lambda\vdash 2m$ when it…

Let $\lambda$ be a symmetric Young diagram that is not a hook, let $G_\lambda$ be the associated group, and let $\dim\lambda$ denote the dimension of the corresponding representati…

Let $\lambda=\mathbf{r}^{\mathbf{p}}$ be a Young diagram in multirectangular coordinates. For a partition $\mu$, let $J^{\star,(\alpha)}_\mu$ be the shifted Jack function and let…

Let $C>0$. A Young diagram $\lambda$ with $n$ boxes has at most $C\sqrt{n}$ rows and columns, and let $\pi\in S_n$ be a permutation. Moore and Russell's character bound. There exis…

Let $Y$ be the Young diagram under consideration, let $S_Y$ denote its transversals, and let $S_Y(213)$ denote the transversals avoiding the pattern $(213)$. For a transversal…

Let $\pi$ be a link pattern described by two Young diagrams $(Y,Y')$, with $r$ and $r'$ arches respectively and $n-r-r'$ parallel arches between them. Let $|Y|$ and $|Y'|$ denote t…

Let $\pi$ be a link pattern described by a Young diagram $Y$, let $|Y|$ be the number of boxes of $Y$, and let $\operatorname{dim}Y$ denote the dimension of the corresponding irred…

Let $\theta$ be a parameter, let $\pi_1$ and $\pi_2$ be rectangular Young diagrams, and let $\pi=\pi_1\cap\pi_2$. Write $I_{\Omega(\pi_1,\pi_2)}(\theta)$ and…

Let $\Lambda=(n_1,n_2,\ldots,n_N)\in\mathbb Z^N$ be an index set with … and let … be the Wronskian–Hermite polynomial for $m=2$. Veselov's conjecture. For any index set $\Lambda$,…

Horizontal-sum existence conjecture. For all other, currently unknown, multiplets, as well as the universal Casimir multiplets, there always exists a pair of representations of…

Let $Y$ be a Young diagram and let $H(Y)$ be the associated 3-partite 3-uniform hypergraph. Write $|Y|$ for the number of cells of $Y$, let $\tau^{(2)}(H(Y))$ denote its second cov…

Let $Y$ be a Young diagram, and let an allocation mean a coarse filling obtained by subdividing $Y$ into subrectangles and partitioning the symbol set into smaller subsets, with pr…

Let $Y$ be a Young diagram with row lengths $l_1,\ldots,l_m$. It is Latin if its cells can be assigned integers so that row $i$ contains $1,\ldots,l_i$, and the entries in each col…

Let $Y$ be a Young diagram, and suppose its cells are filled with elements of the ground set of a matroid $M$ so that each row is independent. A Young diagram is wide if every subd…

Let $H$ be a JUE random matrix with eigenvalues, and let $\widehat H$ be obtained by removing the last row and last column of $H$ and taking its eigenvalues. Construct a random pie…

The generating function for JUE correlators is a tau function of an integrable hierarchy and has a free-fermionic formulation. The skew Howe measure $\mu^{\mathrm{H}}$ has a free-f…

In the asymptotic regime $n,k,p\to\infty$ with $p\sim nk$, let $\lambda$ be distributed according to the skew Howe measure $\mu^{\mathrm{H}}$, and define … In the asymptotic regime…

Let $\lambda$ be a random Young diagram with respect to the skew Howe measure $\mu^{\mathrm{H}}$, and let $\lambda_i$ denote its row lengths. Let $t_+$ be the right edge parameter…

The JUE correlators admit a free-fermionic construction, and the skew Howe measure $\mu^{\mathrm{H}}$ has a free-fermionic construction for its correlation kernel. Free-fermionic c…

Vertical-sum folding conjecture. The vertical sum operation is a kind of the dual version of the folding map of Dynkin diagrams.

Let $n$ and $k$ be integers with $1\leq k\leq n$. Let $\lambda\vdash n$ and $\mu\vdash k$ be Young diagrams with $\mu\subset\lambda$. Write … where $d_\nu$ denotes the number of st…

Let ${\bf O}$ be the orbit representative introduced in the paper's orbit construction. Uniqueness conjecture. We conjecture that ${\bf O}$ is the unique infinite orbit up to some…

McNamara–van Willigenburg's conjecture. The diagrams $\mathcal{D}$ and $\mathcal{E}$ satisfy $\mathcal{D}\sim\mathcal{E}$ if and only if the factors satisfy:…

For a hypergraph $H$, a 2-matching is a set of edges such that every two edges share fewer than two vertices, and let $\nu^{(2)}(H)$ denote its maximum size. For a Young diagram…

A Young diagram $Y$ is wide if, for every subset $S$ of its rows, the diagram formed by $S$ dominates its conjugate; a filling of $Y$ is Latin if it assigns to each row the numbers…