46 problems

Let $\mathfrak{S}_n$ be the symmetric group, let $\mathbb{F}$ be an algebraically closed field of characteristic $p$, and let $\lambda\vdash n$. For $m\in\mathbb{N}$ with $m\leq n$…

James–Mathas conjecture. The Specht module $S^{\lambda}$ remains irreducible over $F$ if and only if $\lambda$ is a $p$-JM partition.

Trivial-source conjecture. Indecomposable, self-dual $k\Sigma_d$-modules with Specht filtrations are trivial source modules. This is a proposed weaker replacement for the earlier c…

Signed Young module conjecture. Indecomposable self-dual $k\Sigma_d$-modules with Specht filtrations should be exactly the signed Young modules. The conjecture is motivated by the…

Let $R_n(\Gamma,p)$ be the quiver Hecke algebra corresponding to a quiver $\Gamma$ over a field of characteristic $p$, and let $\mu,\lambda\vdash n$ be arbitrary partitions. Type A…

Let $e\geq2$, let $i\in\mathbb{Z}_e$, and let $R_\theta({\tt A}_{e-1}^{(1)})$ be the KLR algebra for a root-lattice element $\theta$. Let $\mathcal{MP}_{\mathrm K}^{(i,i)}(\theta)$…

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathcal{NP}_{\mathrm{2res}}(\theta)$ be the set of skew diagrams…

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathsf{D}\in\mathcal{UL}(\theta)$ be an upper ledge diagram. Let…

Let $S^{\lambda}$ be a Specht module, and let $A$ be an upper unitriangular change-of-basis matrix from the polytabloid basis such that $A^{-1}(1+s)A\geq0$ for every simple reflect…

Let $L(r)$ denote the function governing the period in the corresponding result for $2$-part partitions. For integers $n,m$ and fixed $r$, let $\lambda=(n-r,1^r)\vdash n$ and…

For integers $n,k\geq 1$, let $\rho_{n,k}$ be the representation considered in the paper, an $\mathfrak S_{k(n-1)+1}$-module. Column-length conjecture. If $n\leq j\leq k$, then the…

Let $n,k\geq 1$, and let $\gamma_{n,k}$ be an $\mathfrak S_{k(n-1)+1}$-module such that … where every irreducible constituent of $\gamma_{n,k}$ has at most $k-1$ columns. Column-le…

Let $r \geq 3$ and let $\mathscr{S} \subset \Pi(n,d,r)$ denote the collection of set partitions that can be made noncrossing by applying at most $r-3$ adjacent transpositions. For…

Non-unisingularity conjecture. The module $\mathcal{S}^{(n-2,2)'}$ does not afford $1$ as an eigenvalue on the class of such permutations. Every other element of $S_n$ has $1$ as a…

Completeness conjecture. If $e\neq 2$, the cited theorems and the characteristic-$2$ corollary provide a complete list of decomposable Specht modules indexed by bihooks.

Completeness conjecture. In the cases $e\neq 2$ and $p\neq 2$, these are all of the decomposable Specht modules indexed by bihooks.

Let $\mathcal{T}(n,n)$ be the set of standard Young tableaux of shape $(n,n)$, let $\{v_T:T\in\mathcal{T}(n,n)\}$ and $\{w_S:S\in\mathcal{T}(n,n)\}$ be the tableau and web bases of…

For odd $k\in2\mathbb{N}+1$, let $\beta_k$ be the partition defined earlier in the source, and let $\beta_k^R$ and $\beta_k^C$ be the two labels specified there. Let $w(\beta_k)$ d…

For $k\in\mathbb{N}$, let $\alpha_k$ be the partition defined earlier in the source, let $\alpha_k^R$ be its row-labelled constituent, let $\alpha_k^C=(2k+3,2k+1,2k-1,\dots,9,7,5)$…

Let $\mathfrak{S}_n$ be the symmetric group, let $\mathbb{F}$ be an algebraically closed field of characteristic $p$, and let $S^\lambda$ denote the Specht module labelled by a par…

For integers $n$ and $k$, let $\rho_{n,k}$ be the representations in the LAnKe array, and let $W_{k+1}$ denote the Whitehouse module. For each irreducible submodule of $W_{k+1}$, f…

Conjecture on 2-adic forms of hook Specht modules. If $n\equiv 0\pmod{4}$ and $k\in\{2,n-3\}$, then

Fix a multicharge $\kappa\in\mathbb{Z}^l$ and a multipartition $\lambda$ of $n$. Let $\mathcal{S}^{\lambda}_{\kappa}$ be the graded Specht module with standard-tableau vectors…

Let $\lambda\in \mathscr{P}^l_n$, and let $A\in [\lambda]$ be a Garnir node. Suppose $\mathbf{B}^A$ contains $a$ bricks in the first row and $b$ bricks in the second row. For each…

Let $p$ be an odd prime, let $k$ be a positive integer, and let $r$ be an integer with $0\leq r\leq kp-1$. Write $S^{(kp-r,1^r)}$ for the hook Specht module for the symmetric group…