15 problems

Let $d_{\la\mu}$ denote the multiplicity of the simple module $D^\mu$ as a composition factor of the Specht module $S^\la$. Suppose the alcove and reflection-group hypotheses above…

Let ${\cal S}_m$ be the $v$-Schur algebra, let $W(\lambda)$ be the Weyl module for a partition $\lambda$ of $m$, and let $L(\mu)$ be the irreducible module corresponding to a parti…

Let $H_m(v)$ be the Hecke algebra of type $A_{m-1}$, let $S(\lambda)$ be its Specht module, and suppose that $\mu$ is an $n$-regular partition. Let $D(\mu)$ be the simple head of…

Geck–Geck–Hiss conjecture. Assume that $cell$ is not “too small” and that the center of $G$ is connected. For every $varphi\throw in \operatorname{UBr}_{cell}(G^F)$, there is a uni…

Multiplicity formula. The seven asserted formulas in the source hold: for $n\geq9$, $[S_{\mathbb F_2}^{(n-4,2,1^2)}:D_{\mathbb F_2}^{(n-4,4)}]=1$; for $n\geq7$,…

Let $\Lambda_n$ index the cell modules $\Delta(\lambda)$ and simple modules $L(\mu)$ of the orientifold quiver Temperley--Lieb algebra, and let $\langle k\rangle$ denote grading sh…

Let $\Phi_A$ and $\Psi_{\lambda_{\mathbf c}}^+$ be the sets defined by … and … Suppose that $\Phi_A\cap\Psi_{\lambda_{\mathbf c}}^+\neq\emptyset$. The conjecture. Condition is sati…

Weight-three decomposition-number conjecture. The decomposition numbers for every weight-$3$ block of every Ariki–Koike algebra are all either $0$ or $1$.

Let $\bm\lambda,\bm\mu\in\mathcal{P}^r$ lie in a block $B$ of $\mathcal{H}_{r,n}$, with $\bm\mu$ $e$-multiregular. Obtain $\bm\lambda^{+\varnothing}$ and $\bm\mu^{+\varnothing}$ fr…

The Rouquier-block multiplicity conjecture. The multiplicity of $L(\boldsymbol{mu})$ as a composition factor of $Delta(\boldsymbol{lambda})$ is

Let $h=2n+1$, let $\gamma$ be a $w$-Rouquier $h$-bar-core, and let $\alpha$ be a strict partition and $\beta$ a restricted $h$-strict partition, both with $h$-bar-core $\gamma$ and…

Conjecture A. One has $|\operatorname{IBr}(\chi^0)|\le p^2-1$ and $d_{\chi\varphi}\le p$ for every $\varphi\in\operatorname{IBr}(G)$.

Decomposition-number conjecture. The decomposition numbers of cyclotomic walled Brauer algebras over $\mathbb C$ can be determined by those in Theorem 7.19 together with some resul…

Fix a family of groups of Lie type and fix its rank; for example, fix $n$ and consider the unitary groups ${\operatorname{SU}}_n(q)$ as $q$ varies. Let $\ell$ be a prime, and let a…

Let $W$ be a finite Weyl group, let $\mathcal H_k(W,\xi)$ be its Hecke algebra over a field $k$ with parameter $\xi$, and let $e<\infty$ satisfy the standing assumptions…