18 problems

Let $h$ be the quantum characteristic, let $\boldsymbol{S}^{\alpha}$ and $\mathbf{D}^{\beta}$ be the spin modules indexed by an $h$-strict partition $\alpha$ and a restricted $h$-s…

Let $\lambda=(\lambda_1,\ldots,\lambda_r)$ be a partition. Bessenrodt–Morris–Olsson's conjecture. For $p=5$, the labels for irreducible modular spin representations are the partiti…

Let $m$ be a positive integer, let $p$ be an odd prime, and let ${\rm DP}(m)$ denote the relevant set of strict partitions of $m$. Let ${\rm DPR}_p(m)$ denote the indexing set for…

Let $n$ be a positive integer, let $p=2n+1$ be a sufficiently large prime, and consider the level-one $q$-Fock space for the twisted affine algebra $U_q(A^{(2)}_{2n})$, together wi…

Spin Smith normal form conjecture. The Smith normal form of $Z_{s,p}$ should be

Let $W$ be any finite Coxeter group. Let $\operatorname{Irr}(W)$ be its irreducible characters, and call a family $F$ of $\operatorname{Irr}(W)$ good if it contains a character occ…

Let $\mathfrak{g}$ be a simple Lie algebra with Weyl group $W$, reflection representation $V$, and Spaltenstein map $d$ on nilpotent orbits. Let $\Psi$ be the map from genuine irre…

Let $V(\Lambda)$ be a highest weight representation of an affine Lie algebra, with $\Lambda=\Lambda_0$, and let the spin representations be the projective representations of symmet…

Let $\iota:{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes(m-1)})\rightarrow{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes m})$ be given by…

Let $A_m^n$ be a unital ${\mathbb C}(q)$-algebra, and let $\{\mathsf X_r^{(i)}\}_{1\leq r\leq m-1,\,0\leq i\leq n}\subset A_m^n$ satisfy $\mathsf X_r^{(0)}=1$ together with the rel…

Let $I^{>n}\subset U'_{-q^2}(\mathfrak{so}_{m})$ be the two-sided ideal generated by $\mathsf{x}_i^{(n+1)}$ for $i=1,\ldots,m-1$, and set … Let the Wenzl homomorphism be the surjec…

Let $n\geq 1$ and $m\geq 2$. Write $K_0^\tau(-)$ for the weighted Grothendieck group of a category with involution, let $({\EuScript{B}}_{m}^{n})^\tau$ be the equivariant category,…

Homogeneity conjecture for special strict partitions. Suppose $p=3$ and $\lambda$ is a special strict partition, written as $\lambda=\nu+3\alpha$ with $\nu$ a $3$-bar core. Then…

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 15.17. For every $f\in E(G'(m,p,\infty);\chi^0)$, the products $f/F^{\rm VII}$ produce all elements of $E(G'(m,p,\infty);\chi)$, where $\chi=\chi^0\chi^{\rm VII}$.

Let $\widetilde{A}_n$ be the double cover of the alternating group $A_n$, and let $\operatorname{Irr}(\widetilde{A}_n)$ denote its irreducible characters. A character is called spi…

Let $P(n)$ denote the set of partitions of $n$, and let $D^+(n)$ denote the relevant set of strict partitions indexing the spin characters under consideration. For…

Read-inspired isomorphism conjecture. There is a superalgebra isomorphism