18 problems

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 $P$ be the saturation of a weight $\lambda$, namely $P=\{\mu:\mu\nleq\lambda\}$, let $\dot{\mathbb U}_P$ be the corresponding BLN-algebra, and let…

Classical truncated PBW basis conjecture. $S(n,d)$ has a ${\mathbb Q}$-basis consisting of all ordered monomials in $G$ of total degree at most $d$.

Assume that $B\wr \cH_d$ is a polynomial quantum wreath product satisfying conditions … , and write $\bfH_d=B\wr \cH_d$. For each composition $\lambda\vDash d$, let $K_\lambda$ be…

Let $\ell$ be a positive level. Denote by … the set of bounded-partition-enhanced $\ell\times\ell$-block $\mathbb{N}$-matrices, and by … the set of pairs of semistandard tableaux o…

Let $n$ and $r$ be positive integers, let $q$ be the parameter, let $Q$ be the second parameter, and let $\mathcal{L}^n(2r-1)$ denote the algebra under consideration. Quasi-heredit…

Let $K$ be a field, let $S_R(n,d)$ denote the Schur algebra over a commutative ring $R$, and let $V$ be its natural module. For $n,d\in\mathbb{N}$, write…

Let $\dot{\mathbf{U}}_{q}^{\imath}(\mathfrak{sl}_{n})$ be the modified i quantum group and let $\tilde{S}^{\imath}(n,d)$ be the tilde-i Schur algebra. Let … be the algebra homomorp…

Let $n=2r+1$, let $\boldsymbol{c}_{\lambda}^{\jmath}$ be the two-sided cell associated with a special partition $\lambda$ of $2d+1$ with at most $n$ parts, and consider semistandar…

Exceptional wild Schur algebra conjecture. All wild Schur algebras contained in $(\star)$ are $\tau$-tilting finite.

Exceptional-case finiteness conjecture. All wild Schur algebras in the cases $(\star)$ are $\tau$-tilting finite.

Let $D(n,r)^V$ denote the subspace of the parabolic Schur algebra consisting of elements fixed by the relevant $V$-action, and let $\bbC\Psi(\mathfrak{S}_r)$ be the complex span of…

Assume the notation above, and for each $bb\in\Lambda^+(n,r)$ let $\barbb$ be the dominant weight for $SL_n$ determined by $bb$. Let $w_{\barbb}$ be the unique element $x$ of the a…

Fix $e,\ell,m>0$ and let $\nu\in\mathbb{N}^{\ell}(m)$. Let $\mathcal{O}^{\nu}(n)$ be the affine category block decomposition introduced in the source, let $\mathbf{S}^{\nu}(n)$ be…

Let $G=SL_2$ (and, in the proposed extension, consider the type-$A_n$ setting), let ${\mathfrak g}$ be the Lie algebra with nilpotent cone ${\mathcal N}$, and let…

Quantized Schur algebras are considered for types other than $A$, following the decomposition conjecture proved by M. Varagnolo and E. Vasserot in type $A$. Decomposition conjectur…

Let $F$ be the field in which rational representations of $GL_2(F)$ are considered, and let $\underleftarrow{\mathcal{C}}_p(F)\operatorname{-mod}$ denote the corresponding module c…

Saturation conjecture. If $V$ is a finite dimensional irreducible $\mathbf{U}$-module belonging to $\mathcal{C}$, then