15 problems

Let $\boldsymbol{\mathcal V}_{\!\vartriangle\!}(n)$ be the $\mathbb Q(v)$-subspace of $\prod_{r\geqslant 0}\boldsymbol{\mathcal S}_{\!\vartriangle\!}(n,r)$ spanned by all…

Let $k$ be a field of characteristic $\ell$, let $q\in k$ have multiplicative order $e$, and let $\Lambda(n,r)$ label the standard and simple modules of the specialised $q$-Schur a…

Let $S_q=S_q(n,n)$ be the $q$-Schur algebra, where $q=\exp(2\pi i c)$, and let $\mathcal{O}_c$ denote category $\mathcal{O}$ for the rational Cherednik algebra $H_c$. GGOR's equiva…

q-Schur algebra conjecture. The category ${\mathcal O}$ is equivalent to the category of finitely generated modules over the associated $q$-Schur algebra.

Integral Borel basis conjecture. The elements $E_AH_B$ with $A\in{\mathbb N}^{\Phi^+}$, $B\in{\mathbb N}_{i_0}^n$, and $|A|+|B|\le d$ form an $\mathcal A$-basis of…

Quantum Cartan-generator generation conjecture. The elements $\begin{bmatrix}K_i\\m\end{bmatrix}$ lie in the subring generated by the $E_i^{(m)}$ and $F_i^{(m)}$; consequently, one…

Quantum truncated PBW basis conjecture. The monomials

Let $e$ be the quantum characteristic, let $\mathcal{S}_n$ be the level 2 cyclotomic $q$-Schur algebra, and let $B$ be a block of $\mathcal{S}_n$. Schurian-finiteness conjecture. F…

Let $S_{Q,q}^\mathrm{B}(n,d)$ be the type B $q$-Schur algebra, and let $f^\mathrm{B}_d(Q,q)$ denote the parameter whose invertibility is the condition in the preceding quasi-heredi…

Let $n$ and the slim cyclotomic parameters $\boldsymbol{m}$ be fixed. For each $t\geqslant 0$, let … be the corresponding Hopf subalgebra, and let $\xi_{r,\mathbf{m}}^{(t)}$ be the…

Let $\mathcal{S}_{q,k}^{\vartriangle}(n,r)$ be an affine $q$-Schur algebra over a noetherian domain $k$ with $n>r$, and suppose that it has finite global dimension. Evenness conjec…

Let $\mathcal{S}(n,d)$ be the quotient categorification of the quantum $q$-Schur algebra, and let $W_\lambda$ be an irreducible representation associated with…

Krull–Schmidt property. In the case of the categorifications of $\operatorname{Skein}_q(n,N)$ and $S_q(n,\Delta_N)$, the category $\mathcal{X}'$ has the Krull–Schmidt property.

Let $q\in k^\times$, and let $p$ be the least natural number such that … Let $n,w$ be natural numbers with $n\geq w$. Let ${\cal Q}_p(n,w)$ be the quasi-hereditary subquotient of…

Let ${\cal D}_{A_\infty}(w,w)$ be the algebra defined in chapter 8, and let $k{\bf B}_{\rho,w}^{{\cal S}_q}$ denote a Rock block of a $q$-Schur algebra of weight $w$. Schur–Rock co…