20 problems

Let $\mathcal H_n$ be a cyclotomic Hecke algebra of type $G(m,1,n)$, let $S^{\underline\lambda}$ be its Specht module indexed by a multipartition $\underline\lambda$, and define ……

Let $\mathbf G$ be an algebraic group over $\bar{\mathbb F}_q$, let $F$ be a Frobenius map, let $G=\mathbf G^F$, and let $W$ be the $F$-split Weyl group. Let $w\in W$ be a good ele…

Ariki–Koike representation-theory conjecture. These algebras should play a role in the representation theory of reductive groups similar to, but more complicated than, that played…

Let $\mathcal{R}$ be the coefficient ring, let $P_{\mathrm{loc}}=P\otimes_{\mathcal{R}}\operatorname{Frac}\mathcal{R}$ for every $\mathcal{R}$-module $P$, and let … be the unique i…

Let $\mathcal{L}(n,r)$ be the specified closed projective subvariety of the Gieseker space, let $K_{\mathcal{T}}(\mathcal{L}(n,r))$ denote its equivariant $K$-theory, and let the c…

Let $q$ be a parameter with $q\neq 1$, and let the cyclotomic Hecke algebra have its Jucys–Murphy center and center. McGerty's center conjecture. The Jucys–Murphy center of the cyc…

Trinh–Xue's conjecture. There exists a choice of complete HC-parametrisation $\Psi=\{\Psi_{{\mathbf L},\lambda}\}$ such that, for every such pair $({\mathbf L},\lambda)$ and…

McNamara's center conjecture. The center $Z(\mathscr{H}_{n,K}^\Lambda)$ is precisely the set of symmetric polynomials in $\mathcal{L}_1,\ldots,\mathcal{L}_n$.

Finite-Coxeter defect-invariance conjecture. If $V$ and $V'$ belong to the same $\theta$-block, then they have the same $\Phi$-defect.

Let $\mathbb{H}(r,1,d)$ be a cyclotomic Hecke algebra, and let $U_q(\widehat{\mathfrak{sl}}_n)^{(t)}$ be a Hopf subalgebra of $U_q(\widehat{\mathfrak{gl}}_n)$, where $t$ is an inte…

Cyclotomic shuffle spectrum conjecture. The spectrum of left multiplication by ${}^{(2)}\Sha_{1,n-1;q,p}$ in $H(2,1,n)$ consists of the values

Let $e=\infty$ or, more generally, $e>n$, and let $\kappa\in I^\ell$ have no repeated entries. Let $\Bbbk$ be a field of characteristic $p\gg\ell$. For…

Let $e>h\ell$, let $\kappa\in I^\ell$ be an $h$-admissible multicharge, and let $\Bbbk$ be a field of characteristic $p\gg h\ell$. For $\lambda,\mu\in\mathscr{P}^{\ell}_{n}(h)$ in…

Let ${\boldsymbol\lambda},{\boldsymbol\mu}\in\mathscr{P}_{n}$. For the polynomials $d_{{\boldsymbol\lambda}{\boldsymbol\mu}}(q)$ and the defect…

Let $W$ be a complex reflection group. Let $\theta:q\mapsto\xi$ be a specialization of ${\mathcal H}_q(W)$, and let…

Let $W$ be an exceptional complex reflection group of rank $2$, and let ${\mathcal H}_q$ be an associated cyclotomic Hecke algebra over $\mathbb{C}[q,q^{-1}]$. A symmetrizing form…

Let $S^{\boldsymbol\mu}$ be a graded Specht module for a cyclotomic Hecke algebra of type $G(\ell,1,n)$, and let induction be taken from the corresponding Hecke algebra on $n$ lett…

Let $R^{\Lambda}_\alpha$ be the cyclotomic Khovanov–Lauda–Rouquier algebra associated with a dominant weight $\Lambda$ and $\alpha$, and let $F$ be its ground field. Write…

Let $R^{\Lambda}_{\alpha}$ be the cyclotomic Khovanov–Lauda algebra, with generators $y_1,\ldots,y_d$, and let ${\Lambda}$ be a dominant weight of level $l$. Assume that the quantu…

Let $\chi_\phi$ and $\psi_\phi$ be characters belonging to the same Rouquier block of a cyclotomic Hecke algebra associated with an exceptional complex reflection group, and let…