270 problems

Let $C$ be a finite complex reflection group, let $V$ be its reflection representation, and let $H_C(\vec{u})$ be the generic Hecke algebra defined from the braid group of $C$ and…

Let $e$ be the quantum characteristic of a non-trivial invertible element $q\in\mathbb{F}^{\times}$, and let $d_{\lambda,\mu}(v)$ be the integer-coefficient polynomials associated…

James's conjecture. James's conjecture holds for all weight $5$ blocks of $\hhh$.

Let $E$ be the quadratic field under consideration, let $h$ be the Hecke algebra acting on the relevant cohomology, let $\mathfrak{m}\subset h$ be a non-Eisenstein maximal ideal, a…

Haiman's conjecture. For every $z\in S_n$ and $\mu\vdash n$, the nonzero coefficients of $\alpha_{\mu}^{z}$ are all positive and form a unimodal sequence.

Lusztig's boundedness conjecture. The number $N_L(W)$ is a bound for any positively weighted Coxeter group $(W,S,L)$ of finite rank. This is a weighted generalization of Xi's bound…

Lusztig's based-ring conjecture. There should exist a finite $F_{\mathbf c}$-set $Y$ and a bijection $\pi$ from $\mathbf c$ onto the irreducible $F_{\mathbf c}$-vector bundles on…

Let $W$ be a complex reflection group, let $B(W)$ be its braid group, let $\mathcal{H}(W)$ be its generic Hecke algebra, and let $\tau$ be a canonical symmetrising trace on…

Let $K$ be a field and let $\lambda\in K\setminus\{0,1,-1\}$. Let $(W,S)$ be an arbitrary Coxeter system. Write $\mathcal{A}_{\pm 1}=\mathbb{Z}[u,u^{-1},(u+1)^{-1},(u-1)^{-1}]$, le…

Let $H$ be a Hecke algebra of type A, and let $L_1,\ldots,L_n$ denote its Jucys–Murphy elements. A polynomial in these elements is symmetric if it is invariant under permutations o…

Hecke-algebra and uniform eigenspace decomposition conjecture. The subalgebras

Let $G$ be a finite reductive group, and let $(M,\mu)$ be a $\Phi_m$-cuspidal pair with relative Weyl group $W_{M,\mu}^G$. Let $H_{M,\mu}^G(x)$ be the Broué–Malle algebra, and let…

Let $W$ be a complex reflection group and let $\mathcal{H}(W)$ be its Hecke algebra. A symmetrizing trace on an algebra $H$ is a trace function $\tau:H\to R$ for which the bilinear…

Ribet's conjecture. If $\mathfrak{m}$ is an Eisenstein maximal ideal, then

Hida's codimension conjecture. We have

Let $W$ be a complex reflection group, let $R(W)$ be its ring of definition, and let $\mathcal{H}(W)$ be its generic Hecke algebra. Let $B(W)$ be the braid group, let…

Dipper-Du conjecture. The parabolics of $S_n$ occurring as vertices of indecomposable modules in $H_q(S_n)$-mod are exactly the parabolics isomorphic to $S_e^{\times k}$, where…

Let $(W,S)$ be a Coxeter group with weight function $L$, Hecke algebra structure constants $h_{x,y,z}$, Kazhdan–Lusztig $\mathbf{a}$-function, and sets $\mathcal{D}_{\geq N}$ and…

Let $W$ be a complex reflection group, let $c$ be a parameter, and assume the stated freeness and symmetrizing-form hypothesis for the generic Hecke algebra. Write ${\bar Z}_c$ for…

Fan's conjecture. The algebra $TL(H_n)$ is generically semisimple.

Fix $n\geq 2$. Let $V(\Lambda_0)$ be the integrable highest-weight $U_q(\widehat{\mathfrak{sl}}_n)$-module, let $G^{up}(\mu)$ denote its upper global crystal basis, and identify…

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 $b_{w,z}$ be the coefficients defined in Proposition 6.4 for permutations $w,z\in S_n$ occurring in the action of the Hecke algebra generators on the Schubert basis of the coin…

Vigneras's conjecture. This parametrisation depends only on $l$, and not on the coefficient ring $R$.

Let $n$ be a positive integer, let $\widehat{\mathfrak{sl}}_n$ be the affine Lie algebra of type $A^{(1)}_{n-1}$, and let $K_m$ be the complexified Grothendieck group of finitely g…