139 problems

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…

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 $W\subset GL(V)$ be a complex reflection group, let $R(W)$ be its associated Laurent polynomial ring, and let $\mathcal{H}(W)$ be the associated generic Hecke algebra over…

Bonnafé–Geck–Iancu–Lam conjecture. If $s\notin\mathbb{N}$, the Kazhdan–Lusztig left cells coincide with the irreducible combinatorial left cells of rank $\lfloor s\rfloor$. If…

Haiman's conjecture. For any $w\in S_n$, the dual Frobenius character $\operatorname{ch}(q^{\frac{\ell(w)}{2}}C'_w)$ is $h$-positive.

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…

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 $G_r$ be an unramified base-change form of the quasi-split group $G$, with Hecke algebras $\mathcal{H}(G_r)$ and $\mathcal{H}(G)$, and let $\mathfrak{Z}^{\textnormal{st}}(G_r)$…

Let $G$ be a reductive group satisfying the stated restriction that, for every root $\check{\alpha}\in\Lambda^{\vee}$, the map $\Lambda\to\mathbb{Z}$ given by…

Hecke-algebra extension conjecture. Theorem should hold in $\mathcal{H}$ when each $t_\ell$ is replaced by $t_\ell^{\mathcal{H}}$.

Let $\mathbf{G}$ be a reductive group over the nonarchimedean field $k$, let $\mathbf{X}$ be a spherical $\mathbf{G}$-variety, let $K$ be a maximal compact subgroup of…

Dipper–James conjecture. The centre of ${\mathcal H}$ is the set of symmetric polynomials in the Jucys–Murphy operators $L_1,\ldots,L_n$.

The cohomology tensor-product conjecture. There is an equality

Let $W$ be the Weyl group, let $F$ be the Frobenius automorphism, and let ${\boldsymbol\pi}$ be the positive generator of the center of the pure braid group. Let…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, let ${\mathcal H}_q(w)$ be the specialization at $x^{1/a}\mapsto q^{1/a}$ of the cyclotomic Hecke algebra ${\mathcal H}(w)$, and…

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, let $w=\beta({\bf w})$, let $B(w)$ be the braid group of $C_W(wF)$, and let ${\mathcal H}(w)$ be a cyclotomic Hecke algebra for…

Assume that ${\bf G}$ is split of type $A_2$. Let $\varphi:B^+\to\mathbb Z$ be the class function defined in the source by its values on the listed representatives and by…

Integral-closure conjecture. If $k<p-1$, then $\mathbf{T}^{+}$ and $\mathbf{T}^{-}$ are integrally closed. Equivalently, every congruence between distinct eigenforms in…

Let $A=\mathbb{Z}[x,x^{-1}]$, let $R$ be a commutative local $A$-algebra, and let $F$ be its residue field. Let ${\mathcal H}$ be the Hecke algebra of a finite complex reflection g…

Let $W$ be a finite complex reflection group, let $W'$ be a parabolic subgroup, and let ${\mathcal H}(W)$ be the associated Hecke algebra. A canonical symmetrizing form on…

Let $W$ be a finite complex reflection group, let $W'$ be a parabolic subgroup of $W$, and let ${\mathcal H}(W)$ and ${\mathcal H}(W')$ be their Hecke algebras over…

Let $H_{n,p}$ be the Homfly skein algebra with $n$ positive and $p$ negative boundary points, let ${\cal C}$ be the skein of the annulus, and let … be the algebra homomorphism obta…

Let $K$ be a field, let $q\in K$ be an invertible element with $q\ne 1$, and let $(W,S)$ be a finite Weyl group. Let ${\mathcal H}_q(W)$ be the one-parameter Hecke algebra associat…

Let $V$ be a vector space equipped with a Hecke $\check R$-matrix, and let $K^{\bullet\bullet}(V)$ and the complexes $C_V^\bullet(k)$ be the associated bidifferential and one-varia…