117 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 $W$ be a complex reflection group, let ${\mathcal{C}}$ be its Calogero–Moser parameter space, and let $\kappa:{\mathcal{C}}\rightarrow{\mathcal{K}}$ be the parameter correspond…

Let $W$ be a complex reflection group, let ${\mathcal{C}}$ be its parameter space, and let $\kappa:{\mathcal{C}}\rightarrow{\mathcal{K}}$ identify Calogero–Moser parameters with He…

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 $W$ be a well-generated finite complex reflection group, and let $T$ and $T'$ be reflection factorizations of a Coxeter element of $W$. Two factorizations are said to have the…

Calogero–Moser cell coincidence conjecture. If $W$ is a Coxeter group, the Calogero–Moser notions coincide with the corresponding Hecke-algebra notions defined by Kazhdan and Luszt…

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…

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…

Universal freeness conjecture. The assumption that $H_u(W)$ can be generated by $|W|$ elements as an $R$-module is conjectured to hold in all cases.

Topological cyclotomic Hecke algebra isomorphism conjecture. The topological cyclotomic Hecke algebra is conjectured to be isomorphic to the cyclotomic Hecke algebra in all cases.

Bessis's modified-action CSP conjecture. The triple

The hsop existence conjecture. If $p\equiv 1\pmod h$, then $S$ contains an hsop $\mathbf{\Theta}$ carrying $V^*(-p)$; if $p\equiv -1\pmod h$, then $S$ contains an hsop…

Let $W$ be an irreducible complex reflection group of dimension $n$ that can be generated by $n$ reflections. Let $h$ denote its Coxeter number, defined by … where $N$ is the numbe…

Let $W$ be a well-generated finite complex reflection group with degrees $d_1,\dots,d_n$ and Coxeter number $h$, let $d$ be a regular degree, let $W'$ be the centralizer of a $d$-r…

Let $W$ be a finite complex reflection group, let $B$ be its braid group, and let $P$ be the pure braid group. Choose $v\in V^{\operatorname{reg}}$ whose $W$-orbit lies in $L_0$, a…

Let $V$ be a finite-dimensional complex vector space, let $W\subseteq \operatorname{GL}(V)$ be a finite complex reflection group, and let $V^{\operatorname{reg}}$ be the complement…

Shift-functor equivalence conjecture. For general $W$, the shift functor associated to $\zeta$ should give an equivalence whenever $h_\cdot$ and $h_\cdot+\tau$ define the same orde…

Rouquier's derived equivalence conjecture. For every $\tau\in{\mathfrak t}_{\mathbf Z}$, there should be an equivalence

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, put $w=\beta({\bf w})$, and let $C_W(wF)$ be the centralizer of the corresponding regular element of $WF$. Let $B(w)$ be the bra…

Singularity conjecture. The space ${\mathcal{C}}{\mathcal{M}}_c$ is singular for every value of $c$.

Let $W$ be an irreducible complex reflection group that need not be well-generated. The proposed general construction associates a dual braid monoid to a regular degree and maps it…

Let $B$ be the braid group associated with a well-generated complex reflection group, let $\mathcal{R}$ be its set of braid reflections, and let $\delta$ be the braid-group analogu…

Let $V$ be a finite-dimensional complex vector space, let $W\subset \operatorname{GL}(V)$ be a finite complex reflection group, and let $V^{\operatorname{reg}}$ be the complement i…

The $G_{34}$ braid-group conjecture. The braid group associated with $G_{34}$ admits the presentation

The $G_{33}$ braid-group conjecture. The braid group associated with $G_{33}$ admits the presentation