9 problems

Let $G$ be a connected, semisimple, simply connected group defined over a finite field of $q$ elements, and let $F$ be an $F$-split Frobenius automorphism of $G$. The Weyl group of…

Let $W$ be a finite Coxeter group, let $w\in W$, and let $\Omega_W(w;t)$ denote the order polynomial associated with $W$ and $w$. Assume that $t_W$ is a linear function depending o…

Let $W$ be a finite Coxeter group, and let $\Sigma W$ be the Solomon descent algebra of $W$. The Solomon homomorphism $[...]$ maps elements of $\Sigma W$ to class functions on $W$.…

Radical-layer multiplicity conjecture. The multiplicity is ${n\choose i}$ for every radical layer. In particular, the Ext-quiver of $\mathscr{D}_F(\mathbb{B}_n)$ is a single vertex…

Let $n$ and $p$ be as in the descent algebra setting, let mathop{\Lambda}\nolimits^+_p(n) denote the indexing set of $p$-regular partitions, and let $Q_{n,p}$ be the Ext quiver o…

Let $F$ be a field of characteristic $p>0$, let $\mu\in\Lambda^+_p(n)$ be a $p$-regular partition of $n$, and let $q\in\Lambda(n)$ be a partition of $n$. The notation $\Xi^q$,…

Let $\mathsf{Z}$ be the dendriform image of the series $\mathscr{Z}$, let $\mathsf{Z}_n$ be its homogeneous component of degree $n$, and let $b$ be the flow variable. Z-series desc…

Let $W$ be a finite Coxeter group of rank $n$, let $W_L$ denote a parabolic subgroup generated by a subset $L$ of a Coxeter generating set, and call a conjugacy class cuspidal if i…

Let $D_n$ be the Weyl group of type $D$ with descent algebra $\Sigma_{D_n}$, let $\mathcal{C}(n)$ denote the relevant class of compositions, and let $P^{(\kappa)}$ and $B_\nu$ be t…