8 problems

Let $(W,S,*)$ be a triple consisting of a Coxeter system $(W,S)$ and an $S$-preserving involution $*\in\operatorname{Aut}(W)$, with corresponding twisted involutions $\mathbf I_*$.…

Twisted property B-prime. The polynomials $P^\pm_{y,w}$ are decreasing for fixed $w$: whenever $y,z,w\in\mathbf I_*$ and $y\leq z$, both differences $P^+_{y,w}-P^+_{z,w}$ and…

Bruhat covering property. If $y\in I_*$ is a twisted involution in an arbitrary Coxeter group and $t\in T$, then there exists at most one $z\in I_*$ such that

Let $(W,S)$ be a Coxeter system with an involutive automorphism $\t$, and let $w\in\mathfrak{I}(\t)$. A reduced $\underline{S}$-expression for $w$ is an expression of minimal lengt…

Conjecture on basic braid transformations. There exists a finite set of basic braid $I_\ast$-transformations that spans and preserves the reduced $I_\ast$-expressions for every twi…

Strong exchange conjecture. Under this condition,

QP1–QP2 conjecture. Every conjugacy class in $I^+$ satisfying (QP1) also satisfies (QP2), and hence is quasiparabolic.

Let $(W,S,*)$ be a triple in which $(W,S)$ is a Coxeter system and $*\in\operatorname{Aut}(W)$ is an $S$-preserving involution. Let Properties A-prime, B-prime, C-prime, and D-prim…