6 problems

Let $A^{c,p}_{n}$ be the set of fully commutative elements of size $p$ in $W(A_n)$, and let $\mathcal C_n^p=|A^{c,p}_{n}|$ for $0\le p\le n$, with $\mathcal C_n^p=0$ when $n<0$,…

Let $a$ and $b$ be positive integers. In the notation $a[b]a$ for the rank-two Shephard group with endpoint parameters $a,a$ and braid parameter $b$, and $b[a]b$ for the group with…

Consider the complex reflection group $G(m,m,3)$ with its affine generating set, and let $\ell$ denote the length of an element. Constant-length enumeration conjecture. When…

Let $a_m$ denote the number of fully commutative elements in $G(m,m,n)$, for $n\geq 3$, with the affine generating set. Affine enumeration conjecture. There is a positive integer…

Let $W(A_n)$ be the Coxeter group of type $A_n$, identified with the symmetric group $S_{n+1}$, and let $w\in W(A_n)$ correspond to a permutation with disjoint cycles…

Let $W_c(\widetilde{C}_n)$ denote the set of fully commutative elements in the affine Coxeter group of type $\widetilde{C}_n$. Assume that $n$ is even, and consider the elements…