All Questions

Let $(W,S)$ be a finite Coxeter system. Let us consider the following situation: Let $v_1,v_2,w\in W$ such that $v_1=wv_2w^{-1}$. Let $s_{\beta_r}\ldots s_{\beta_1}$ be a reduced expression of $v_2$. ...

Let $w\in S_n$ be a permutation. Is there a reasonable "formula" for the number of elements of the initial interval $[e,w]$ of weak (Bruhat) order from the identity to $w$? In terms of what such a "...

Let $W$ be the symmetric group on $n+1$ letters. Let $\ell$ be the length function on $W$. As the title says, can we characterize all $v\in W$ such that there exists a $w\in W$ such that for all ...

Let $\sigma\in S_n$ be a permutation on $n$ elements, and $\mathrm{Inv}(\sigma):=\{(i,j) : 1\leq i<j\leq n\text{ and }\sigma(i)>\sigma(j)\}$ be its set of inversions. In the weak order on ...

For those who are unfamiliar with the terminology, I'll explain a little. The symmetric group $S_n$, as a type A Coxeter group, has generators $\{s_1,\ldots,s_{n-1}\}$ with relations (1) $s_i^2$ for ...

Let $\pi\in S_n$. I recently needed to understand the permutations $\rho$ such that $\rho\not\leq\pi$ in Bruhat order. Since there are $O(n!)$ of those I really wanted a description of the $O(n^2)$ ...