Questions tagged [bruhat-order]

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11
votes
2answers
578 views

Principal Order Ideals in the Weak Bruhat Order

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 ...
3
votes
2answers
219 views

Partial ordering on $\mathfrak{h}^*$ and Bruhat ordering

In section 5.2 (p.95) of Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$. Let $\mu\le \lambda$ if $\lambda-\mu\in \Gamma$, where $\Gamma$ is the set of $\mathbb{Z}^{\ge 0}$...
8
votes
1answer
265 views

Formula for number of permutations less than a given permutation in weak order

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 "...
8
votes
1answer
141 views

How many maximal length Bruhat paths from $u$ to $w$ can there be?

I've been doing some work with saturated Bruhat paths in a Coxeter group between two elements $u\leq w$. It seems to me that if $\ell(u) =0$, then there are at most $\ell(w)! $. I haven't tried to ...
4
votes
1answer
157 views

points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of $...
3
votes
1answer
142 views

Bruhat ordering and non-vanishing Extension groups

Let $P_{x,w}(q)$ be the Kazhdan Lusztig polynomial. It is well-known that $P_{x,w}(q)\neq 0\iff x\le w$. By the interpretation of the Kazhdan Lusztig polynomial in terms of extension group, it holds ...