# Questions tagged [bruhat-order]

The tag has no usage guidance.

32 questions
Filter by
Sorted by
Tagged with
35 views

### Intersection of certain subsets in a split connected reductive group $G$ related to affine open cover of $G/B$

Let $k$ be a field of characteristic zero and $G$ a split connected reductive group over $k$. Moreover, let $T$ be a split maximal torus of $G$ and $B\supset T$ a Borel subgroup. Additionally, we ...
1 vote
137 views

252 views

### Number of paths in the Bruhat order in the symmetric group

Let $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...
185 views

### Two algebraic guises of Alternating Sign Matrices: any connection?

Alternating Sign Matrices (ASMs) have a famous history: they were discovered by Mills, Robbins, and Rumsey, who conjectured a product formula for their enumeration; this product formula was first ...
192 views

### Is the order complex of open Bruhat intervals polytopal?

Let $P$ be the Bruhat order of a Coxeter group, and let $s<t$ in $P$. The set $\Delta(s,t)$ of all chains of the open interval $(s,t)$ (called the order complex of $(s,t)$) is a simplicial complex. ...
363 views

### Rank matrices for type $D$ Bruhat order

Roughly, this question asks how the Bruhat (strong) order in type $D$ can be understood like the Bruhat orders in types A and B=C. I'll review how types A and B work before asking my question. As a ...
148 views

### Consequence of Lifting property of Bruhat ordering

I am reading the book: Anders Björner, Francesco Brenti --- Combinatorics of Coxeter Groups. I would like to know whether a variation of Corollary 2.2.8 is true. In other words, does the following ...
158 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 ...
192 views

### Reduced expression and Bruhat order

For $n\ge 3$. Let $s_1\cdots s_n$ be a reduced expression of $x$. Suppose $s_1\cdots s_{n-1}\le w$ and $s_2\cdots s_{n}\le w$. Does this imply $x\le w$?
304 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 $\mathbb{Z}^{\ge 0}$...
1 vote
55 views

358 views

### A hard Lefschetz theorem for nilCoxeter algebras

Let $W$ be a finite Coxeter group and $\mathcal{N}(W)$ its nilCoxeter algebra (over the reals, say), as defined at https://en.wikipedia.org/wiki/Nil-Coxeter_algebra. $\mathcal{N}(W)$ has a natural ...
423 views

274 views

Let $\Gamma$ be a Coxeter group on some generating set $S$, with reflection representation $V$. Then $\Gamma$ has two standard partial orders, the weak and strong Bruhat orders. Moreover, if $\lambda ... 20 votes 1 answer 949 views ### Bruhat order and the Robinson-Schensted correspondence The Robinson-Schensted correspondence is a bijection between elements of the symmetric group$S_n$and pairs of standard tableaux of the same shape. The symmetric group is partially ordered by the ... 15 votes 3 answers 769 views ### Kazhdan-Luzstig Polynomials and Lower Intervals in the Bruhat Order I have read in a number of places that the lower Bruhat interval$[e, w]$is rank-symmetric if and only if the KL-polynomial$P_{e, w}(q) = 1$. All of the proofs I've come across use "rationally ... 6 votes 1 answer 373 views ### Efficient enumeration of Bruhat intervals Hi everyone. I'm currently programming some stuff for Hecke algebras. My current implementations have several bottlenecks and I'd like to improve that as much as I can so that I can use stuff like$...
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 $O(n^2)$ ...