Questions tagged [bruhat-order]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
169 views

Visualizing the affine Bruhat decomposition for $\operatorname{SL}_2$

$ \newcommand\Fl{\mathcal{F}\!\ell} \newcommand\numC{\mathbb{C}} \newcommand\numZ{\mathbb{Z}} \newcommand\ringO{\mathbb{O}} \newcommand\ringK{\mathbb{K}} \newcommand\power{\...
Gaussler's user avatar
  • 295
2 votes
0 answers
194 views

Geometric or combinatorial interpretations of the (weak) Bruhat order?

$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two ...
Brendan Mallery's user avatar
4 votes
0 answers
75 views

Parabolic Bruhat graphs for exceptional types

I am looking for some computer software or a reference for some parabolic Bruhat graphs. In particular, what I really need $E_8 \setminus E_7$. Does anyone know where or how I'd find this?
Chris Bowman's user avatar
  • 1,191
2 votes
0 answers
72 views

On $\Psi$-generating paths in the Bruhat order of a Weyl group

Let $W$ be a Weyl group with roots $R$ and positive roots $R^+$. Let $v\in W$ of length $r$. We call $\mathbb{m}=(\alpha_1,\ldots,\alpha_r)\in(R^+)^r$ a Bruhat path from $1$ to $v$ if $1\lessdot s_{\...
Christoph Mark's user avatar
4 votes
1 answer
274 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 ...
Christoph Mark's user avatar
8 votes
0 answers
189 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 ...
Sam Hopkins's user avatar
  • 22.5k
8 votes
0 answers
194 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. ...
Richard Stanley's user avatar
8 votes
2 answers
379 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 ...
David E Speyer's user avatar
2 votes
1 answer
152 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 ...
James Cheung's user avatar
  • 1,855
3 votes
1 answer
159 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 ...
James Cheung's user avatar
  • 1,855
2 votes
1 answer
194 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 $x\le w$?
James Cheung's user avatar
  • 1,855
3 votes
2 answers
341 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 $\mathbb{Z}^{\ge 0}$...
James Cheung's user avatar
  • 1,855
1 vote
0 answers
55 views

$\mathrm{Ext}^1$-ordering on ${}^IW^{\Sigma_\mu}$

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+...
James Cheung's user avatar
  • 1,855
0 votes
1 answer
96 views

In Type $A$, if the Bruhat graph of an element $w$ in the Weyl group is regular, then to show that $l(w)=$ # $ \{\alpha \in R^+| s_{\alpha} \le w\}$

I am trying to prove that for type $A$ , rational smoothness of Schubert varieties implies smoothness. So suppose we are in Type $A_{n-1}$, so let $G=Sl(n,\mathbb C)$, $B=$ the group of upper ...
user102248's user avatar
1 vote
0 answers
146 views

A certain kind of permutations and transport of Bruhat chains under conjugation

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 $v_2$. ...
Christoph Mark's user avatar
8 votes
1 answer
350 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 "...
Sam Hopkins's user avatar
  • 22.5k
8 votes
1 answer
160 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 ...
Matt Samuel's user avatar
  • 1,978
2 votes
0 answers
79 views

Characterization of permutations which have at most one successor in the covering relation of the weak Bruhat order

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 ...
Christoph Mark's user avatar
8 votes
1 answer
803 views

There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$

Let $G$ be a split reductive algebraic group (over a local field if you like), $B$ be a fixed Borel subgroup, and $P$ be a fixed standard parabolic subgroup. Let $W$ be the Weyl group of $G$. For $w\...
Q. Zhang's user avatar
  • 960
13 votes
0 answers
361 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 ...
Richard Stanley's user avatar
9 votes
1 answer
445 views

Bruhat order of reflection subgroups

Let $(W,S)$ be a Coxeter group, $T=\bigcup_{w\in W}wSw^{-1}$ its set of reflections, and $A\subseteq T$. From results of Dyer and Deodhar, we know that the subgroup $W_A$ generated by the elements of $...
Balazs Elek's user avatar
3 votes
0 answers
75 views

Can Bruhat cells in semi simple groups be induced from matrices?

Let $G$ be a semisimple Lie group. Embed it as a subgroup into a special linear group of suitable rank, $SL(n)$ (real or complex). The question is: is it always possible to find such an embedding, ...
user59308's user avatar
12 votes
2 answers
720 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 ...
Gwyn Whieldon's user avatar
5 votes
4 answers
572 views

Bruhat order and Schubert cycles

I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ ...
Misha's user avatar
  • 31k
4 votes
1 answer
177 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 $...
Rami's user avatar
  • 2,571
5 votes
1 answer
284 views

Edge graph of the polytope of a Bruhat interval

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 ...
Allen Knutson's user avatar
20 votes
1 answer
967 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 ...
M T's user avatar
  • 2,681
15 votes
3 answers
780 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 ...
xpilot's user avatar
  • 341
6 votes
1 answer
386 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 $...
Johannes Hahn's user avatar
10 votes
0 answers
298 views

Are plactic classes convex under the right weak Bruhat order?

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 ...
Kurt Luoto's user avatar
3 votes
1 answer
270 views

Reference for: the Bruhat-minimal permutations not less than a fixed permutation pi?

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 $O(n^2)$ ...
Allen Knutson's user avatar