# Questions tagged [bruhat-order]

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### 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 ...
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### 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?
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### 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 ...
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### 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 $v_2$. ...
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### 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 "...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...