8
votes
Accepted
Number of paths in the Bruhat order in the symmetric group
$\ell(v)!$ is of course even if $\ell(v)>1$, so the statement is really that $N_v$ is even for $\ell(v)>1$. We find a fixed-point free involution on the set of such Bruhat paths. Suppose that $...
- 40.2k
7
votes
Accepted
Formula for number of permutations less than a given permutation in weak order
By Dittmer and Pak - Counting linear extensions of restricted posets (Theorem 1.4), computing the size of $[e,w]$ is $\#$P-complete. Thus, a nice formula like the suggested $n \times n$ determinant ...
- 511
5
votes
Accepted
Partial ordering on $\mathfrak{h}^*$ and Bruhat ordering
This question was answered correctly by Sam Hopkins. Let me just abuse this space for answers for pointing out that James Humphreys keeps up to date corrections on the AMS webpage of the book. Direct ...
- 8,597
4
votes
Accepted
Reduced expression and Bruhat order
Let $W = S_5$ (as a side note, we could let $W = S_3 \times S_2$). Let $w = (132)(45), x = (123)(45) \in W$; we write $x = (12)(45)(23)$ as a reduced expression. Then the subexpressions written in the ...
- 4,991
4
votes
Accepted
Consequence of Lifting property of Bruhat ordering
No, it is not true. Take $\mathfrak{sl}_3$ with simple reflections $r, s$. Then the longest element $t := rsr=srs$ is also a reflection. Put $w=e$.
Then $w=e < sw=s < stw = rs$, but $tw = srs \...
- 1,368
4
votes
Accepted
There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$
Your condition $C(w)\subseteq Pw_1P$ implies $PwP=Pw_1P\Rightarrow\overline{PwP}=\overline{Pw_1P}$.
Moreover $w\ge w'\Rightarrow\overline{BwB}\supseteq\overline{Bw'B}\Rightarrow\overline{PwP}\...
- 15.5k
3
votes
Accepted
How many maximal length Bruhat paths from $u$ to $w$ can there be?
If you'll allow me to ignore the restriction to finite groups, there is a conjectured upper bound, phrased in terms of polytopes, depending on both $\ell(u)$ and $\ell(w)$: Conjecture 7.3 of the ...
- 1,846
3
votes
points with small U stabilizer on a spherical variety
This set is a special case of regular semisimple elements on spherical varieties.
In particular it is open and dense.
- 2,639
2
votes
Accepted
Rank matrices for type $D$ Bruhat order
Here are some more thoughts about how the Type D Bruhat order is more complicated than the Type A and Type B/C orders. These ideas might even suggest that giving a "rank matrix"-like description of ...
- 24.2k
2
votes
Rank matrices for type $D$ Bruhat order
The answer to my questions is that I was wrong. I'll switch to the more standard notation of having $D_n$ act on $\{ -n, -n+1, \ldots, -2, -1, 1, 2, \ldots, n-1, n \}$ in order to match the reference ...
2
votes
Accepted
Bruhat ordering and non-vanishing Extension groups
It is not true in general. Take $\mathfrak{sl}_3$, with simple reflections $s,t$, such that $s$ is singular. Put $x=e$, $w= st$; both are in $W^\Sigma$.
I claim that $Ext_\mathcal{O}^i(M(\mu),L(st \...
- 1,368
2
votes
Partial ordering on $\mathfrak{h}^*$ and Bruhat ordering
I think the implication you are asking about might not in fact hold.
If my computations are correct, we can see this already for $S_4$.
Let me ignore the dot action part of the question (I don't ...
- 24.2k
1
vote
Accepted
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\}$
Unless I'm missing something obvious:
Consider the degree of the identity element $e$ in $\Gamma(w)$. On the one hand, by the assumption you've made it is $\ell(w)$. On the other hand, it is clearly ...
- 24.2k
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