P_J$

Current License: CC BY-SA 4.0

10 events

when toggle format	what		by	license	comment
S Aug 12, 2022 at 5:03	history	bounty ended	CommunityBot
S Aug 12, 2022 at 5:03	history	notice removed	CommunityBot
S Aug 4, 2022 at 3:33	history	bounty started	Suzet
S Aug 4, 2022 at 3:33	history	notice added	Suzet		Draw attention
Jul 6, 2022 at 13:18	comment	added	LSpice		Ah, so we are saying different things. I was saying that one should take $w$ to be maximal length, and I misread you to be saying the same. I agree that the minimal-length $w$ will usually give an intersection $I \cap w J w^{-1}$. In fact, I think (without having thought very hard) that $\ell(w) + \dim(G/P_{I \cap w J w^{-1}})$ might be independent of the choice of representative $w$, so that it's right for all choices or wrong for all choices. I am suggesting that thinking of the longest element might make it easiest to prove.
Jul 6, 2022 at 4:20	comment	added	Suzet		@LSpice (I deleted my previous comment because it was wrong). I don't think that $I\cap wJw^{-1}$ is necessarily empty. Consider the case $W = \mathfrak S_n$ with simple reflections $s_i = (i, i+1)$ for $1\leq i \leq n-1$. Let $I = J = \{s_2,\ldots ,s_{n-1}\}$ and let $w = s_1$. Then $s_1Is_1 = \{s_1s_2s_1, s_3, \ldots ,s_{n-1}\}$. In particular $I \cap s_1Is_1$ contains $s_3,\ldots ,s_{n-1}$.
Jul 6, 2022 at 3:16	comment	added	LSpice		Ah, I had missed that condition. But doesn't that mean that $I \cap w J w^{-1}$ is empty?
Jul 6, 2022 at 2:14	comment	added	Suzet		@LSpice Since I assume my element $w$ to be $I$-reduced-$J$, I think it is already uniquely determined inside its double coset $W_IwW_J$. In fact, it has minimal length.
Jul 6, 2022 at 1:53	comment	added	LSpice		Whatever the answer is, it had better be invariant under replacing $w$ by an element in its $(W_I, W_J)$-double coset, so we might as well assume that $w$ is the longest element in its double coset. Then I think that $I \cap w J w^{-1}$ is empty, so you are conjecturing that the dimension is $\ell(w) + \operatorname{dim}(G/B)$. That is, you're proposing that the dimension is the same as that of $\mathcal O_{\emptyset, \emptyset}(w)$. (Just some observations, not a proof.)
Jul 6, 2022 at 1:17	history	asked	Suzet	CC BY-SA 4.0