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Slightly smoothing notation

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edited Oct 1, 2022 at 0:48

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As I mentioned in a comment, this is a special case of finding a distinguished representative for a coset of a parabolic subgroup. Let's inductively define elements $w_n$, with the convenient starting point $w_n = w_0$ when $n = 0$. Having defined $w_n$ in general, one of two things can happen: either $w_n^{-1}K$ consists of positive roots, or there is some $\alpha \in K$ such that $w_n^{-1}\alpha$ is negative. In the former case, we stop, and put $v_K = w_n$. In the latter case, we put $w_{n + 1} = s_\alpha w_n$. It is part of the general theory of length in Coxeter groups that $\ell(w_{n + 1}) = \ell(w_n) - 1$, and hence that $\ell(w_n) = \ell(w_0) - n$ for all natural numbers $n$ such that $w_n$ is defined.

Suppose that $w^K \mathrel{:=} w_0 v_K^{-1}$ is not the long element $w_0^K$ of the parabolic subgroup $W_K$ of the Weyl group generated by $K$. Then $\ell(w_0^K)$ is strictly greater than $\ell(w^K)$; but then $\ell(w_0^K v_K)$, which equals $\ell(w_0^K) + \ell(v_K)$ by Proposition 1.10(c) of Humphreys - Reflection groups …, is strictly larger than $\ell(w^K) + \ell(v_K) = \ell(w^K v_K) = \ell(w_0)$. This is a contradiction.

As I mentioned in a comment, this is a special case of finding a distinguished representative for a coset of a parabolic subgroup. Let's inductively define elements $w_n$, with the convenient starting point $w_n = w_0$ when $n = 0$. Having defined $w_n$ in general, one of two things can happen: either $w_n^{-1}K$ consists of positive roots, or there is some $\alpha \in K$ such that $w_n^{-1}\alpha$ is negative. In the former case, we stop, and put $v_K = w_n$. In the latter case, we put $w_{n + 1} = s_\alpha w_n$. It is part of the general theory of length in Coxeter groups that $\ell(w_{n + 1}) = \ell(w_n) - 1$, and hence that $\ell(w_n) = \ell(w_0) - n$ for all natural numbers $n$ such that $w_n$ is defined.

Suppose that $w^K \mathrel{:=} w_0 v_K^{-1}$ is not the long element $w_0^K$ of the parabolic subgroup $W_K$ of the Weyl group. Then $\ell(w_0^K)$ is strictly greater than $\ell(w^K)$; but then $\ell(w_0^K v_K)$, which equals $\ell(w_0^K) + \ell(v_K)$ by Proposition 1.10(c) of Humphreys - Reflection groups …, is strictly larger than $\ell(w^K) + \ell(v_K) = \ell(w^K v_K) = \ell(w_0)$. This is a contradiction.

As I mentioned in a comment, this is a special case of finding a distinguished representative for a coset of a parabolic subgroup. Let's inductively define elements $w_n$, with the convenient starting point $w_n = w_0$ when $n = 0$. Having defined $w_n$ in general, one of two things can happen: either $w_n^{-1}K$ consists of positive roots, or there is some $\alpha \in K$ such that $w_n^{-1}\alpha$ is negative. In the former case, we stop, and put $v_K = w_n$. In the latter case, we put $w_{n + 1} = s_\alpha w_n$. It is part of the general theory of length in Coxeter groups that $\ell(w_{n + 1}) = \ell(w_n) - 1$, and hence that $\ell(w_n) = \ell(w_0) - n$ for all natural numbers $n$ such that $w_n$ is defined.

Suppose that $w^K \mathrel{:=} w_0 v_K^{-1}$ is not the long element $w_0^K$ of the parabolic subgroup of the Weyl group generated by $K$. Then $\ell(w_0^K)$ is strictly greater than $\ell(w^K)$; but then $\ell(w_0^K v_K)$, which equals $\ell(w_0^K) + \ell(v_K)$ by Proposition 1.10(c) of Humphreys - Reflection groups …, is strictly larger than $\ell(w^K) + \ell(v_K) = \ell(w^K v_K) = \ell(w_0)$. This is a contradiction.

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created Sep 30, 2022 at 19:59

12.9k
4
45
69

As I mentioned in a comment, this is a special case of finding a distinguished representative for a coset of a parabolic subgroup. Let's inductively define elements $w_n$, with the convenient starting point $w_n = w_0$ when $n = 0$. Having defined $w_n$ in general, one of two things can happen: either $w_n^{-1}K$ consists of positive roots, or there is some $\alpha \in K$ such that $w_n^{-1}\alpha$ is negative. In the former case, we stop, and put $v_K = w_n$. In the latter case, we put $w_{n + 1} = s_\alpha w_n$. It is part of the general theory of length in Coxeter groups that $\ell(w_{n + 1}) = \ell(w_n) - 1$, and hence that $\ell(w_n) = \ell(w_0) - n$ for all natural numbers $n$ such that $w_n$ is defined.

Suppose that $w^K \mathrel{:=} w_0 v_K^{-1}$ is not the long element $w_0^K$ of the parabolic subgroup $W_K$ of the Weyl group. Then $\ell(w_0^K)$ is strictly greater than $\ell(w^K)$; but then $\ell(w_0^K v_K)$, which equals $\ell(w_0^K) + \ell(v_K)$ by Proposition 1.10(c) of Humphreys - Reflection groups …, is strictly larger than $\ell(w^K) + \ell(v_K) = \ell(w^K v_K) = \ell(w_0)$. This is a contradiction.