Timeline for Reference request: Weyl group action on the power set of positive roots

6 events

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Oct 23, 2021 at 8:39	comment	added	Jianrong Li		@AndreiSmolensky, yes, you are right. I would like to define a map to send ${\bf p}_{\bf i, \bf j}$ to ${\bf p}_{w \cdot \bf i, \bf j}$ for every $w \in S_m$. Maybe this is not necessarily an action. I would like to have a similar result as the Lemma for other Weyl groups.
Oct 23, 2021 at 8:34	comment	added	Andrei Smolensky		Yes, but $w$ comes from $S_m$, not from $S_{n+1}$.
Oct 23, 2021 at 8:32	history	edited	Jianrong Li	CC BY-SA 4.0	added 18 characters in body
Oct 23, 2021 at 8:31	comment	added	Jianrong Li		@AndreiSmolensky, thank you very much. In Equation (1), ${\bf p}_{{\bf i}, {\bf j}}$ is identified with a set of positive roots, $(i,j)$ is identified with $\alpha_i+\cdots + \alpha_j$. The action is $(w, {\bf p}_{{\bf i}, {\bf j}}) \mapsto {\bf p}_{w \cdot {\bf i}, {\bf j}}$.
Oct 23, 2021 at 8:19	comment	added	Andrei Smolensky		I do not follow, where is the action of the Weyl group here? $W(\mathsf{A}_n)\cong S_{n+1}$, and here the power-set is acted on by $S_{n(n+1)/2}$.
Oct 23, 2021 at 7:19	history	asked	Jianrong Li	CC BY-SA 4.0