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@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.
