Séminaires : Séminaire d'Algèbre

Every cluster variety admits a faithful regular group action scaling all cluster variables. Given a partial compactification of a cluster variety, a natural question is whether this action extends to a regular action on the compactification. I will explain that, in the case of braid varieties for simple Lie groups, the answer is affirmative for brick compactifications. If the braid variety is an open Richardson variety in the flag variety, the same result holds for the corresponding closed Richardson variety. In the latter case, the faithfully acting group is a torus of rank the Kazhdan-Lusztig d-invariant. The same result holds for projected Richardson varieties. I will also explain that, in general, a braid variety is a torus if and only if each of its brick compactifications is a toric variety with respect to this extended action, if and only if the polar dual of the moment polytope for this action realizes the corresponding subword complex. The braid words satisfying this property turn out to be precisely the double root free words of V. Pilaud and C. Stump. This provides a novel approach to the longstanding open question of the polytopality of spherical subword complexes asked by A. Knutson and E. Miller in 2004, and in particular gives infinite families of subword complexes admitting polytopal realizations in dimension higher than the rank of the corresponding Coxeter group. The talk is based on joint work with Lara Bossinger and José Simental, arXiv:2609.12414.

This talk will take place in hybrid mode at the Institut Henri Poincaré.