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- Title
Torus orbit closures in flag varieties and retractions on Weyl groups.
- Authors
Lee, Eunjeong; Masuda, Mikiya; Park, Seonjeong
- Abstract
A finite Coxeter group W has a natural metric d and if ℳ is a subset of W, then for each u ∈ W , there is q ∈ ℳ such that d (u , q) = d (u , ℳ). Such q is not unique in general but if ℳ is a Coxeter matroid, then it is unique, and we define a retraction ℛ ℳ m : W → ℳ ⊂ W so that ℛ ℳ m (u) = q. The T-fixed point set Y T of a T-orbit closure Y in a flag variety G/B is a Coxeter matroid, where G is a semi-simple algebraic group, B is a Borel subgroup, and T is a maximal torus of G contained in B. We define a retraction ℛ Y g : W → Y T ⊂ W geometrically, where W is the Weyl group of G , and show that ℛ Y g = ℛ Y T m . We introduce another retraction ℛ ℳ a : W → ℳ ⊂ W algebraically for an arbitrary subset ℳ of W when W is a Weyl group of classical Lie type, and show that ℛ ℳ a = ℛ ℳ m when ℳ is a Coxeter matroid.
- Subjects
WEYL groups; COXETER groups; TORUS; BOREL subgroups; FINITE groups; MATROIDS; BOREL sets; MAXIMAL subgroups
- Publication
International Journal of Mathematics, 2022, Vol 33, Issue 4, p1
- ISSN
0129-167X
- Publication type
Article
- DOI
10.1142/S0129167X22500288