Fixed points of involutive automorphisms of the Bruhat order

Details Fixed points of involutive automorphisms of the Bruhat order Fixed points of involutive automorphisms of the Bruhat order

2004-03-04

arxiv.org/abs/math/0403095v2

Mathematics

Combinatorics

16 pages. Appendix added, minor revisions; to appear in Adv. Math

Scientific paper

Applying a classical theorem of Smith, we show that the poset property of being Gorenstein$^*$ over $\mathbb{Z}_2$ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as a Coxeter group in its own right.

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