Axel Hultman

Criteria for rational smoothness of some symmetric orbit closures
dmtcs:2852 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

https://doi.org/10.46298/dmtcs.2852
Criteria for rational smoothness of some symmetric orbit closures
Authors: Axel Hultman ^{1}
NULL
Axel Hultman
1 Department of Mathematics
Let $G$ be a connected reductive linear algebraic group over $ℂ$ with an involution $θ$ . Denote by $K$ the subgroup of fixed points. In certain cases, the $Korbits$ in the flag variety $G/B$ are indexed by the twisted identities $ι (θ ) = {θ (w^{1})w  w∈W}$ in the Weyl group $W$. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph'' whose vertices form a subset of $ι (θ )$. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on $ι (θ )$ is rank symmetric. In the special case $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one'', needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.