Rigidity of Bott-Samelson-Demazure-Hansen variety for \(PSO(2n+1, \mathbb{C})\)

Let \(G=PSO(2n+1, \mathbb{C}) (n \ge 3)\) and \(B\) be the Borel subgroup of \(G\) containing maximal torus \(T\) of \(G.\) Let \(w\) be an element of Weyl group \(W\) and \(X(w)\) be the Schubert variety in the flag variety \(G/B\) corresponding to \(w.\) Let \(Z(w, \underline{i})\) be the Bott-Sam...

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Bibliographic Details
Published in:arXiv.org 2019-08
Main Authors: Kannan, S Senthamarai, Saha, Pinakinath
Format: Article
Language:English
Subjects:
Homology
Modules
Subgroups
Theorems
Toruses
Online Access:Get full text
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Summary:Let \(G=PSO(2n+1, \mathbb{C}) (n \ge 3)\) and \(B\) be the Borel subgroup of \(G\) containing maximal torus \(T\) of \(G.\) Let \(w\) be an element of Weyl group \(W\) and \(X(w)\) be the Schubert variety in the flag variety \(G/B\) corresponding to \(w.\) Let \(Z(w, \underline{i})\) be the Bott-Samelson-Demazure-Hansen variety (the desingularization of \(X(w)\)) corresponding to a reduced expression \(\underline{i}\) of \(w.\) In this article, we study the cohomology modules of the tangent bundle on \(Z(w_{0}, \underline{i}),\) where \(w_{0}\) is the longest element of the Weyl group \(W.\) We describe all the reduced expressions of \(w_{0}\) in terms of a Coxeter element such that all the higher cohomology modules of the tangent bundle on \(Z(w_{0}, \underline{i})\) vanish (see Theorem \ref{theorem 8.1}).
ISSN:2331-8422