A criterion for smoothness of Schubert varieties in \(Sp(2n)/B\)
type: Item
instance of: scholarly article
title: A criterion for smoothness of Schubert varieties in \(Sp(2n)/B\)
zbMATH Open document ID: 0891.14015
author: O. Diekmann
published in: Journal of Algebra
publication date: 1998-07-19T00:00:00Z
review text: Let \(k\) be a field, \(V=k^n\), \(B\) the Borel subgroup of \(SL(n)\) consisting of upper triangular matrices. The space \(SL(n)/B\) can be identified with the flag manifold of complete flags in \(V\). The Weyl group in this case is the symmetric group \(S_n\). For each permutation \(w\in S_n\) one defines a Schubert variety \(X(w)\) in \(SL(n)/B\). -- \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45-52 (1990; Zbl 0714.14033)] gave a criterion for smoothness of \(X(w)\) in terms of \(w\). In this paper, the authors consider the same problem for the group \(Sp(2n)\). Let \(V=k^{2n}\) and consider on \(V\) the nondegenerate, skew-symmetric bilinear form \((-,-)\) defined, for \(1\leq i<j\leq 2n\), by \((e_i, e_j)=1\) if \(i+j= 2n+1\) and 0 otherwise, where \(e_1, \dots, e_{2n}\) is the canonical basis of \(V\). \(Sp(2n)\) is, by definition, the subgroup of \(SL(2n)\) consisting of the matrices which invariate \((-,-)\). Alternatively, \(Sp(2n)= SL(2n)^\sigma\) where \(\sigma\) is the involution defined by \(\sigma (A)= E(^tA)^{-1} E^{-1}\), where \(E\) is the matrix of \((-,-)\). \(B^\sigma\) is a Borel subgroup of \(Sp(2n)\) and the space \(Sp(2n)/B^\sigma\) can be identified to the variety of complete flags in \(V\) invariant under the involution which associates to each flag its orthogonal with respect to \((-,-)\). The Weyl group, in this case, is the subgroup \(W\) of \(S_{2n}\) consisting on the permutations \(w\) such that \(w(i)+ w(2n+1-i) =2n+1\) for all \(i\). To each \(w\in W\) one associates a Schubert variety \(Y(w)\) in \(Sp(2n)/B^\sigma\). In fact, \(Y(w)= X(w)^\sigma\). The authors give a criterion for \(Y(w)\) to be smooth in terms of \(w\). In particular, they show that \(Y(w)\) is smooth if \(X(w)\) is smooth. The proof is based on the description of the tangent spaces to a Schubert variety obtained by \textit{V. Lakshmibai} and \textit{C. S. Seshadri} [Bull. Am. Math. Soc., New Ser. 11, 363-366 (19…
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14L35
zbMATH DE Number: 1013988
zbMATH Keywords: flag manifold
zbMATH Keywords: Schubert variety
reviewed by: Iustin Coandă
MaRDI profile type: Publication
full work available at URL: jabr.1996.6839
OpenAlex ID: W2068072274
cites work: Q1089060
cites work: Singular loci of Schubert varieties for classical groups
cites work: Tangent spaces to Schubert varieties
cites work: Q3892353
cites work: Criterion for smoothness of Schubert varieties in Sl(n)/B
cites work: Geometry of G/P-II [The work of De Concini and Procesi and the basic conjectures]
cites work: Singular locus of a Schubert variety
cites work: Geometry of G/P. V
cites work: Classical Bruhat orders and lexicographic shellability
DOI: 10.1006/JABR.1996.6839
description: scientific article; zbMATH DE number 1013988