A criterion for smoothness of Schubert varieties in \(Sp(2n)/B\)
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 (1984; Zbl 0549.14016)].
scientific article; zbMATH DE number 1013988