The singular locus of a Schubert variety
The authors describe completely the singular loci of Schubert varieties in flag manifolds. Such Schubert varieties are indexed by the symmetric group \(\mathfrak S_n\). The singular locus of a Schubert variety \(X_\mu\) corresponding to an element \(\mu\in \mathfrak S_n\) is the union of smaller Schubert varieties \(X_\nu\) where \(\nu\) runs over a set of permutations \(\text{Sg}(\mu)\). The maximal elements of \(\text{Sg}(\mu)\) correspond to the irreducible components of \(X_\mu\). The main result of the article is to characterize the maximal elements of this set in terms of certain planar representations. It is shown that the maximal elements fall into three types, each of them being invariant under inverses, and under conjugation by the longest element of \(\mathfrak S_n\). With the same techniques the authors prove a conjecture of \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, 45-52 (1990; Zbl 0714.14033)] stating that a certain subset \(Z_\mu\) of \(\mathfrak S_n\) has the same maximal elements as \(\text{Sg}(\mu)\). \textit{V. Gasharov} [Compos. Math. 126, 47-56 (2001; Zbl 0983.14038)] verified that the singuarity condition of Lakshmibai and Sandhya was necessary. In this article it is also proved that it is sufficient. The singularities of Schubert varieties in flags have been much studied recently [see \textit{S. Billey} and \textit{V. S. Lakshmibai}, ``Singular loci of Schubert varieties'', Prog. Math. 182 (2000; Zbl 0959.14032)]. Independently the main results of this article have been proved \textit{S. C. Billey} and \textit{G. S. Warrington} [Trans. Am. Math. Soc. 355, 3915-3945 (2003; Zbl 1037.14020)], and by \textit{L. Manivel} [Int. Math. Res. Not. 2001, 849-871 (2001; Zbl 1023.14022)].
scientific article; zbMATH DE number 2005901