The singular locus of Schubert varieties (Q2761856)

The author describes the irreducible components of the singular locus of Schubert varieties in flag manifolds. The full flag variety of flags in a vector space of dimension \(n\) has a cellular decomposition corresponding to Schubert varieties \(X_w\) indexed by the permutations \(\mathfrak S_n\) of \(1,\dots, n\). For a fixed flag \(V_1 \subset \cdots \subset V_n\) the Schubert variety \(X_w\) is given by the flags \(W_1\subset \cdots \subset W_n\) such that \(\dim(W_p\cap V_q)\geq r_w(p,q)\) for \(1\leq p,q\leq n\), where \(r_w(p,q) =\#\{i\leq p, w(i)\leq q\}\). By a result by \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci. Math. Sci. 100, 45--52 (1990; Zbl 0714.14033)] the variety \(X_w\) is smooth if and only if there is no quadruple of integers \(i<j<k<l\) such that there are configurations \(w(l)< w(j)< w(k)< w(i)\) or \(w(k) <w(l)< w(i) <w(j)\). The author describes an explicit process that from minimal configurations of the above type produces a set \(C(w)\) such that the irreducible components of the singular locus of \(X_w\) are exactly the Schubert varieties \(X_v\) such that \(v\in C(w)\). From this result the author can describe the irreducible components of the Schubert varieties in manifolds of not necessarily complete flags.NEWLINENEWLINENEWLINEThe results have been obtained independently by \textit{S. C. Billey} and \textit{G. S. Warrington} [Trans. Am. Math. Soc. 355, 3915--3945 (2003; Zbl 1037.14020)], and by \textit{C. Kassel, A. Lascoux}, and \textit{C. Reutenauer} [Adv. Math. 150, 1--35 (2000; Zbl 0981.15009)].