## Maximal Singular Loci of Schubert Varieties in SL(n)/B

### Sara C. Billey Gregory S. Warrington

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001

## Abstract

In this paper we give an explicit combinatorial description of the maximal singular locus maxsing(X_w) of a Schubert variety X_w for an element w \in S_n. With our description, the computation of maxsing(X_w) becomes computationally efficient (O(n^6)). The key to our result is a map R(y,w) --> R(yt,w) where yt < y < w, t is a reflection, and R(y,w) is the set of reflections t' such that y < yt' \leq w. Our result depends on a characterization of maxsing(X_w) in terms of the cardinality of R(y,w) due to Lakshmibai and Seshadri.

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