Combinatorial Results on (1, 2, 1, 2)-Avoiding GL(p, ℂ) × GL(q, ℂ)-Orbit Closures on GL(p+q, ℂ)/B. (3rd April 2015) - Woo, Alexander; Wyser, Benjamin J.
Abstract : Using recent results of the second author which explicitly identify the "(1, 2, 1, 2) -avoiding" GL(p, C) × GL(q, C) -orbit closures on the flag manifold GL(p+q, C)/B as certain Richardson varieties, we give combinatorial criteria for determining smoothness, lci-ness, and Gorensteinness of such orbit closures. (In the case of smoothness, this gives a new proof of a theorem of W.M. McGovern.) Going a step further, we also describe a straightforward way to compute the singular locus, the non-lci locus, and the non-Gorenstein locus of any such orbit closure. We then describe a manifestly positive combinatorial formula for the Kazhdan–Lusztig–Vogan polynomial P_τ, γ (q) in the case where γ corresponds to the trivial local system on a (1, 2, 1, 2) -avoiding orbit closure Q and τ corresponds to the trivial local system on any orbit Q' contained in Q . This combines the aforementioned result of the second author, results of Knutson et al., and a formula of Lascoux and Schützenberger which computes the ordinary (type A ) Kazhdan–Lusztig polynomial P_x, w(q) whenever w S_n is cograssmannian.