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" ${\rm GL}(p, {{ \mathbb C}}) × {\rm GL}(q, {{ \mathbb C}})$ -orbit closures on the flag manifold ${\rm GL}(p+q, {{ \mathbb C}})/B$ 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 $\bar {Q}$ . 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 $w \in S_n$ is cograssmannian.