A Robinson-Schensted Correspondence for Partial Permutations

Abstract

We study the Steinberg variety associated to matrix Schubert varieties, and develop a Robinson-Schensted type correspondence, $\tau \leftrightarrow (\Lambda ,\mathsf{Q},\mathsf{P})$. Here $\tau$ is a partial permutation of size $p\times q$, $\Lambda$ an admissible signed Young diagram of size $p+q$, and $\mathsf{P}$ (resp. $\mathsf{Q}$) a standard Young tableau of size $p$ (resp. $q$) whose shape is determined by $\Lambda$. By embedding the matrix Schubert variety into a Schubert variety, we find a close relationship between the combinatorics of the classical Robinson-Schensted-Knuth correspondence and our bijection. We also show that an involution $(\Lambda ,\mathsf{Q},\mathsf{P})\mapsto ({\Lambda }^{\vee },\mathsf{P},\mathsf{Q})$ corresponds to projective duality on matrix Schubert varieties.

20 pages, 3 figures