A Robinson-Schensted Correspondence for Partial Permutations

TL;DR

This paper extends the Robinson-Schensted correspondence to partial permutations via matrix Schubert varieties, revealing new combinatorial structures and symmetries related to duality.

Contribution

It introduces a Robinson-Schensted type correspondence for partial permutations and links it to matrix Schubert varieties and projective duality.

Findings

Established a bijection between partial permutations and Young tableaux with signed diagrams.

Connected the new correspondence to classical Robinson-Schensted-Knuth combinatorics.

Linked involutions in the correspondence to projective duality on matrix Schubert varieties.

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.