Classic and mirabolic Robinson-Schensted-Knuth correspondence for partial flags

TL;DR

This paper extends the Robinson-Schensted-Knuth correspondence to partial flags and uses it to generalize the mirabolic correspondence involving two partial flags and a line, advancing combinatorial and geometric understanding.

Contribution

It generalizes classical results relating flags and irreducible components to partial flags and extends the mirabolic RSK correspondence accordingly.

Findings

Generalization of the flag position and irreducible component relation to partial flags

Extension of the mirabolic RSK correspondence to partial flags and a line

New combinatorial-geometric connections in flag varieties

Abstract

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.