Mirabolic Robinson-Schensted-Knuth correspondence

TL;DR

This paper introduces a bijective Mirabolic RSK correspondence linking decorated permutations to triples of Young tableaux and partitions, providing a combinatorial classification of certain orbits and conjectural connections to representation theory.

Contribution

It develops a new Mirabolic RSK correspondence and explores its implications for orbit classification and connections to Kazhdan-Lusztig theory.

Findings

Partition of orbits into combinatorial cells

Equivalence of orbit partition and conormal vector types

Conjectural links to bimodule Kazhdan-Lusztig cells

Abstract

The set of orbits of $GL(V)$ in $Fl(V)\times Fl(V)\times V$ is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of $GL(V)$ arising from $Fl(V)\times Fl(V)\times V$. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on $GL(V)\times V$.