Irreducible Components of Exotic Springer fibres II: The Exotic Robinson-Schensted Algorithm

TL;DR

This paper introduces a new combinatorial version of the Robinson-Schensted algorithm based on Kato's exotic nilpotent cone, providing a novel bijection that extends classical type A results in a non-trivial way.

Contribution

It presents the exotic Robinson-Schensted bijection, a new combinatorial description using the exotic nilpotent cone, expanding the understanding of geometric correspondences beyond classical type A.

Findings

The exotic Robinson-Schensted bijection differs from the classical one.

Provides a combinatorial framework for the exotic nilpotent cone.

Connects geometric and combinatorial aspects of type C representations.

Abstract

Kato's exotic nilpotent cone was introduced as a substitute for the ordinary nilpotent cone of type C with cleaner properties. The geometric Robinson-Schensted correspondence is obtained by parametrizing the irreducible components of the Steinberg variety (the conormal variety for the action of a semisimple group on two copies of its flag variety); in type A the bijection coincides with the classical Robinson-Schensted algorithm for the symmetric group. Here we give a combinatorial description of the bijection obtained by using the exotic nilpotent cone instead of ordinary type C nilpotent cone in the geometric Robinson-Schensted correspondence; we refer this as the "exotic Robinson-Schensted bijection". This is interesting from a combinatorial perspective, and not a naive extension of the type A Robinson-Schensted bijection.