Exotic Spaltenstein varieties

TL;DR

This paper introduces exotic Spaltenstein varieties, extending classical Springer fibers to a new setting, and establishes a combinatorial description of their top-dimensional components for certain nilpotent endomorphisms.

Contribution

It defines exotic Spaltenstein varieties and provides an explicit bijection between their top components and semi-standard Young bitableaux for order-two nilpotent endomorphisms.

Findings

Top-dimensional components correspond to semi-standard Young bitableaux.

A combinatorial formula for the top dimension is provided.

Conjecture extends the description to arbitrary nilpotent orders.

Abstract

We define a new family of algebraic varieties, called exotic Spaltenstein varieties. These generalise the notion of Spaltenstein varieties (which are the partial flag analogues to classical Springer fibres) to the case of exotic Springer fibres. We show that, for self-adjoint nilpotent endomorphisms of order two, the top-dimensional irreducible components are in bijection with semi-standard Young bitableaux, via constructing an explicit map. Moreover, we are able to give a combinatorial formula for this top dimension. We conjecture that this description of the irreducible components holds for nilpotent endomorphisms of arbitrary order. Finally, we mention some connections to the Robinson-Schensted-Knuth correspondence.