The nullcone in the multi-vector representation of the symplectic group and related combinatorics

TL;DR

This paper investigates the nullcone in the multi-vector representation of the symplectic group, revealing its algebraic structure, toric degeneration, and standard monomial theory through combinatorial and geometric methods.

Contribution

It introduces a novel combinatorial and geometric framework for understanding the nullcone in symplectic group representations, including a toric degeneration and standard monomial theory.

Findings

Describes the algebra over a distributive lattice structure

Establishes a toric degeneration of the nullcone

Develops a standard monomial theory using double tableaux

Abstract

We study the nullcone in the multi-vector representation of the symplectic group with respect to a joint action of the general linear group and the symplectic group. By extracting an algebra over a distributive lattice structure from the coordinate ring of the nullcone, we describe a toric degeneration and standard monomial theory of the nullcone in terms of double tableaux and integral points in a convex polyhedral cone.