A Demazure Character Formula for the Product Monomial Crystal

Joel Gibson

doi:10.5802/alco.156

Abstract

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type $A$, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on $\mathbf{R}$.

Highlights

Let G be a semisimple -laced algebraic group over C, for example SLn, SO(2n), or one of the exceptional types E6, E7, E8
The aim of this paper is to study a family of finite dimensional representations of G which appear in the study of slices to Schubert varieties in the affine Grassmannian
In type A we show that these representations are related to generalised Schur modules, and give an explicit realisation for the crystal of a generalised Schur module

Summary

Introduction

Let G be a semisimple -laced algebraic group over C, for example SLn (type An−1), SO(2n) (type Dn), or one of the exceptional types E6, E7, E8. In [9] the authors initiate a program to construct quantisations of Wλμ, and study the representation theory of the resulting algebras These algebras are called truncated shifted Yangians and denoted Yμλ(R). Modules over truncated shifted Yangians naturally afford a highest weight theory, leading to a category O(Yμλ(R)), which is the “algebraic category O” in the sense of [1] This category plays an important role in the “symplectic duality” program of Braden–Licata–Proudfoot–Webster. Our first main result provides an explicit character formula for the crystal B(R), in terms of multiplications by dominant weights and application of isobaric Demazure operators (Theorem 5.9). Theorem 6.23 implies that in type A, the category V(R) defined by the truncated shifted Yangians are categorifications of generalised Schur modules associated to column-convex diagrams. We note that our result shows that any skew Schur module is categorified by V(R), for some R

Background

Definition of the product monomial crystal

Analysis of the product monomial crystal

Truncations and a character formula

The product monomial crystal in type A