Combinatorial characterization of the weight monoids of smooth affine spherical varieties

TL;DR

This paper provides a combinatorial characterization of the weight monoids of smooth affine spherical varieties, enhancing understanding of their structure through representation theory and smoothness criteria.

Contribution

It introduces a new combinatorial criterion for identifying the weight monoids of smooth affine spherical varieties, building on Losev's theorem and Camus's smoothness criterion.

Findings

Characterization of weight monoids using combinatorial methods

Application of Camus's smoothness criterion to spherical varieties

Enhanced classification framework for smooth affine spherical varieties

Abstract

Let G be a connected complex reductive group. A well known theorem of I. Losev's says that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.