Higher-order congruence relations on affine moment graphs: The subgeneric case

TL;DR

This paper investigates the structure algebra of the stable moment graph for affine root system A1, introducing higher-order congruence relations that generalize traditional moment graph relations and analyzing the module structure.

Contribution

It constructs a basis for the structure algebra and introduces higher-order congruence relations generalizing moment graph relations for affine root system A1.

Findings

The structure algebra $\\mathcal{Z}$ is a module over a symmetric algebra.

A basis for the module $\\mathcal{Z}$ is constructed.

Higher-order congruence relations generalize ordinary moment graph relations.

Abstract

We study the structure algebra $\mathcal{Z}$ of the stable moment graph for the case of the affine root system $A_{1}$. The structure algebra $\mathcal{Z}$ is an algebra over a symmetric algebra and in particular, it is a module over a symmetric algebra. We study this module structure on $\mathcal{Z}$ and we construct a basis. By "setting $c$ equal to zero" in $\mathcal{Z}$, we obtain the module $\mathcal{Z}_{c=0}$. This module can be described in terms of the finite root system $A_{1}$ and we show that it is determined by a set of certain divisibility relations. These relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs, and because of this we call them higher-order congruence relations.