Bricks over preprojective algebras and join-irreducible elements in Coxeter groups

TL;DR

This paper explicitly describes the bijections between join-irreducible elements of Coxeter groups and bricks over preprojective algebras for types A and D, using combinatorial and representation-theoretic methods.

Contribution

It provides explicit combinatorial descriptions of these bijections and the canonical join representations for types A and D Coxeter groups.

Findings

Explicit description of bricks for join-irreducible elements in types A and D.

Connection between canonical join representations and semibricks.

Use of Young diagram-like notation for describing bricks.

Abstract

A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $\Delta$, there is a bijection from the Coxeter group $W$ of type $\Delta$ to the set of semibricks over the preprojective algebra $\Pi$ of type $\Delta$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $\Pi$. This paper is devoted to giving an explicit description of these bijections in the case $\Delta=\mathbb{A}_n$ or $\mathbb{D}_n$. First, for each join-irreducible element $w \in W$, we describe the corresponding brick $S(w)$ in terms of "Young diagram-like" notation. Next, we determine the canonical join representation $w=\bigvee_{i=1}^m w_i$ of an arbitrary element $w \in W$ based on Reading's work, and prove that $\bigoplus_{i=1}^n S(w_i)$ is the semibrick corresponding to $w$.