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

A (semi)brick over an algebra is a module S such that its endomorphism ring EndA(S) (product of) division algebra. For each Dynkin diagram Δ, there bijection from the Coxeter group W of type Δ to set semibricks preprojective Π which restricted join-irreducible elements bricks Π. This paper devoted giving explicit description these bijections in case Δ=An or Dn. First, for element w∈W, we describe corresponding brick S(w) terms “Young diagram-like” notation. Next, determine canonical join representation w=⋁i=1mwi arbitrary w∈W based on Reading's work, and prove ⨁i=1mS(wi) semibrick w.