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Bricks over preprojective algebras and join-irreducible elements in Coxeter groups. (arXiv:1712.08311v1 [math.RT])
来源于:arXiv
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 $w$. 查看全文>>