Preprojective partitions and the determinant of the Hom matrix

Igusa, K.; Todorov, G.

doi:10.1090/S0002-9939-1985-0784160-0

Preprojective partitions and the determinant of the Hom matrix
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Proc. Amer. Math. Soc. 94 (1985), 189-197 Request permission

Abstract:

If $\Lambda$ is an artin algebra and $\Lambda$ is the set of isomorphism classes of indecomposable finitely generated $\Lambda$-modules, then there is a partition $\operatorname {ind}\Lambda = { \cup _{i \geqslant 0}}{\underline {\underline P} _i}$, called the preprojective partition. We give an algorithm for computing this partition, which is given only in terms of numerical properties of the Auslander-Reiten quiver of $\Lambda$. If $\Lambda$ is of finite representation type, then there are two essentially different proofs that the matrix $\operatorname {Hom} = ({\text {lengt}}{{\text {h}}_{{\text {End(}}N)/{\text {rEnd(}}N{\text {)}}}}\operatorname {Hom}_{\Lambda }(M,N))$, where $M,n \in \operatorname {ind}\Lambda$ has determinant $+ 1[{\mathbf {IT1}},{\mathbf {W1}},{\mathbf {Z1}}]$. We show that the paths between the Hom matrix and the identity matrix in $\operatorname {GL}_{n}(\mathbf {R})$ given by these two proofs are homotopic.

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