Coxeter matrices and homological quadratic forms of $\mathbf{n}$-hereditary algebras

Abstract

We study the Coxeter matrices and the homological quadratic forms of $n$-hereditary algebras within the framework of higher dimensional Auslander--Reiten theory. Let $\Lambda$ be a finite dimensional $n$-hereditary algebra with the Coxeter matrix $\Phi$ and the homological quadratic form $\chi$. We prove that if $\Lambda$ is $n$-representation finite, then there exists a positive integer $d$ such that ${\Phi }^d=1$. In the case $n$ is an odd number, we show that if there exists a positive integer $d$ such that ${\Phi }^d=1$, then $\Lambda$ is $n$-representation finite. Let ${\mathcal{C}}^0$ be the subcategory of $\moddΛ$ which is a higher analogue of the module category in the context of higher dimensional Auslander--Reiten theory. We introduce a Grothendieck group ${\mathrm{K}}_0({\mathcal{C}}^0)$ associated with ${\mathcal{C}}^0$ and show that it is isomorphic to the Grothendieck group of $\Lambda$. We further prove that if the restriction of $\chi$ to $\mathrm{K}_0(\mcc^0)$ is positive definite, then $\Lambda$ is $n$-representation finite for odd $n$. To prove these results, we first show that indecomposable $n$-preprojective and $n$-preinjective modules are uniquely determined up to isomorphism by their dimension vectors for odd $n$. We also provide examples of $n$-representation finite algebras that the restriction of $\chi$ to $\mathrm{K}_0(\mcc^0)$ is not positive definite.