Automorphism Groups of Finite-Dimensional Algebras Acting on Subalgebra Varieties

Abstract

Let $k$ be an algebraically-closed field, and $B$ a unital, associative $k$-algebra with $n := \dim_kB < \infty$. For each $1 \le m \le n$, the collection of all $m$-dimensional subalgebras of $B$ carries the structure of a projective variety, which we call $\operatorname{AlgGr_m(B)}$. The group $\operatorname{Aut}_k(B)$ of all $k$-algebra automorphisms of $B$ acts regularly on $\operatorname{AlgGr}_m(B)$. In this paper, we study the problem of explicitly describing $\operatorname{AlgGr}_m(B)$, and classifying its $\operatorname{Aut}_k(B)$-orbits. Inspired by recent results on maximal subalgebras of finite-dimensional algebras, we compute the homogeneous vanishing ideal of $\operatorname{AlgGr}_{n-1}(B)$ when $B$ is basic, and explictly describe its irreducible components. We show that in this case, $\operatorname{AlgGr}_{n-1}(B)$ is a finite union of $\operatorname{Aut}_k(B)$-orbits if $B$ is monomial or its Ext quiver is Schur, but construct a class of examples to show that these conditions are not necessary.