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 k B < ∞. For each 1 ≤ m ≤ n, the collection of all m-dimensional subalgebras of B carries the structure of a projective variety, which we call AlgGr m (B). The group Aut k (B) of all k-algebra automorphisms of B acts regularly on AlgGr m (B). In this paper, we study the problem of explicitly describing AlgGr m (B), and classifying its Aut k (B)-orbits. Inspired by recent results from [22], we compute the homogeneous vanishing ideal of AlgGr n−1 (B) when B is basic, and explictly describe its irreducible components. We show that in this case, AlgGr n−1 (B) is a finite union of 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.