On representation schemes and Grassmanians of finite dimensional algebras and a construction of Lusztig

TL;DR

This paper proves that certain moduli spaces of finite-dimensional modules over algebras are isomorphic to Grassmannian varieties, resolving an open question about the isomorphism of Lusztig's related varieties in representation theory.

Contribution

It establishes the isomorphism between Lusztig's two varieties, confirming a long-standing open question in the representation theory of finite-dimensional algebras.

Findings

Moduli spaces of modules are isomorphic to Grassmannian varieties.

Lusztig's two varieties are proven to be isomorphic as algebraic varieties.

Addresses an open problem raised by Savage and Tingley.

Abstract

Let I be a finite set and CI be the algebra of functions on I. For a finite dimensional C algebra A with \CI contained in A we show that certain moduli spaces of finite dimsional modules are isomorphic to certain Grassmannian (quot-type) varieties. There is a special case of interest in representation theory. Lusztig defined two varieties related to a quiver and gave a bijection between their C-points (citation in article). Savage and Tingley raised the question (citation in article) of whether these varieties are isomorphic as algebraic varieties. This question has been open since Lusztig's original work. It follows from the result of this note that the two varieties are indeed isomorphic.