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

Let $I$ be a finite set and $\CI$ be the algebra of functions on $I$. For a finite dimensional $\C$ algebra $A$ with $\CI \subset A$ 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, \cite[Theorem 2.20]{L1}. Savage and Tingley raised the question \cite[Remark 4.5]{ST} 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.

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Published 1 January 2010