MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Algebra A with Spec(A) reduced and Rep_n(A) non-reduced

As always, corrections to my misconceptions/misstatements are appreciated. This question is related to the following one, but in this question the algebras considered are commutative: http://mathoverflow.net/questions/9738/non-smooth-algebra-with-smooth-representation-variety

The "commuting scheme" X is defined intuitively as "pairs of nxn commuting matrices." More precisely, it's the subscheme of the affine space $M_n(\mathbb{C})\times M_n(\mathbb{C})$ defined by the equations in the entries of the matrices corresponding to the matrix equations "XY = YX," which are $n^2$ homogenous equations of degree n. To the best of my knowledge, the question of whether X is reduced is an old problem that is still open.

The commuting scheme is naturally isomorphic to $Rep^n_\mathbb{C}(\mathbb{C}[x,y])$. This leads to my question: Is there an algebra A such that $Spec(A)$ is reduced but $Rep_\mathbb{C}^n(A)$ is not reduced? Even stronger, is there an A with $Spec(A)$ smooth and $Rep_\mathbb{C}^n(A)$ non-reduced? My guess would be yes for both, but I'm not sure how to find one for either one.

-