Let $G$ be a (discrete, say) group and $\mathbb K$ a field. The *regular representation* $G^{\mathbb K}$ is the vector space of all functions $G \to \mathbb K$. It is a (left, say) $G$-module: given $g\in G$ and $f: G \to \mathbb K$, the action is $g\cdot f: x \mapsto f(xg)$. Then $G^{\mathbb K}$ is a commutative algebra object in $G\text{-rep}\_{\mathbb K}$, the symmetric monoidal category of $\mathbb K$-valued $G$-representations, under pointwise multiplication $f\_1f\_2: x \to f\_1(x)f\_2(x)$.


But the pointwise product is not necessarily the only commutative algebra (in $G\text{-rep}$) structure that can be put on $G^{\mathbb K}$. For example, when $\mathbb K = \mathbb R$ and $G = \mathbb Z/2$, as an algebra $G^{\mathbb K} \cong \mathbb R[\epsilon]/(\epsilon^2 = 1)$, with the $G$-action corresponding to conjugation $\epsilon \mapsto -\epsilon$. The same $G$-module supports the algebra structure $\mathbb R[\epsilon]/(\epsilon^2 = -1) = \mathbb C$, which is patently a different algebra.


My question is whether there are any examples with $\mathbb K = \mathbb C$? I.e.:



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> Does there exist a group $G$ so that there is a commutative algebra object in $G\text{-rep}\_{\mathbb C}$ that is isomorphic to $G^{\mathbb C}$ as a representation but not as an algebra?
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> 
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I believe that any such group must be rather large: in particular, I'm sure that it cannot be finite.



Are there other algebra structures on the regular representation of a group?
**Summary** Yes, there are many such examples, even for finite groups.


**Construction** Let $W<GL(V)$ be a complex reflection group, $A=\mathbb{C}[V]$ be the algebra of polynomial functions on $V$ and $A^W$ be the subalgebra of $W$-invariants. Then by the [Chevalley–Shephard–Todd theorem](http://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem), $A^W$ is a polynomial algebra and $A$ is a free $A^W$-module. This may be viewed as a deformation of $\mathbb{C}^W$ as follows. For any $z\in\text{Spec}A^W,$ consider the fiber $A\_z=A/zA.$ By the freeness property, each $A\_z$ carries the regular representation of $W.$ For a regular $z,$ corresponding to a $W$-orbit of a regular point in $V,$ the algebra $A\_z$ may be identified with the algebra of functions on the orbit, which consists of $|W|$ points; in particular, $A\_z\cong \mathbb{C}^W$ as a $W$-algebra. But for values of $z$ corresponding to singular orbits, algebras $A\_z$ are not reduced. The most singular fiber is $A\_0=\mathbb{C}[V]/(\mathbb{C}[V]^W\_{+})$ and is a graded nilpotent Frobenius algebra, with one-dimensional socle and radical, called the *covariant algebra* of $W.$


**Example** Let $W=\mathbb{Z}\_n$ acting on the one-dimensional vector space with coordinate $x$ by the $n$th roots of unity. Then regular fibers $\mathbb{C}[x]/(x^n-a)$ with $a\ne 0$ are semisimple and isomorphic to $\mathbb{C}^{\mathbb{Z}\_n}$ (explicitly, $x$ is mapped to $b\sum\_k \zeta^{-k}\delta\_k,$ where $b^n=a$), whereas the singular fiber $\mathbb{C}[x]/(x^n)$ is a graded nilpotent algebra.


