Let $S$ be a finite set. Let $R$ be a complex vector space with basis indexed by subsets of $S$. Define a product on $R$ by defining it on the basis elements as $1\_A\cdot 1\_B=1\_{A\Delta B}$, where $A\Delta B$ is the symmetric difference of $A$ and $B$. This gives $R$ the structure of a commutative and associative $C$-algebra.


Is this a well-understood algebra? Does it have a name?



An algebra constructed from symmetric differences
It is the complex group algebra over $((\mathbb{Z}/2)^S,+)$. This may be also described as the tensor product of $S$ copies of $\mathbb{C}[\mathbb{Z}/2] = \mathbb{C} \times \mathbb{C}$.


