Question

Let $S$ be a finite set. Now $\mathop{End}(S)$ is a monoid, and we may build a ring $R$ by allowing formal sums of functions.

Preliminary questions, since $R$ is surely well-known: What is it called? In a general category, what is the name of the construction that builds a ring out of the endomorphisms of an object?

My real question is about $A = R \otimes \mathbb{C}$. Restricting to the subalgebra generated by the automorphisms of $S$, it is clear that understanding the representation theory of symmetric groups is a prerequisite for understanding $A$. Are the symmetric groups on sets of size smaller than $| S |$ responsible for all nontrivial properties of $A$? If so, what suitably strong formulation of inclusion/exclusion is at work?

Answer 1

The monoid of all maps on $n$ letters is denoted $T_n$ and called the full transformation monoid. Your intuition is both right and wrong. The irreducible representations of $T_n$ are in bijection with irreducible representations of all symmetric groups of degree at most $n$. The character table is block upper triangular with diagonal blocks character tables of symmetric groups of degree at most $n$. Thus inverting the table, which is how you decompose characters into irreducibles, is a sort of generalized inclusion-exclusion. Putcha computed the character table, although I think Zuckermann stated the result without complete proof long ago. But this is only the semisimple part of the story.

The algebra of $T_n$ is not semisimple. For example, nobody knows what the projective indecomposables look like in general. The algebra is quasihereditary and so has finite global dimension. Putcha computed the quiver. Ponizovsky showed $T_3$ has finite representation type, Ringel showed $T_4$ has finite representation type and Putcha showed the representation type is infinite in all other cases. I think this part of the representation theory is not completely controlled by symmetric groups.