Some examples of semigroup algebras of finite representation type

I. S. Ponizovskii

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 8, Tome 160 (1987), pp. 229-238

Résumé

The semigroup algebras over a field $K$ of the semigroups $T_n$ of all permutations of a set of $n$ elements are considered. It is proved: if $n\leq3$ and $(n!)^{-1}\in K$ then the algebra $KT_n$ has a finite representation type. Also the finiteness of the representation type of the semigroup algebra $KS$ is established, where $S$ is the sub-semigroup of $T_n$ ($n$ is arbitrary) such that $S=J_n\cup G$ where $J_n=\{x\in T_n|\operatorname{rank}x=1\}$, while $G$ is a doubly transitive subgroup of the symmetric group $S_n$, the order of $G$ being invertible in $K$.