The representation type of the full transformation semigroup \(T_4\)
Let \(K\) be field of characteristic zero, let \(T_n\) be the full transformation semigroup on the set \(\{1,\dots,n\}\), and let \(KT_n\) be the semigroup algebra of \(T_n\) over \(K\). The reviewer showed that \(KT_n\) is of finite representation type for \(n\leq 3\); on the other hand Putcha showed that \(KT_n\) is of infinite representation type for \(n\geq 5\). In the present paper the only remaining case is settled: the algebra \(KT_4\) turns out to be of finite representation type. The quiver \(Q\) of \(KT_4\) was counted by Putcha. The author calculates the relations for \(Q\). With this at hand, the author proves the main theorem stating that the algebra \(KT_4\) is Morita equivalent to the path algebra of \(Q\) modulo three relations. Corollary 1 states that \(KT_4\) is a string algebra with 36 isomorphism classes of indecomposable left modules; particularly, \(KT_4\) is of finite representation type. Some remarkable properties of \(KT_4\) are also discussed.
scientific article; zbMATH DE number 1545254