Plactic algebra of rank 3

Abstract

The structure of the algebra K [ M ] of the plactic monoid M of rank 3 over a field K is studied. The minimal prime ideals of K [ M ] are described. There are only two such ideals and each of them is a principal ideal determined by a homogeneous congruence on M . Moreover, in case K is uncountable and algebraically closed, the left and right primitive spectrum and the corresponding irreducible representations of the algebra K [ M ] are described. All these representations are monomial. As an application, a new proof of the semiprimitivity of K [ M ] is given.