Irreducible representations of the plactic algebra of rank four

Irreducible representations of the plactic monoid M of rank four are studied. Certain concrete families of simple modules over the plactic algebra K[M] over a field K are constructed. It is shown that the Jacobson radical J(K[M]) of K[M] is nilpotent. Moreover, the congruence ρ on M determined by J(K[M]) coincides with the intersection of the congruences determined by the primitive ideals of K[M] corresponding to the constructed simple modules. In particular, M/ρ is a subdirect product of the images of M in the corresponding endomorphism algebras.