Irreducible representations of the Chinese monoid

All irreducible representations of the Chinese monoid CnCn, of any rank n, over a nondenumerable algebraically closed field K  , are constructed. It turns out that they have a remarkably simple form and they can be built inductively from irreducible representations of the monoid C2C2. The proof shows also that every such representation is monomial. Since CnCn embeds into the algebra K[Cn]/J(K[Cn])K[Cn]/J(K[Cn]), where J(K[Cn])J(K[Cn]) denotes the Jacobson radical of the monoid algebra K[Cn]K[Cn], a new representation of CnCn as a subdirect product of the images of CnCn in the endomorphism algebras of the constructed simple modules follows.