Irreducible representations of the Chinese monoid

TL;DR

This paper classifies all irreducible representations of the Chinese monoid over an algebraically closed field, showing they are simple, monomial, and can be constructed inductively from the case n=2.

Contribution

It provides a complete construction and classification of irreducible representations of the Chinese monoid for any rank n, revealing their simple, monomial structure and inductive nature.

Findings

All irreducible representations are constructed explicitly.

Representations are monomial and can be built from those of C_2.

The Chinese monoid embeds into a subdirect product of endomorphism algebras.

Abstract

All irreducible representations of the Chinese monoid $C_n$, 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 $C_2$. The proof shows also that every such representation is monomial. Since $C_n$ embeds into the algebra $K[C_n]/J(K[C_n])$, where $J(K[C_n])$ denotes the Jacobson radical of the mooned algebra $K[C_n]$, a new representation of $C_n$ as a subdirect product of the images of $C_n$ in the endomorphism algebras of the constructed simple modules follows.