# Quiver and relations for a monoid related to Catalan numbers

Let $$C_n$$ be the monoid consisting of monotone maps $$\{1,...,n\} \rightarrow \{1,...,n\}$$ with $$f(i) \leq i$$ for all $$i$$. The cardinality of $$C_n$$ is given by the Catalan numbers.

Consider $$A_n= \mathbb{C} C_n$$ , the monoid algebra over the complex numbers.

Some experiments suggest that $$A_n$$ is (isomorphic to) a quiver algebra $$KQ/I$$ with admissible ideal $$I$$ having $$2^{n-1}$$ simple modules. (It seems $$A_n/rad(A_n) \cong \mathbb{C}^{2^{n-1}}$$, which would imply this. This is true for $$n \leq 5$$.)

Question: Is such an explicit description via quiver and admissible relations known for this monoid algebra?

Answers for small $$n$$ would also be interesting. For $$n=2$$ the algebra is isomorphic to $$\mathbb{C} \times \mathbb{C}$$.

This monoid is called the Catalan monoid $$C_n$$ and all the things you wish to know are well understood. It was first observed by Hivert and Thiery that its algebra is isomorphic to the incidence algebra of the Grassmann poset $$P_{n-1}$$. The easiest proof is in my unpublished paper with Margolis, https://arxiv.org/abs/1806.06531, which shows the isomorphism already over $$\mathbb Z$$.
The Grassmann poset $$P_n$$ has elements subsets of $$\{1,\ldots, n\}$$ and $$X\leq Y$$ if $$|X|=|Y|$$ and when you list them in order, then $$i^{th}$$ element of $$X$$ is less than or equal to the $$i^{th}$$ element of $$Y$$. For more on the history of this, see the paper.
So the isomorphism of $$KC_n$$ with the incidence algebra of $$P_{n-1}$$ gives you it is a basic algebra with $$2^{n-1}$$ simples, quiver the Hasse diagram of $$P_{n-1}$$ and relations saying coterminal paths are equal.