Planar rook algebra with colors and Pascal's simplex

TL;DR

This paper introduces a colored diagram algebra called $P_{n,c}$, analyzes its irreducible representations, and links its structure to Pascal's $(c+1)$-simplex, providing new insights into multinomial coefficients.

Contribution

It defines the planar rook algebra with colors, finds its irreducible representations, and connects its Bratteli diagram to Pascal's simplex, offering a novel algebraic perspective.

Findings

Complete set of irreducible representations of $ ext{C} P_{n,c}$

Bratteli diagram of $ ext{C} P_{n,c}$ is Pascal's $(c+1)$-simplex

Provides an alternative proof of multinomial coefficient recursion

Abstract

We define $P_{n,c}$ to be the set of all diagrams consisting of two rows of $n$ vertices with edges, each colored with an element in a set of $c$ possible colors, connecting vertices in different rows. Each vertex can have at most one edge incident to it, and no edges of the same color can cross. In this paper, we find a complete set of irreducible representations of $\C P_{n,c}$. We show that the Bratteli diagram of $\C P_{0,c} \subseteq \C P_{1,c} \subseteq \C P_{2,c} \subseteq...$ is Pascal's $(c+1)$-simplex, and use this to provide an alternative proof of the well-known recursive formula for multinomial coefficients.