Planar rook algebra with colors and Pascal's simplex
Sarah Mousley, Nathan Schley, Amy Shoemaker

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 to be the set of all diagrams consisting of two rows of vertices with edges, each colored with an element in a set of 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 . We show that the Bratteli diagram of is Pascal's -simplex, and use this to provide an alternative proof of the well-known recursive formula for multinomial coefficients.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Combinatorial Mathematics · graph theory and CDMA systems · Mathematical Dynamics and Fractals
