The Planar Rook Algebra and Pascal's Triangle

TL;DR

This paper explores the combinatorial representation theory of the planar rook algebra, revealing that its Bratteli diagram forms Pascal's triangle and connecting binomial identities to its representations.

Contribution

It constructs irreducible representations of the planar rook algebra and demonstrates that its Bratteli diagram is Pascal's triangle, linking algebraic structure to combinatorial identities.

Findings

Bratteli diagram of $P_n$ is Pascal's triangle

Explicit decomposition of regular representation into irreducibles

Binomial identities interpreted through representation theory

Abstract

We study the combinatorial representation theory of the ``planar rook algebra" $P_n$. This algebra has a basis consisting of planar rook diagrams and multiplication given by diagram concatenation. For each integer $0 \le k \le n$, we construct natural representations $V^n_k$ which form a complete set of non-isomorphic, irreducible $P_n$-representations. We explicitly decompose the regular representation of $P_n$ into a direct sum of irreducible modules. We compute the Bratteli diagram for the tower of algebras $P_0 \subseteq P_1 \subseteq P_2 \subseteq ...$ and show that this Bratteli diagram is Pascal's triangle. In fact, we show that many of the binomial identities, both additive and multiplicative, have interpretations in terms of the representation theory of the planar rook algebra.