Mathematics – Representation Theory
Scientific paper
2008-06-24
Mathematics
Representation Theory
14 pages
Scientific paper
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.
Flath Daniel
Halverson Tom
Herbig Kathryn
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