RSK Insertion for Set Partitions and Diagram Algebras

2005/07/01 by Tom Halverson, Halverson, Tom, Tim Lewandowski +1 · 2 citations
Mathematics · #05E10 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:05E10

paper · pdf · doi:10.48550/arxiv.math/0507026

24 pages

arxiv created 2005/07/01 · arxiv updated 2009/12/01

Abstract

We give combinatorial proofs of two identities from the representation theory of the partition algebra C A_k(n), n ≥ 2k. The first is n^k = ∑_λf^λm_k^λ, where the sum is over partitions λ of n, f^λ is the number of standard tableaux of shape λ, and m_k^λ is the number of "vacillating tableaux" of shape λ and length 2k. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is B(2k) = ∑_λ(m_k^λ)², where B(2k) is the number of set partitions of \1, >..., 2k\. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.

Cited by

• Half-Diagrams in Partition Algebras: A Geometric Perspective on Multiplicities
• Semi-simple partition algebras as centralizers of representations for rook monoids

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