Abstract
A Gelfand model for a semisimple algebra $\mathsf{A}$ over $\mathbb{C}$ is a complex linear representation that contains each irreducible representation of $\mathsf{A}$ with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via ``signed conjugation" on the linear span of their vertically symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group, and, in fact, our method is to use the Jones basic construction to lift the Saxl model from the symmetric group to each diagram algebra. In the case of the planar diagram algebras, our construction exactly produces the irreducible representations of the algebra.
Highlights
A famous consequence of Robinson-Schensted-Knuth (RSK) insertion is that the set of standard Young tableaux with k boxes is in bijection with the set of involutions in the symmetric group Sk
These standard Young tableux index the bases for the irreducible CSk modules, so it follows that the sum of the degrees of the irreducible Sk modules equals the number of involutions in Sk
A consequence [?, (5.5)] of this algorithm is that the sum of the degrees of the irreducible representations of each of these algebras equals the number of horizontally symmetric basis diagrams in the algebra
Summary
The ideal Jk−1 is in Schur-Weyl duality with one of Ak−1 or Ak−2 (depending on the specific diagram algebra) In this setup, we are able to take a model for each Cr, 0 ≤ r ≤ k, and lift them to a module for Ak. For the planar diagram algebras – the Temperley-Lieb, Motzkin, and planar rook monoid algebras – the algebra C ∼= C1k is trivial and the model is trivial. For the planar diagram algebras – the Temperley-Lieb, Motzkin, and planar rook monoid algebras – the algebra C ∼= C1k is trivial and the model is trivial It follows that MrAk is irreducible and that signed conjugation produces a complete set of irreducible modules for the planar algebras. In practice, the model provides a natural and easy way to compute the explicit action of basis diagrams on irreducible representations. (3) Gelfand models are useful in the study of Markov chains on related combinatorial objects; see, for example, Chapter 3F of [?] and the references therein, as well as [?], [?]
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