Abstract
Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph.
Highlights
Schur–Weyl duality forms an archetypal situation in combinatorial representation theory involving two actions that complement each other
The general linear group GLn(C) of n × n matrices over the field C of complex numbers acts on the tensor space V ⊗k of an n dimensional vector space V, and its centralizer algebra is the symmetric group Sk on the k tensor factors
1. the Brauer algebra is the centralizer of the symplectic and orthogonal groups acting on the tensor space (Cn)⊗k [9]; 2. the Temperley-Lieb algebra is the centralizer of the special linear Lie group of degree two acting on the tensor space (C2)⊗k [14]; 3. the partition algebra is the centralizer of the symmetric group acting on the tensor space V ⊗k of its permutation representation V [13]
Summary
Schur–Weyl duality forms an archetypal situation in combinatorial representation theory involving two actions that complement each other. M ⊗k ∼= Gλ ⊗ Zkλ as a (G, Zk)-bimodule λ the electronic journal of combinatorics 26(2) (2019), #P2.12 where the Gλ are irreducible G-modules and the Zkλ are irreducible Zk-modules This bimodule decomposition intimately relates the irreducible representations of G with the irreducible representations of Zk. In the classical situation, the general linear group GLn(C) of n × n matrices over the field C of complex numbers acts on the tensor space V ⊗k of an n dimensional vector space V , and its centralizer algebra is the symmetric group Sk on the k tensor factors. The partition algebra is the centralizer of the symmetric group acting on the tensor space V ⊗k of its permutation representation V [13]. As opposed to the representation theory of the symmetric group, they depend on the embedding of Un−1 in Un. We use the branching rules to create a graph that encodes the decomposition of V ⊗k known as the Bratteli diagram. The shell combinatorics developed from this paper may help compute in other algebraic structures related to the supercharacter theory of Un, such as the Hopf algebra of symmetric functions in noncommuting variables
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