On the structures of hive algebras and tensor product algebras for general linear groups of low rank

Research output: Contribution to journalArticle

Abstract

The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of GL(n). Using the hive model for the Littlewood-Richardson (LR) coefficients, we provide a finite presentation of the algebra TA(n) for n = 2, 3, 4 in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of LR coefficients.

Original languageEnglish
JournalInternational Journal of Algebra and Computation
DOIs
Publication statusPublished - 2019 Jan 1

Fingerprint

General Linear Group
Littlewood-Richardson Coefficients
Tensor Product
Algebra
Irreducible polynomial
Irreducible Representation
Generating Function
Generator
Decompose
Model

Keywords

  • General linear group
  • highest weight vector
  • Hilbert-Poincaré series
  • hive
  • Littlewood-Richardson coefficients
  • tensor product algebra
  • tensor product decomposition

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "On the structures of hive algebras and tensor product algebras for general linear groups of low rank",
abstract = "The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of GL(n). Using the hive model for the Littlewood-Richardson (LR) coefficients, we provide a finite presentation of the algebra TA(n) for n = 2, 3, 4 in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of LR coefficients.",
keywords = "General linear group, highest weight vector, Hilbert-Poincar{\'e} series, hive, Littlewood-Richardson coefficients, tensor product algebra, tensor product decomposition",
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