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Ribbon Hopf algebras from group character rings


Fauser, B and Jarvis, PD and King, RC, Ribbon Hopf algebras from group character rings, Linear and Multilinear Algebra, 62, (6) pp. 749-775. ISSN 0308-1087 (2014) [Refereed Article]

Copyright Statement

Copyright 2014 Taylor and Francis

DOI: doi:10.1080/03081087.2013.790385


We study the diagram alphabet of knot moves associated with the character rings of certain matrix groups. The primary object is the Hopf algebra Char-GL of characters of the finite dimensional polynomial representations of the complex group GL(n) in the inductive limit, realized as the ring of symmetric functions (X) on countably many variables X = {x1, x2, . . .}. Isomorphic as spaces are the character rings Char-O and Char-Sp of the classical matrix subgroups of GL(n), the orthogonal and symplectic groups. We also analyse the formal character rings Char-Hπ of algebraic subgroups of GL(n), comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type π, which have been introduced earlier (Fauser et al.) (these include the orthogonal and symplectic groups as special cases). The set of tangle diagrams encoding manipulations of the group and subgroup characters has many elements deriving from products, coproducts, units and counits as well as different types of branching operators. From these elements we assemble for each π a crossing tangle which satisfies the braid relation and which is nontrivial, in spite of the commutative and co-commutative setting. We identify structural elements and verify the axioms to establish that each Char-Hπ ring is a ribbon Hopf algebra. The corresponding knot invariant operators are rather weak, giving merely a measure of the writhe.

Item Details

Item Type:Refereed Article
Keywords:symmetric functions, group character ring, Hopf algebra, group branching, knot invariant
Research Division:Mathematical Sciences
Research Group:Pure mathematics
Research Field:Group theory and generalisations
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the mathematical sciences
UTAS Author:Jarvis, PD (Dr Peter Jarvis)
ID Code:98780
Year Published:2014
Web of Science® Times Cited:1
Deposited By:Mathematics and Physics
Deposited On:2015-03-02
Last Modified:2017-11-01

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