Ribbon Hopf algebras from group character rings

Abstract

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 in the inductive limit, realized as the ring of symmetric functions on countably many variables . Isomorphic as spaces are the character rings Char-O and Char-Sp of the classical matrix subgroups of , the orthogonal and symplectic groups. We also analyse the formal character rings Char-H of algebraic subgroups of , comprised of matrix transformations leaving invariant a fixed but arbitrary tensor of Young symmetry type , 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 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-H ring is a ribbon Hopf algebra. The corresponding knot invariant operators are rather weak, giving merely a measure of the writhe.