Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

Abstract

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co-and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the Appendix we discuss a relation to formal group laws.

Plethystic vertex operators and boson-fermion correspondences

“…The present paper gives further development of the algebraic-combinatorial context for the description of the general classes of symmetric functions of π-type, introduced in our previous papers [8,9,10]. Specifically, further to the previously-derived π-type 'plethystic' vertex operators given as the algebraic tools for deriving the π-type symmetric functions as modal products, we have identified in this work the dual counterparts of these objects.…”

“…Our results from [7][8][9][10] are as follows. For each π we have identified in the ring of symmetric functions, the linear basis of symmetric functions of type π which plays the role of the universal character ring for the group H π , analogously to the way in which the standard basis of Schur functions, and Schur functions of symplectic and orthogonal type, provide the structure of the universal character rings in the general linear, orthogonal and symplectic cases (for the Hopf structure of the character rings in the latter cases see also [11]).…”

“…A detailed reconciliation of the categorical, Hopf-algebra inspired, but in principle basisfree, methods underlying our present derivations (as in [7,8,10]), with the adapted bases of Jing and Rozhkowskaya [19], better matched to the physical point of view using creation and annihilation of composite or effective particles, is beyond the scope of this paper, and a subject for further investigation 6 .…”