Plethystic Vertex Operators and Boson-Fermion Correspondences

TL;DR

This paper extends the boson-fermion correspondence to plethystic vertex operators linked with symmetric functions of arbitrary shape, revealing their algebraic structure and Clifford relations.

Contribution

It introduces an extension of the boson-fermion correspondence involving plethystic vertex operators for arbitrary partitions, expanding the algebraic framework of symmetric functions.

Findings

Plethystic vertex operators satisfy Clifford algebra relations.

Extended boson-fermion correspondence for arbitrary partitions.

Utilization of Hopf algebraic structures in symmetric functions.

Abstract

We study the algebraic properties of plethystic vertex operators, introduced in J. Phys. A: Math. Theor. 43 405202 (2010), underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group, defined to stabilize tensors of Young symmetry type characterized by a partition of arbitrary shape \pi. Here we establish an extension of the well-known boson-fermion correspondence involving Schur functions and their associated (Bernstein) vertex operators: for each \pi, the modes generated by the plethystic vertex operators and their suitably constructed duals, satisfy the anticommutation relations of a complex Clifford algebra. The combinatorial manipulations underlying the results involve exchange identities exploiting the Hopf-algebraic structure of certain symmetric function series and their plethysms.