Plethystic vertex operators and boson-fermion correspondences

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 π. Here we establish an extension of the well-known boson-fermion correspondence involving Schur functions and their associated (Bernstein) vertex operators: for each π, 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 Hopfalgebraic structure of certain symmetric function series and their plethysms. ) * (π) λ (X), in terms of products of our dual vertex operators acting on the identity. This is done in § 4, in which the π-Schur functions and their duals are then evaluated in terms of ordinary Schur functions by exploiting a powerful normal ordering lemma proved in Appendix C.The paper concludes in § 5 with a brief summary, a short survey of related work, and some discussion of implications of the results and future work. Symmetric functions of π-typeAs is well known, in 1 + 1-dimensional quantum field theory, field mode expansions in advanced and retarded variables x ± ct, become in the Euclidean picture, Laurent expansions in a complex variable z. In the simplest ("chiral scalar") case, the mode operators α n , α −n ≡ α n † (n ∈ Z >0 ) fulfil the quantum mechanical commutation relations of an infinite Heisenberg algebra, [α m , α n ] = m δ m+n,0 , m, n ∈ Z =0 . In the combinatorial equivalent, these operators are in turn realized on the ring Λ(X) of symmetric functions of a countably infinite alphabet of indeterminates X = {x 1 , x 2 , · · · } , and it is in this setting that we wish to investigate the calculus of vertex operators.In order to develop this, we begin with some notational preliminaries (following [3]). The ring Λ(X) has various distinguished algebraic generating sets, of which we will need the power sum symmetric

The Orthogonal and Symplectic Schur Functions, Vertex Operators and Integrable Hierarchies

“…where h n is the nth complete symmetric function of the form in equation (2). Define vertex operators…”

Π-Type FERMIONS AND Π-Type KP HIERARCHY

“…We begin this section with some notational preliminaries [10]. Let Λ(x) be the ring of symmetric functions of a countably infinite alphabet of variables x = {x 1 , x 2 , • • • }.…”

“…The linear basis of π-type symmetric functions provides the structure of the universal character ring of group H π (subgroup of GL(n)) [7,8,9]. Like Schur functions, π-type symmetric functions can also be realized from vertex operators which are constructed in [10]. Then free Fermions can be constructed and there exists for sure an integrable system.…”

A note on Cauchy's formula

“…The action of the half-vertex operator φ + (x) on Schur functions can be used to derive Macdonald's skew Schur functions. Bernstein operator can also be formulated in plethystic manner [2,3,6,12,18], and another combinatorial formulation can be found in [7,15].…”