On a Categorical Boson–Fermion Correspondence

Abstract

Abstract. We propose a categorical version of the Boson-Fermion correspondence and its twisted version. One can view it as a relative of the Frenkel-Kac-Segal construction of quantum affine algebras.

A graphical calculus for the Jack inner product on symmetric functions

“…On the other hand, allowing the Nakayama automorphism to be nontrivial (e.g. equal to the parity involution) would introduce twisted Heisenberg algebras into the picture (see [CS15,HS16] and [RS17, Rem. 6.2]).…”

Towards a categorical boson-fermion correspondence

“…The element b λµ induces a generator of Hom B (V λ , V µ ), which is still denoted by b λµ . The map b λµ is the image of the map f λµ under the isomorphism (6). The equation (12) According to [1,Lemma 2.4], {1 µ ; µ ∈ Par} is a complete set of nonisomorphic primitive orthogonal idempotents of the quiver algebra F .…”

The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph

“…A twisted version of Khovanov's Heisenberg category was introduced by Cautis and Sussan in [6]. The twisted Heisenberg category H tw is a C-linear additive monoidal category, with an additional Z/2Z-grading.…”

“…As a Corollary to part 3. of the above Proposition, we have another formula for the value of p ρ . [6]. It is a Z/2Z-graded additive monoidal category whose morphisms are described diagrammatically as oriented compact 1-manifolds immersed in R × [0, 1].…”

“…Diagrams as bimodule homomorphisms. An action of H tw on the category S whose objects are compositions of induction and restriction functors between Z/2Z-graded finite dimensional S n -supermodules, for all n ≥ 0, is described in [6,Section 6.3]. Because induction and restriction functors for the algebras S n can be written as…”