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

TL;DR

This paper links the center of the twisted Heisenberg category with symmetric functions generated by odd power sums, providing graphical descriptions and connections to transition functions on the Schur graph, and explores algebra actions.

Contribution

It establishes an isomorphism between the twisted Heisenberg category's center and a subalgebra of symmetric functions, with graphical and algebraic descriptions of factorial Schur Q-functions.

Findings

Center of the twisted Heisenberg category is isomorphic to a subalgebra of symmetric functions.

Graphical representation of factorial Schur Q-functions as diagrams.

Action of the W-algebra on the symmetric functions.

Abstract

We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra of the symmetric functions $\Gamma$ generated by odd power sums. We give a graphical description of the factorial Schur $Q$-functions as closed diagrams in the twisted Heisenberg category and show that the bubble generators of the center correspond to two sets of generators of $\Gamma$ which encode data related to up/down transition functions on the Schur graph. Finally, we describe an action of the trace of the twisted Heisenberg category, the $W$-algebra $W^-\subset W_{1+\infty}$, on $\Gamma$.