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

We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra of the symmetric functions $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ 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 $W^{-}\subset W_{1+\mathrm{\infty }}$, on $\mathrm{\Gamma }$.