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

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