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  1. The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra $$W_{1+\infty }$$. This induces an action of $$W_{1+\infty }$$ on the center of the categorified Fock space representation, which can be identified with the action of $$W_{1+\infty }$$ on symmetric functions. 
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