The Trace and Center of the Twisted Heisenberg Category

Khovanov's categorification of the Heisenberg algebra has many interesting representation theoretic and algebro-combinatorial properties. This Heisenberg category was constructed so that its Grothendieck group contains (and is conjecturally equal to) the Heisenberg algebra, but applying alternative decategorification functors reveals additional information. One such functor, the trace, yields $W_{1+\infty}$ at level one, a large and rich algebra which contains the Heisenberg algebra. Another such functor, the center, gives an algebra of shifted symmetric functions with connections to the asymptotic representation theory of symmetric groups.

In this dissertation, we investigate the trace and center of a twisted version of the Heisenberg category, which was defined by Cautis and Sussan to categorify the twisted Heisenberg algebra. We show that its trace is isomorphic to a distinguished subalgebra of $W_{1+\infty}$ at level one introduced by Kac, Wang, and Yan. The center of the category is then shown to be a subalgebra of the symmetric functions generated by odd power sums. There is a natural action of the trace of a category on its center. We describe this action for the twisted Heisenberg category, which is a twisted version of a representation of $W_{1+\infty}$ introduced by Lascoux and Thibon.