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Khovanov's Heisenberg category, moments in noncommutative probability, and shifted symmetric functions

Student-Run Applied & Math Seminar

Speaker: Henry Kvinge
Location: 2112 MSB
Start time: Wed, Oct 26 2016, 12:10PM

Khovanov proposed a monoidal category $\mathcal{H}$ to categorify the Heisenberg algebra. $\mathcal{H}$ can be elegantly defined via a graphical calculus of planar diagrams modulo local relations. The center $Z(\mathcal{H})$ of this category is, by definition, the commutative algebra of closed diagrams. In a recent work with Licata and Mitchell, we show that $Z(\mathcal{H})$ is isomorphic to the algebra of shifted symmetric functions of Okounkov-Olshanski. We give a graphical description of the shifted power and Schur bases of $\Lambda^*$ as elements of the center, and describe Khovanov's curl generators of $Z(\mathcal{H})$ in the language of shifted symmetric functions. This latter description makes use of the transition and co-transition measures of Kerov and the noncommutative probability spaces of Biane.

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