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Séminaire Lotharingien de Combinatoire, 78B.63 (2017), 12 pp.

# Henry Kvinge, Anthony M. Licata
and Stuart Mitchell

# Khovanov's Heisenberg Category, Moments in Free Probability, and Shifted Symmetric Functions

**Abstract.**
We establish an isomorphism between the center
End_{H'}(**1**) of Khovanov's Heisenberg
category **H**' and the algebra Λ^{*} of shifted
symmetric functions defined by Okounkov-Olshanski. We give a
graphical description of the shifted power and Schur bases of
Λ^{*} as elements of
End_{H'}(**1**),
and describe the curl generators of
End_{H'}(**1**) 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.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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