Khovanov’s Heisenberg category, moments in free probability, and shifted symmetric functions

Kvinge, Henry; Licata, Anthony M.; Mitchell, Stuart

Algebraic Combinatorics, Tome 2 (2019) no. 1, pp. 49-74.

We establish an isomorphism between the center ${\text{End}}_{{\mathscr{H}}^{\prime }}\left(\mathbf{1}\right)$ of the Heisenberg category defined by Khovanov in [13] and the algebra ${\Lambda }^{*}$ of shifted symmetric functions defined by Okounkov–Olshanski in [18]. We give a graphical description of the shifted power and Schur bases of ${\Lambda }^{*}$ as elements of ${\text{End}}_{{\mathscr{H}}^{\prime }}\left(\mathbf{1}\right)$, and describe the curl generators of ${\text{End}}_{{\mathscr{H}}^{\prime }}\left(\mathbf{1}\right)$ in the language of shifted symmetric functions. This latter description makes use of the transition and co-transition measures of Kerov [10] and the noncommutative probability spaces of Biane [2]

Classification : 05E05, 20B30, 18D10
Mots clés : Symmetric functions, asymptotic representation theory, Heisenberg categorification, graphical calculus

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