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

Kvinge, Henry¹ ; Licata, Anthony M.² ; Mitchell, Stuart²

¹University of California Davis Department of Mathematics Davis, CA USA
²Mathematical Sciences Institute Australian National University Canberra, Australia

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

Résumé

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]

Détail

Reçu le : 2017-08-19
Révisé le : 2018-05-11
Accepté le : 2018-07-09
Publié le : 2019-02-04
DOI : 10.5802/alco.32

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits