ALGEBRAIC COMBINATORICS

no. 1

Khovanov’s Heisenberg category, moments in free probability, and shifted symmetric functions
Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 49-74.

We establish an isomorphism between the center End ℋ ' (1) of the Heisenberg category defined by Khovanov in [13] and the algebra Λ * of shifted symmetric functions defined by Okounkov–Olshanski in [18]. 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 [10] and the noncommutative probability spaces of Biane [2]

Received: 2017-08-19
Revised: 2018-05-11
Accepted: 2018-07-09
Published online: 2019-02-04
DOI: https://doi.org/10.5802/alco.32
Classification: 05E05,  20B30,  18D10
Keywords: Symmetric functions, asymptotic representation theory, Heisenberg categorification, graphical calculus 
@article{ALCO_2019__2_1_49_0,
     author = {Kvinge, Henry and Licata, Anthony M. and Mitchell, Stuart},
     title = {Khovanov's Heisenberg category, moments in free probability, and shifted symmetric functions},
     journal = {Algebraic Combinatorics},
     pages = {49--74},
     publisher = {MathOA foundation},
     volume = {2},
     number = {1},
     year = {2019},
     doi = {10.5802/alco.32},
     zbl = {1405.05188},
     mrnumber = {3912168},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2019__2_1_49_0/}
}
Kvinge, Henry; Licata, Anthony M.; Mitchell, Stuart. Khovanov’s Heisenberg category, moments in free probability, and shifted symmetric functions. Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 49-74. doi : 10.5802/alco.32. https://alco.centre-mersenne.org/item/ALCO_2019__2_1_49_0/

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