ALGEBRAIC COMBINATORICS

Khovanov’s Heisenberg category, moments in free probability, and shifted symmetric functions
Algebraic Combinatorics, Volume 2 (2019) no. 1, p. 49-74
We establish an isomorphism between the center ${\text{End}}_{{ℋ}^{\text{'}}}\left(\mathbf{1}\right)$ 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 ${\text{End}}_{{ℋ}^{\text{'}}}\left(\mathbf{1}\right)$, and describe the curl generators of ${\text{End}}_{{ℋ}^{\text{'}}}\left(\mathbf{1}\right)$ 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]
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},
publisher = {MathOA foundation},
volume = {2},
number = {1},
year = {2019},
pages = {49-74},
doi = {10.5802/alco.32},
language = {en},
url = {https://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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