ALGEBRAIC COMBINATORICS

no. 4
On the definition of Heisenberg category
Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 523-544.

We revisit the definition of the Heisenberg category of central charge k∈ℤ. For central charge -1, this category was introduced originally by Khovanov, but with some additional cyclicity relations which we show here are unnecessary. For other negative central charges, the definition is due to Mackaay and Savage, also with some redundant relations, while central charge zero recovers the affine oriented Brauer category of Brundan, Comes, Nash and Reynolds. We also discuss cyclotomic quotients.

Received: 2017-10-10
Revised: 2018-05-22
Accepted: 2018-06-12
Published online: 2018-09-10
DOI: https://doi.org/10.5802/alco.26
Classification: 17B10,  18D10
Keywords: Heisenberg category, string calculus 
@article{ALCO_2018__1_4_523_0,
     author = {Brundan, Jonathan},
     title = {On the definition of Heisenberg category},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {1},
     number = {4},
     year = {2018},
     pages = {523-544},
     doi = {10.5802/alco.26},
     mrnumber = {3875075},
     zbl = {06963903},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2018__1_4_523_0/}
}
Brundan, Jonathan. On the definition of Heisenberg category. Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 523-544. doi : 10.5802/alco.26. https://alco.centre-mersenne.org/item/ALCO_2018__1_4_523_0/

[1] Ariki, Susumu On the decomposition numbers of the Hecke algebra of G(m,1,n), J. Math. Kyoto Univ., Volume 36 (1996) no. 4, pp. 789-808 | Article | MR 1443748 | Zbl 0888.20011

[2] Brundan, Jonathan On the definition of Kac-Moody 2-category, Math. Ann., Volume 364 (2016) no. 1-2, pp. 353-372 | Article | MR 3451390 | Zbl 06540658

[3] Brundan, Jonathan Representations of oriented skein categories (2017) (https://arxiv.org/abs/1712.08953 )

[4] Brundan, Jonathan; Comes, Jonathan; Nash, David; Reynolds, Andrew A basis theorem for the affine oriented Brauer category and its cyclotomic quotients, Quantum Topol., Volume 8 (2017) no. 1, pp. 75-112 | Article | MR 3630282 | Zbl 06718140

[5] Brundan, Jonathan; Davidson, Nicholas Categorical actions and crystals, Categorification and higher representation theory (Contemporary Mathematics) Volume 684, American Mathematical Society, 2017, pp. 116-159

[6] Brundan, Jonathan; Kleshchev, Alexander Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math., Volume 222 (2009) no. 6, pp. 1883-1942 | Article | MR 2562768 | Zbl 1241.20003

[7] Brundan, Jonathan; Savage, Alistair On the definition of quantum Heisenberg category (in preparation)

[8] Cautis, Sabin; Lauda, Aaron; Licata, Anthony; Samuelson, Peter; Sussan, Joshua The elliptic Hall algebra and the deformed Khovanov Heisenberg category (2016) (https://arxiv.org/abs/1609.03506 ) | Zbl 06976959

[9] Cautis, Sabin; Lauda, Aaron; Licata, Anthony; Sussan, Joshua W-algebras from Heisenberg categories, J. Inst. Math. Jussieu (2016), pp. 1-37 | Article | Zbl 06963839

[10] Cautis, Sabin; Licata, Anthony Heisenberg categorification and Hilbert schemes, Duke Math. J., Volume 161 (2012) no. 13, pp. 2469-2547 | Article | MR 2988902 | Zbl 1263.14020

[11] Comes, Jonathan; Kujawa, Jonathan Higher level twisted Heisenberg supercategories (in preparation)

[12] Hill, David; Sussan, Joshua A categorification of twisted Heisenberg algebras, Adv. Math., Volume 295 (2016), pp. 368-420 | Article | MR 3488039 | Zbl 06570861

[13] Khovanov, Mikhail Heisenberg algebra and a graphical calculus, Fundam. Math., Volume 225 (2014), pp. 169-210 | Article | MR 3205569 | Zbl 1304.18019

[14] Khovanov, Mikhail; Lauda, Aaron A categorification of quantum 𝔰𝔩(n), Quantum Topol., Volume 1 (2010) no. 1, pp. 1-92 | Article | MR 2628852 | Zbl 1206.17015

[15] Kleshchev, Alexander Linear and projective representations of symmetric groups, Cambridge University Press, 2005, xiv+277 pages | MR 2165457 | Zbl 1080.20011

[16] Licata, Anthony; Savage, Alistair Hecke algebras, finite general linear groups, and Heisenberg categorification, Quantum Topol., Volume 4 (2013) no. 2, pp. 125-185 | Article | MR 3032820 | Zbl 1279.20006

[17] Mackaay, Marco; Savage, Alistair Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification, J. Algebra, Volume 505 (2018), pp. 150-193 | Article | MR 3789909 | Zbl 06893263

[18] Queffelec, Hervé; Savage, Alistair; Yacobi, Oded An equivalence between truncations of categorified quantum groups and Heisenberg categories, J. Éc. Polytech., Math., Volume 5 (2018), pp. 192-238 | MR 3738513 | Zbl 06988578

[19] Rosso, Daniele; Savage, Alistair A general approach to Heisenberg categorification via wreath product algebras, Math. Z., Volume 286 (2017) no. 1-2, pp. 603-655 | Article | MR 3648512 | Zbl 1366.18006

[20] Rouquier, Raphael 2-Kac-Moody algebras (2008) (https://arxiv.org/abs/0812.5023 )

[21] Rouquier, Raphael Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq., Volume 19 (2012) no. 2, pp. 359-410 | Article | MR 2908731 | Zbl 1247.20002

[22] Rui, Hebing; Su, Yucai Affine walled Brauer algebras and super Schur-Weyl duality, Adv. Math., Volume 285 (2015), pp. 28-71 | Article | MR 3406495 | Zbl 1356.17012

[23] Savage, Alistair Frobenius Heisenberg categorification (2018) (https://arxiv.org/abs/1802.01626 )

[24] Webster, Ben Canonical bases and higher representation theory, Compos. Math., Volume 151 (2015) no. 1, pp. 121-166 | Article | MR 3305310 | Zbl 06417584