Browse issues
no. 4
On the definition of Heisenberg category
Brundan, Jonathan1
1 Department of Mathematics University of Oregon Eugene OR 97403 USA
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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.26
Classification: 17B10, 18D10
Keywords: Heisenberg category, string calculus

Brundan, Jonathan 1

1 Department of Mathematics University of Oregon Eugene OR 97403 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2018__1_4_523_0,
     author = {Brundan, Jonathan},
     title = {On the definition of {Heisenberg} category},
     journal = {Algebraic Combinatorics},
     pages = {523--544},
     publisher = {MathOA foundation},
     volume = {1},
     number = {4},
     year = {2018},
     doi = {10.5802/alco.26},
     zbl = {06963903},
     mrnumber = {3875075},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.26/}
}
TY  - JOUR
AU  - Brundan, Jonathan
TI  - On the definition of Heisenberg category
JO  - Algebraic Combinatorics
PY  - 2018
SP  - 523
EP  - 544
VL  - 1
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.26/
DO  - 10.5802/alco.26
LA  - en
ID  - ALCO_2018__1_4_523_0
ER  - 
%0 Journal Article
%A Brundan, Jonathan
%T On the definition of Heisenberg category
%J Algebraic Combinatorics
%D 2018
%P 523-544
%V 1
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.26/
%R 10.5802/alco.26
%G en
%F 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/articles/10.5802/alco.26/

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

Cited by Sources: