Frobenius Heisenberg categorification

Savage, Alistair

Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 937-967.

We associate a graded monoidal supercategory $\mathscr{H}ei{s}_{F,k}$ to every graded Frobenius superalgebra $F$ and integer $k$. These categories, which categorify a broad range of lattice Heisenberg algebras, recover many previously defined Heisenberg categories as special cases. In this way, the categories $\mathscr{H}ei{s}_{F,k}$ serve as a unifying and generalizing framework for Heisenberg categorification. Even in the case of previously defined Heisenberg categories, we obtain new, more efficient, presentations of these categories, based on an approach of Brundan. When $k=0$, our construction yields new versions of the affine oriented Brauer category depending on a graded Frobenius superalgebra.

Classification : 18D10, 17B10, 17B65
Mots clés : Categorification, graded Frobenius superalgebra, Heisenberg algebra, diagrammatic calculus

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

[2] Brundan, Jonathan On the definition of Heisenberg category, Algebr. Comb., Volume 1 (2018) no. 4, pp. 523-544 (https://arxiv.org/abs/1709.06589, https://doi.org/10.5802/alco.26) | DOI | MR | Zbl

[3] Brundan, Jonathan; Comes, Jonathan; Kujawa, Jonathan A basis theorem for the degenerate affine oriented Brauer–Clifford supercategory (To appear in Canad. J. Math. https://arxiv.org/abs/1706.09999, https://doi.org/10.4153/CJM-2018-030-8 ) | DOI

[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 (https://arxiv.org/abs/1404.6574, https://doi.org/10.4171/QT/87) | DOI | MR | Zbl

[5] Brundan, Jonathan; Ellis, Alexander P. Monoidal supercategories, Comm. Math. Phys., Volume 351 (2017) no. 3, pp. 1045-1089 (https://arxiv.org/abs/1603.05928, https://doi.org/10.1007/s00220-017-2850-9) | DOI | MR | Zbl

[6] Brundan, Jonathan; Savage, Alistair Quantum Frobenius Heisenberg categorification (In preparation)

[7] Brundan, Jonathan; Savage, Alistair; Webster, Ben The degenerate Heisenberg category and its Grothendieck ring (2018) (https://arxiv.org/abs/1812.03255)

[8] Brundan, Jonathan; Savage, Alistair; Webster, Ben On the definition of quantum Heisenberg category (2018) (https://arxiv.org/abs/1812.04779)

[9] Cautis, Sabin; Licata, Anthony Heisenberg categorification and Hilbert schemes, Duke Math. J., Volume 161 (2012) no. 13, pp. 2469-2547 (https://arxiv.org/abs/1009.5147, https://doi.org/10.1215/00127094-1812726) | DOI | MR | Zbl

[10] Comes, Jonathan; Kujawa, Jonathan Higher level twisted Heisenberg supercategories (In preparation)

[11] Hill, David; Sussan, Joshua A categorification of twisted Heisenberg algebras, Adv. Math., Volume 295 (2016), pp. 368-420 (https://arxiv.org/abs/1501.00283, https://doi.org/10.1016/j.aim.2016.03.033) | DOI | MR | Zbl

[12] Khovanov, Mikhail Heisenberg algebra and a graphical calculus, Fund. Math., Volume 225 (2014) no. 1, pp. 169-210 (https://arxiv.org/abs/1009.3295, https://doi.org/10.4064/fm225-1-8) | DOI | MR | Zbl

[13] Licata, Anthony; Rosso, Daniele; Savage, Alistair A graphical calculus for the Jack inner product on symmetric functions, J. Combin. Theory Ser. A, Volume 155 (2018), pp. 503-543 (https://arxiv.org/abs/1610.01862, https://doi.org/10.1016/j.jcta.2017.11.020) | DOI | MR | Zbl

[14] Licata, Anthony; Savage, Alistair Hecke algebras, finite general linear groups, and Heisenberg categorification, Quantum Topol., Volume 4 (2013) no. 2, pp. 125-185 (https://arxiv.org/abs/1101.0420, https://doi.org/10.4171/QT/37) | DOI | MR | Zbl

[15] Mackaay, Marco; Savage, Alistair Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification, J. Algebra, Volume 505 (2018), pp. 150-193 (https://arxiv.org/abs/1705.03066, https://doi.org/10.1016/j.jalgebra.2018.03.004) | DOI | MR | Zbl

[16] 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 (https://arxiv.org/abs/1507.06298, https://doi.org/10.1007/s00209-016-1776-9) | DOI | MR | Zbl

[17] Rosso, Daniele; Savage, Alistair Quantum affine wreath algebras (2019) (https://arxiv.org/abs/1902.00143)

[18] Savage, Alistair Affine wreath product algebras (To appear in Int. Math. Res. Not. IMRN https://arxiv.org/abs/1709.02998, https://doi.org/10.1093/imrn/rny092) | DOI