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Hoel Queffelec; Alistair Savage; Oded Yacobi
An equivalence between truncations of categorified quantum groups and Heisenberg categories
(Une équivalence entre des troncations de groupes quantiques catégorifiés et des catégories de Heisenberg)
Journal de l'École polytechnique — Mathématiques, 5 (2018), p. 197-238, doi: 10.5802/jep.68
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Class. Math.: 17B10, 17B65, 20C30, 16D90
Keywords: Categorification, Heisenberg algebra, Fock space, basic representation, principal realization, symmetric group

Résumé - Abstract

We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty $. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov–Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty $, and also to a truncation of Khovanov’s Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty $. As a result of the categorical equivalences described above, certain actions of $\mathscr{H}$ induce actions of $\mathscr{U}$, and vice versa. In particular, we obtain an explicit action of $\mathscr{U}$ on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of $\mathscr{H}$. The 2-category $\mathscr{A}$ can be viewed as a graphical calculus describing the functors of $i$-induction and $i$-restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities.

Bibliography

[BHLW17] A. Beliakova, K. Habiro, A. D. Lauda & B. Webster, “Current algebras and categorified quantum groups”, J. London Math. Soc. (2) 95 (2017) no. 1, p. 248-276
[BK09a] J. Brundan & A. Kleshchev, “Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras”, Invent. Math. 178 (2009) no. 3, p. 451-484
[BK09b] J. Brundan & A. Kleshchev, “Graded decomposition numbers for cyclotomic Hecke algebras”, Adv. Math. 222 (2009) no. 6, p. 1883-1942
[CKL13] S. Cautis, J. Kamnitzer & A. Licata, “Coherent sheaves on quiver varieties and categorification”, Math. Ann. 357 (2013) no. 3, p. 805-854
[CL11] S. Cautis & A. Licata, “Vertex operators and 2-representations of quantum affine algebras”, arXiv:1112.6189 2011
[CL12] S. Cautis & A. Licata, “Heisenberg categorification and Hilbert schemes”, Duke Math. J. 161 (2012) no. 13, p. 2469-2547
[CL15] S. Cautis & A. D. Lauda, “Implicit structure in 2-representations of quantum groups”, Selecta Math. (N.S.) 21 (2015) no. 1, p. 201-244
[CLLS16] S. Cautis, A. D. Lauda, A. Licata & J. Sussan, “W-algebras from Heisenberg categories”, J. Inst. Math. Jussieu (2016), online, doi:10.1017/S1474748016000189
[CR08] J. Chuang & R. Rouquier, “Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification”, Ann. of Math. (2) 167 (2008) no. 1, p. 245-298
[FH91] W. Fulton & J. Harris, Representation theory, Graduate Texts in Math. 129, Springer-Verlag, New York, 1991
[Kac90] V. G. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990
[Kho14] M. Khovanov, “Heisenberg algebra and a graphical calculus”, Fund. Math. 225 (2014) no. 1, p. 169-210
[KL10] M. Khovanov & A. D. Lauda, “A categorification of quantum ${\rm sl}(n)$”, Quantum Topol. 1 (2010) no. 1, p. 1-92
[Kle05] A. Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics 163, Cambridge University Press, Cambridge, 2005
[Kle14] A. Kleshchev, “Modular representation theory of symmetric groups”, arXiv:1405.3326 2014
[Lem16] J. Lemay, “Geometric realizations of the basic representation of $\widehat{\mathfrak{gl}}_r$”, Selecta Math. (N.S.) 22 (2016) no. 1, p. 341-387
[LRS] A. Licata, D. Rosso & A. Savage, “Categorification and Heisenberg doubles arising from towers of algebras”, J. Combinatorial Theory Ser. A , to appear, arXiv:1610.01862
[LS10] A. Licata & A. Savage, “Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves”, Selecta Math. (N.S.) 16 (2010) no. 2, p. 201-240
[LS13] A. Licata & A. Savage, “Hecke algebras, finite general linear groups, and Heisenberg categorification”, Quantum Topol. 4 (2013) no. 2, p. 125-185
[MS17] M. Mackaay & A. Savage, “Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification”, arXiv:1705.03066 2017
[MSV13] M. Mackaay, M. Stošić & P. Vaz, “A diagrammatic categorification of the $q$-Schur algebra”, Quantum Topol. 4 (2013) no. 1, p. 1-75
[Nag09] K. Nagao, “Quiver varieties and Frenkel-Kac construction”, J. Algebra 321 (2009) no. 12, p. 3764-3789
[Nak98] H. Nakajima, “Quiver varieties and Kac-Moody algebras”, Duke Math. J. 91 (1998) no. 3, p. 515-560
[QR16] H. Queffelec & D. E. V. Rose, “The $\mathfrak{sl}_n$ foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality”, Adv. Math. 302 (2016), p. 1251-1339
[Rou08] R. Rouquier, “2-Kac-Moody algebras”, arXiv:0812.5023v1 2008
[RS17] D. Rosso & A. Savage, “A general approach to Heisenberg categorification via wreath product algebras”, Math. Z. 286 (2017) no. 1-2, p. 603-655
[Sav06] A. Savage, “A geometric boson-fermion correspondence”, C. R. Math. Rep. Acad. Sci. Canada 28 (2006) no. 3, p. 65-84
[SVV17] P. Shan, M. Varagnolo & E. Vasserot, “On the center of quiver Hecke algebras”, Duke Math. J. 166 (2017) no. 6, p. 1005-1101
[VV11] M. Varagnolo & E. Vasserot, “Canonical bases and KLR-algebras”, J. reine angew. Math. 659 (2011), p. 67-100
[Web12] B. Webster, “A categorical action on quantized quiver varieties”, arXiv:1208.5957 2012
[Zhe14] H. Zheng, “Categorification of integrable representations of quantum groups”, Acta Mech. Sinica (English Ed.) 30 (2014) no. 6, p. 899-932