Cyclotomic quiver Hecke algebras and Hecke algebra of 𝐺(𝑟,𝑝,𝑛)

Rostam, Salim

doi:10.1090/tran/7485

Cyclotomic quiver Hecke algebras and Hecke algebra of $G(r,p,n)$
HTML articles powered by AMS MathViewer

by Salim Rostam PDF

Trans. Amer. Math. Soc. 371 (2019), 3877-3916 Request permission

Abstract:

Given a quiver automorphism with nice properties, we give a presentation of the fixed point subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type $G(r, 1, n)$ and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type $G(r,p,n)$ which looks very similar to the one of a cyclotomic quiver Hecke algebra. In addition, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki–Koike algebras.

References

Susumu Ariki, Representation theory of a Hecke algebra of $G(r,p,n)$, J. Algebra 177 (1995), no. 1, 164–185. MR 1356366, DOI 10.1006/jabr.1995.1292
Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MR 1443748, DOI 10.1215/kjm/1250518452
Susumu Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type $G(m,1,n)$ and Kleshchev multipartitions, Osaka J. Math. 38 (2001), no. 4, 827–837. MR 1864465
Susumu Ariki and Kazuhiko Koike, A Hecke algebra of $(\textbf {Z}/r\textbf {Z})\wr {\mathfrak {S}}_n$ and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216–243. MR 1279219, DOI 10.1006/aima.1994.1057
Susumu Ariki and Andrew Mathas, The number of simple modules of the Hecke algebras of type $G(r,1,n)$, Math. Z. 233 (2000), no. 3, 601–623. MR 1750939, DOI 10.1007/s002090050489
C. Boys, Alternating quiver Hecke algebras, Ph.D. thesis, University of Sydney (2014).
Clinton Boys and Andrew Mathas, Quiver Hecke algebras for alternating groups, Math. Z. 285 (2017), no. 3-4, 897–937. MR 3623735, DOI 10.1007/s00209-016-1732-8
Michel Broué, Gunter Malle, and Raphaël Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127–190. MR 1637497
Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), no. 3, 451–484. MR 2551762, DOI 10.1007/s00222-009-0204-8
Maria Chlouveraki and Nicolas Jacon, Schur elements for the Ariki-Koike algebra and applications, J. Algebraic Combin. 35 (2012), no. 2, 291–311. MR 2886292, DOI 10.1007/s10801-011-0314-4
Maria Chlouveraki and Loïc Poulain d’Andecy, Markov traces on affine and cyclotomic Yokonuma-Hecke algebras, Int. Math. Res. Not. IMRN 14 (2016), 4167–4228. MR 3556417, DOI 10.1093/imrn/rnv257
Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic $q$-Schur algebras, Math. Z. 229 (1998), no. 3, 385–416. MR 1658581, DOI 10.1007/PL00004665
Richard Dipper and Andrew Mathas, Morita equivalences of Ariki-Koike algebras, Math. Z. 240 (2002), no. 3, 579–610. MR 1924022, DOI 10.1007/s002090100371
Meinolf Geck, On the representation theory of Iwahori-Hecke algebras of extended finite Weyl groups, Represent. Theory 4 (2000), 370–397. MR 1780716, DOI 10.1090/S1088-4165-00-00093-5
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
Gwenaëlle Genet and Nicolas Jacon, Modular representations of cyclotomic Hecke algebras of type $G(r,p,n)$, Int. Math. Res. Not. , posted on (2006), Art. ID 93049, 18. MR 2276351, DOI 10.1155/IMRN/2006/93049
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
I. Grojnowski, Affine $\widehat {\mathfrak {sl}_p}$ controls the modular representation theory of the symmetric group and related Hecke algebras, arXiv:math/9907129v1.
Jun Hu, Crystal bases and simple modules for Hecke algebras of type $G(p,p,n)$, Represent. Theory 11 (2007), 16–44. MR 2295687, DOI 10.1090/S1088-4165-07-00313-5
Jun Hu, Modular representations of Hecke algebras of type $G(p,p,n)$, J. Algebra 274 (2004), no. 2, 446–490. MR 2043358, DOI 10.1016/j.jalgebra.2003.10.029
Jun Hu and Andrew Mathas, Decomposition numbers for Hecke algebras of type $G(r,p,n)$: the $(\epsilon ,q)$-separated case, Proc. Lond. Math. Soc. (3) 104 (2012), no. 5, 865–926. MR 2928331, DOI 10.1112/plms/pdr047
Jun Hu and Andrew Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type $A$, Adv. Math. 225 (2010), no. 2, 598–642. MR 2671176, DOI 10.1016/j.aim.2010.03.002
Jun Hu and Andrew Mathas, Morita equivalences of cyclotomic Hecke algebras of type $G(r,p,n)$, J. Reine Angew. Math. 628 (2009), 169–194. MR 2503239, DOI 10.1515/CRELLE.2009.022
Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215–236 (1964). MR 165016
Seok-Jin Kang and Masaki Kashiwara, Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras, Invent. Math. 190 (2012), no. 3, 699–742. MR 2995184, DOI 10.1007/s00222-012-0388-1
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309–347. MR 2525917, DOI 10.1090/S1088-4165-09-00346-X
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685–2700. MR 2763732, DOI 10.1090/S0002-9947-2010-05210-9
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572
Sinéad Lyle and Andrew Mathas, Blocks of cyclotomic Hecke algebras, Adv. Math. 216 (2007), no. 2, 854–878. MR 2351381, DOI 10.1016/j.aim.2007.06.008
Loic Poulain d’Andecy, Invariants for links from classical and affine Yokonuma-Hecke algebras, Algebraic modeling of topological and computational structures and applications, Springer Proc. Math. Stat., vol. 219, Springer, Cham, 2017, pp. 77–95. MR 3748891, DOI 10.1007/978-3-319-68103-0_{4}
Salim Rostam, Cyclotomic Yokonuma-Hecke algebras are cyclotomic quiver Hecke algebras, Adv. Math. 311 (2017), 662–729. MR 3628228, DOI 10.1016/j.aim.2017.03.004
R. Rouquier, 2–Kac–Moody algebras, arXiv:0812.5023v1.
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
C. Stroppel and B. Webster, Quiver Schur algebras and $q$-Fock space, arXiv:1110.1115v2.
Denis Uglov, Canonical bases of higher-level $q$-deformed Fock spaces and Kazhdan-Lusztig polynomials, Physical combinatorics (Kyoto, 1999) Progr. Math., vol. 191, Birkhäuser Boston, Boston, MA, 2000, pp. 249–299. MR 1768086