On a conjecture about cellular characters for the complex reflection group G(d,1,n)
Annales mathématiques Blaise Pascal, Volume 27 (2020) no. 1, pp. 37-64.

We propose a conjecture relating two different sets of characters for the complex reflection group G(d,1,n). From one side, the characters are afforded by Calogero–Moser cells, a conjectural generalisation of Kazhdan–Lusztig cells for a complex reflection group. From the other side, the characters arise from a level d irreducible integrable representations of 𝒰 q (𝔰𝔩 ). We prove this conjecture in some cases: in full generality for G(d,1,2) and for generic parameters for G(d,1,n).

Published online:
DOI: 10.5802/ambp.390
Classification: 20F55, 20G42
Keywords: Cellular characters, Complex reflection groups
Abel Lacabanne 1

1 Institut de Recherche en Mathématique et Physique Université Catholique de Louvain Chemin du Cyclotron 2 1348 Louvain-la-Neuve, Belgium
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Abel Lacabanne. On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$. Annales mathématiques Blaise Pascal, Volume 27 (2020) no. 1, pp. 37-64. doi : 10.5802/ambp.390. https://ambp.centre-mersenne.org/articles/10.5802/ambp.390/

[1] Gwyn Bellamy Endomorphisms of Verma modules for rational Cherednik algebras, Transform. Groups, Volume 19 (2014) no. 3, pp. 699-720 | DOI | MR | Zbl

[2] Cédric Bonnafé Constructible characters and b-invariant, Bull. Belg. Math. Soc. Simon Stevin, Volume 22 (2015) no. 3, pp. 377-390 | MR | Zbl

[3] Cédric Bonnafé On the Calogero–Moser space associated with dihedral groups, Ann. Math. Blaise Pascal, Volume 25 (2018) no. 2, pp. 265-298 | DOI | MR | Zbl

[4] Cédric Bonnafé; Raphaël Rouquier Cherednik algebras and Calogero–Moser cells (2017) (https://arxiv.org/abs/1708.09764)

[5] Cédric Bonnafé; Ulrich Thiel Calogero-Moser families and cellular characters: computational aspects (in preparation)

[6] Michel Broué Introduction to complex reflection groups and their braid groups, Lecture Notes in Mathematics, 1988, Springer, 2010, xii+138 pages | DOI | MR | Zbl

[7] Charles W. Curtis; Irving Reiner Methods of representation theory: With applications to finite groups and orders, Pure and Applied Mathematics, Vol. I, John Wiley & Sons, 1981, xxi+819 pages | MR | Zbl

[8] Meinolf Geck; Nicolas Jacon Representations of Hecke algebras at roots of unity, Algebra and Applications, 15, Springer, 2011, xii+401 pages | DOI | MR | Zbl

[9] Masaki Kashiwara On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | DOI | MR

[10] Bernard Leclerc; Hyohe Miyachi Constructible characters and canonical bases, J. Algebra, Volume 277 (2004) no. 1, pp. 298-317 | DOI | MR | Zbl

[11] Bernard Leclerc; Philippe Toffin A simple algorithm for computing the global crystal basis of an irreducible U q ( sl n )-module, Int. J. Algebra Comput., Volume 10 (2000) no. 2, pp. 191-208 | DOI | MR | Zbl

[12] George Lusztig Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc., Volume 3 (1990) no. 2, pp. 447-498 | DOI | MR | Zbl

[13] George Lusztig Hecke algebras with unequal parameters, CRM Monograph Series, 18, American Mathematical Society, 2003, vi+136 pages | MR | Zbl

[14] Raphaël Rouquier q-Schur algebras and complex reflection groups, Mosc. Math. J., Volume 8 (2008) no. 1, p. 119-158, 184 | MR | Zbl

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