We propose a conjecture relating two different sets of characters for the complex reflection group . 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 irreducible integrable representations of . We prove this conjecture in some cases: in full generality for and for generic parameters for .
CC-BY 4.0
@article{AMBP_2020__27_1_37_0,
author = {Abel Lacabanne},
title = {On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {37--64},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {27},
number = {1},
year = {2020},
doi = {10.5802/ambp.390},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.390/}
}
TY - JOUR AU - Abel Lacabanne TI - On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$ JO - Annales mathématiques Blaise Pascal PY - 2020 SP - 37 EP - 64 VL - 27 IS - 1 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.390/ DO - 10.5802/ambp.390 LA - en ID - AMBP_2020__27_1_37_0 ER -
%0 Journal Article %A Abel Lacabanne %T On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$ %J Annales mathématiques Blaise Pascal %D 2020 %P 37-64 %V 27 %N 1 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.390/ %R 10.5802/ambp.390 %G en %F AMBP_2020__27_1_37_0
Abel Lacabanne. On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$. Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 37-64. doi : 10.5802/ambp.390. https://ambp.centre-mersenne.org/articles/10.5802/ambp.390/
[1] Endomorphisms of Verma modules for rational Cherednik algebras, Transform. Groups, Volume 19 (2014) no. 3, pp. 699-720 | DOI | MR | Zbl
[2] Constructible characters and -invariant, Bull. Belg. Math. Soc. Simon Stevin, Volume 22 (2015) no. 3, pp. 377-390 | MR | Zbl
[3] 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] Cherednik algebras and Calogero–Moser cells (2017) (https://arxiv.org/abs/1708.09764)
[5] Calogero-Moser families and cellular characters: computational aspects (in preparation)
[6] Introduction to complex reflection groups and their braid groups, Lecture Notes in Mathematics, 1988, Springer, 2010, xii+138 pages | DOI | MR | Zbl
[7] 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] Representations of Hecke algebras at roots of unity, Algebra and Applications, 15, Springer, 2011, xii+401 pages | DOI | MR | Zbl
[9] On crystal bases of the -analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | DOI | MR
[10] Constructible characters and canonical bases, J. Algebra, Volume 277 (2004) no. 1, pp. 298-317 | DOI | MR | Zbl
[11] A simple algorithm for computing the global crystal basis of an irreducible -module, Int. J. Algebra Comput., Volume 10 (2000) no. 2, pp. 191-208 | DOI | MR | Zbl
[12] Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc., Volume 3 (1990) no. 2, pp. 447-498 | DOI | MR | Zbl
[13] Hecke algebras with unequal parameters, CRM Monograph Series, 18, American Mathematical Society, 2003, vi+136 pages | MR | Zbl
[14] -Schur algebras and complex reflection groups, Mosc. Math. J., Volume 8 (2008) no. 1, p. 119-158, 184 | MR | Zbl
Cité par Sources :