The Calogero-Moser partition for \(G(m,d,n)\).
type: Item
instance of: scholarly article
zbMATH Open document ID: 1259.20003
publication date: 2012-09-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0911.0066
full work available at URL: https://projecteuclid.org/euclid.nmj/1343309818
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 16S99
zbMATH DE Number: 6081393
zbMATH Keywords: Calogero-Moser partitions
zbMATH Keywords: complex reflection groups
zbMATH Keywords: irreducible representations
zbMATH Keywords: cyclotomic Hecke algebras
zbMATH Keywords: rational Cherednik algebras
zbMATH Keywords: Rouquier blocks
author: Gwyn Bellamy
MaRDI profile type: Publication
arXiv ID: 0911.0066
arXiv classification: math.RT
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title: The Calogero-Moser partition for \(G(m,d,n)\).
published in: Nagoya Mathematical Journal
review text: Let \(W\) be a finite complex reflection group. The rational Cherednik algebras associated to \(W\) depend on two parameters \(t\) and \(\mathbf c\). The restricted rational Cherednik algebra is a certain finite-dimensional quotient of the rational Cherednik algebra at \(t=0\). The blocks of the restricted rational Cherednik algebra induce a partitioning of the set \(\text{Irr}(W)\) of irreducible \(W\)-modules, called Calogero-Moser partition. Using the geometry of certain quiver varieties, \textit{I. G. Gordon} and \textit{M. Martino} gave an explicit combinatorial description of the Calogero-Moser partition [in Math. Res. Lett. 16, No. 2-3, 255-262 (2009; Zbl 1178.16030)] when \(W=G(m,1,n)\).NEWLINENEWLINE The present paper shows that Clifford theoretic arguments can be used to extend this result to the normal subgroups \(G(m,d,n)\) of \(G(m,1,n)\). Gordon and Martino conjectured [in loc. cit.] that the Calogero-Moser partition should be related in some precise way to the Rouquier blocks of a particular Hecke algebra associated to the same complex reflection group \(W\). This conjecture was refined by \textit{M. Martino} [in J. Algebra 323, No. 1, 193-205 (2010; Zbl 1219.20006)] and was verified in the case of \(W=G(m,1,n)\). A consequence of the main result of the present paper is that the conjecture as stated by \textit{M. Martino} [loc. cit., Conjecture 2.7 (i)] is true for all \(G(m,d,n)\). However, it should be noted that, when \(n=2\) and \(d\) is even, there are certain unequal parameter cases where the method used in the present paper fails. In these cases, the Calogero-Moser partition is not known.
description: scientific article; zbMATH DE number 6081393