Morita equivalences of cyclotomic Hecke algebras of type G(r,p,n)

Jun Hu; Andrew Mathas

arXiv:0707.3316·math.RT·April 23, 2010

Morita equivalences of cyclotomic Hecke algebras of type G(r,p,n)

Jun Hu, Andrew Mathas

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TL;DR

This paper establishes a Morita equivalence reduction for cyclotomic Hecke algebras of type G(r,p,n), simplifying the computation of their decomposition numbers by relating them to smaller, more manageable cases.

Contribution

It proves a Morita reduction theorem for cyclotomic Hecke algebras of type G(r,p,n), enabling decomposition number calculations to be reduced to simpler subcases.

Findings

01

Reduction of decomposition number computation to p'-splittable cases

02

Parameters Q' lie in a single $( heta,q)$-orbit

03

Simplification of representation theory for cyclotomic Hecke algebras

Abstract

We prove a Morita reduction theorem for the cyclotomic Hecke algebras H_{r,p,n}({q,Q}) $o f t y p e G (r, p, n) . A s a co n se q u e n ce, w es h o w t ha t co m p u t in g t h e d eco m p os i t i o nn u mb er so f H_{r, p, n} (Q) r e d u ces t oco m p u t in g t h e p^{'} - s pl i tt ab l e d eco m p os i t i o nn u mb er so f t h ecy c l o t o mi cH ec k e a l g e b r a s H_{r^{'}, p^{'}, n^{'}} (Q^{'}), w h er e$ 1\le r'\le r $,$ 1\le n'\le n $,$ p'\mid p $an d w h er e t h e p a r am e t er s Q^{'} a r eco n t ain e d ina s in g l e$ (\epsilon,q) $- or bi t an d$ \epsilon$ is a primitive p'th root of unity.

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