Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Brou\'e-Malle-Rouquier conjecture for the exceptional groups of rank
  2   Between 1994 and 1998, the work of M. Brou\'e, G. Malle, and R. Rouquier
generalized in a natural way the definition of the Hecke algebra associated to
a finite Coxeter group, for the case of an arbitrary complex reflection group.
Attempting to also generalize the properties of the Coxeter case, they stated a
number of conjectures concerning these Hecke algebras. One specific example of
importance regarding those yet unsolved conjectures is the so-called BMR
freeness conjecture. This conjecture is known to be true apart from 16 cases,
that are almost all the exceptional groups of rank 2. These exceptional groups
of rank 2 fall into three families: the tetrahedral, octahedral and icosahedral
family. We prove the validity of the BMR freeness conjecture for the
exceptional groups belonging to the first two families, using a case-by-case
analysis and we give a nice description of the basis, similar to the classical
case of the finite Coxeter groups. We also give a new consequence of this
conjecture, by obtaining the classification of irreducible representations of
the braid group on 3 strands in dimension at most 5, recovering results of Tuba
and Wenzl.

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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th