Dominant regions in noncrystallographic hyperplane arrangements

Details Dominant regions in noncrystallographic hyperplane arrangements Dominant regions in noncrystallographic hyperplane arrangements

2006-01-09

arxiv.org/abs/math/0601191v2

Mathematics

Combinatorics

29 pages, 5 figures

Scientific paper

For a crystallographic root system, dominant regions in the Catalan hyperplane arrangement are in bijection with antichains in a partial order on the positive roots. For a noncrystallographic root system, the analogous arrangement and regions have importance in the representation theory of an associated graded Hecke algebra. Since there is also an analogous root order, it is natural to hope that a similar bijection can be used to understand these regions. We show that such a bijection does hold for type $H_3$ and for type $I_2(m)$, including arbitrary ratio of root lengths when $m$ is even, but does not hold for type $H_4$. We give a criterion that explains this failure and a list of the 16 antichains in the $H_4$ root order which correspond to empty regions.

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