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Painlevé equations and complex reflections

Philip Boalch (2003)

Annales de l’institut Fourier

We will explain how some new algebraic solutions of the sixth Painlevé equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-- Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.

Periodic subgroups of projective linear groups in positive characteristic

Alla Detinko, Dane Flannery (2008)

Open Mathematics

We classify the maximal irreducible periodic subgroups of PGL(q, 𝔽 ), where 𝔽 is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and 𝔽 × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, 𝔽 ) containing the centre 𝔽 ×1q of GL(q, 𝔽 ), such that G/ 𝔽 ×1q is a maximal periodic subgroup of PGL(q, 𝔽 ), and if H is another group of this kind then H is GL(q, 𝔽 )-conjugate to a group in the list. We give criteria for determining...

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