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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Symmetries of the hypergeometric function $ \phantom{}_mF_{m-1}$


Author: Oleg Gleizer
Journal: Trans. Amer. Math. Soc. 360 (2008), 2547-2580
MSC (2000): Primary 33C20
Published electronically: November 28, 2007
MathSciNet review: 2373325
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Abstract: In this paper, we show that the generalized hypergeometric function $ \phantom{}_mF_{m-1}$ has a one parameter group of local symmetries, which is a conjugation of a flow of a rational Calogero-Mozer system. We use the symmetry to construct fermionic fields on a complex torus, which have linear-algebraic properties similar to those of the local solutions of the generalized hypergeometric equation. The fields admit a nontrivial action of the quaternions based on the above symmetry. We use the similarity between the linear-algebraic structures to introduce the quaternionic action on the direct sum of the space of solutions of the generalized hypergeometric equation and its dual. As a side product, we construct a ``good'' basis for the monodromy operators of the generalized hypergeometric equation inspired by the study of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by Magyar, Weyman, and Zelevinsky. As an example of computational effectiveness of the basis, we give a proof of the existence of the monodromy invariant hermitian form on the space of solutions of the generalized hypergeometric equation (in the case of real local exponents) different from the proofs of Beukers and Heckman and of Haraoka. As another side product, we prove an elliptic generalization of Cauchy identity.


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Additional Information

Oleg Gleizer
Affiliation: Apartment 302, 309 S. Sherbourne Drive, Los Angeles, California 90048
Email: ogleizer@mac.com

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04369-3
PII: S 0002-9947(07)04369-3
Received by editor(s): March 21, 2005
Received by editor(s) in revised form: February 20, 2006
Published electronically: November 28, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.