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Monodromy of A-hypergeometric functions

  • Frits Beukers EMAIL logo
Published/Copyright: July 19, 2014
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Journal für die reine und angewandte Mathematik
From the journal Journal für die reine und angewandte Mathematik Volume 2016 Issue 718

Abstract

Using Mellin–Barnes integrals we give a method to compute elements of the monodromy group of an A-hypergeometric system of differential equations. The method works under the assumption that the A-hypergeometric system has a basis of solutions consisting of Mellin–Barnes integrals. Hopefully these elements generate the full monodromy group, but this has only been verified in some special cases.

A Appendix

We reproduce the six matrices obtained from the monodromy calculation of Appell F2:

G1=(00100001-(1+c)/ab′)c/a⁢b′1/b′+1/a+c/a-c/a-1/a⁢b′01/a1/b′),
G2=(00100001-b/a⁢b′-1/c′b/c′1+b/a+1/c′b⁢(-1+1/b′-1/c′)-1/a⁢b′01/a1/b′),
G3=(00100001-1/c′01+1/c′00-1/c′01+1/c′),
G4=(0100-b′/a⁢b-1/c1+b′/a+1/cb′/cb′⁢(-1+1/b-1/c)0001-1/a⁢b1/a01/b),
G5=(0100-1/c1+1/c00000100-1/c1+1/c)
G6=(0100-(1+c′)/a⁢b1/b+1/a+c′/ac′/a⁢b-c′/a0001-1/a⁢b1/a01/b).

Now let g1,g2,g3,g4,g5 be the monodromy matrices defined in [20, formulas (2.7)–(2.11)] where our symbols a,b,b′,c,c′ are Kato’s symbols e⁢(a),e⁢(b),e⁢(b′),e⁢(c),e⁢(c′). Define the conjugation matrix

S=(-1cc′-c⁢c′-11c′-c′-1c1-c-111-1).

Then the relations between the Gi and gj are given by

G1=S-1⁢g2⁢g3⁢g5⁢S,G2=S-1⁢g2⁢g5⁢S,G3 =S-1g2S,
G4=S-1⁢g1⁢g4⁢S,G5=S-1⁢g1⁢S,G6 =S-1g1g3g4S.

From these relations it follows that the group we computed and the group computed in [20] are conjugate.

Acknowledgements

Many thanks to the referees whose detailed and important remarks have improved the presentation of the paper significantly.

References

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Received: 2013-06-26
Revised: 2014-03-11
Published Online: 2014-07-19
Published in Print: 2016-09-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Tight contact structures on the Brieskorn spheres -Σ(2,3,6n-1) and contact invariants
  3. Some examples of quasiisometries of nilpotent Lie groups
  4. Spans of special cycles of codimension less than 5
  5. The braided Thompson's groups are of type F∞
  6. Fonctions régulues
  7. A Dixmier--Douady theory for strongly self-absorbing C*-algebras
  8. Monodromy of A-hypergeometric functions
  9. Primitive ideals, twisting functors and star actions for classical Lie superalgebras
https://doi.org/10.1515/crelle-2014-0054

Articles in the same Issue

  1. Frontmatter
  2. Tight contact structures on the Brieskorn spheres -Σ(2,3,6n-1) and contact invariants
  3. Some examples of quasiisometries of nilpotent Lie groups
  4. Spans of special cycles of codimension less than 5
  5. The braided Thompson's groups are of type F∞
  6. Fonctions régulues
  7. A Dixmier--Douady theory for strongly self-absorbing C*-algebras
  8. Monodromy of A-hypergeometric functions
  9. Primitive ideals, twisting functors and star actions for classical Lie superalgebras
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