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The \({{\mathbb {F}}}_p\)-Selberg Integral

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Abstract

We prove an \({{\mathbb {F}}}_p\)-Selberg integral formula, in which the \({{\mathbb {F}}}_p\)-Selberg integral is an element of the finite field \({{\mathbb {F}}}_p\) with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p.

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Notes

  1. In [23] Selberg remarks: “This paper was published with some hesitation, and in Norwegian, since I was rather doubtful that the results were new. The journal is one which is read by mathematics-teachers in the gymnasium, and the proof was written out in some detail so it should be understandable to someone who knew a little about analytic functions and analytic continuation.” See more in [11].

References

  1. Anderson, G.W.: The evaluation of Selberg sums. C. R. Acad. Sci. Paris Sér. I Math. 311(8), 469–472 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  3. Aomoto, K.: Jacobi polynomials associated with Selberg integral. SIAM J. Math. 18(2), 545–549 (1987)

    Article  MathSciNet  Google Scholar 

  4. Askey, R.: Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11, 938–951 (1980)

    Article  MathSciNet  Google Scholar 

  5. Cherednik, I.: From double Hecke algebra to analysis. Doc. Math. J. DMV Extra Vol. ICM II, 527–531 (1998)

  6. Etingof, P., Frenkel, I., Kirillov, A.: Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations. Mathematical Surveys and Monographs, vol. 58. AMS, Providence (1998). ISBN:0-8218-0496-0

  7. Evans, R.J.: The evaluation of Selberg character sums. L’Enseign. Math. 37, 235–248 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Evans, R.J.: Selberg–Jack character sums of dimension 2. J. Number Theory 54(1), 1–11 (1995)

    Article  MathSciNet  Google Scholar 

  9. Felder, G., Stevens, L., Varchenko, A.: Elliptic Selberg integrals and conformal blocks. Math. Res. Lett. 10(5–6), 671–684 (2003)

    Article  MathSciNet  Google Scholar 

  10. Felder, G., Varchenko, A.: Integral representation of solutions of the elliptic Knizhnik–Zamolodchikov–Bernard equations. Int. Math. Res. Not. 5, 221–233 (1995)

    Article  MathSciNet  Google Scholar 

  11. Forrester, P.J., Warnaar, S.O.: The importance of the Selberg integral. Bull. Am. Math. Soc. (N.S.) 45, 489–534 (2008)

    Article  MathSciNet  Google Scholar 

  12. Habsieger, L.: Une \(q\)-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19, 1475–1489 (1988)

    Article  MathSciNet  Google Scholar 

  13. Kaneko, J.: \(q\)-Selberg integrals and Macdonald polynomials. Ann. Sci. Ecole Norm. Sup. 29, 583–637 (1996)

    Article  MathSciNet  Google Scholar 

  14. Knizhnik, V., Zamolodchikov, A.: Current algebra and the Wess–Zumino model in two dimensions. Nucl. Phys. B 247, 83–103 (1984)

    Article  MathSciNet  Google Scholar 

  15. Lucas, E.: Theorie des Fonctions Numeriques Simplement Periodiques. American Journal of Mathematics. 1(2), 184–196 (1878). https://doi.org/10.2307/2369308, JSTOR 2369308, MR 1505161

  16. Morris, W.G.: Constant term identities for finite and affine root system. Ph.D. Thesis, University of Wisconsin, Madison (1982)

  17. Mukhin, E., Varchenko, A.: Remarks on critical points of phase functions and norms of Bethe vectors. Adv. Stud. Pure Math. 27, 239–246 (2000)

    Article  MathSciNet  Google Scholar 

  18. Opdam, E.M.: Some applications of hypergeometric shift operators. Invent. Math. 98, 1–18 (1989)

    Article  MathSciNet  Google Scholar 

  19. Rains, E.: Multivariate quadratic transformations and the interpolation kernel. Contribution to the special issue on elliptic hypergeometric functions and their applications. SIGMA 14, 019 (2018). https://doi.org/10.3842/SIGMA.2018.019

  20. Rimányi, R., Tarasov, V., Varchenko, A., Zinn-Justin, P.: Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral. J. Geom. Phys. 62, 2188–2207 (2012)

    Article  MathSciNet  Google Scholar 

  21. Rimányi, R., Varchenko, A.: The \({\mathbb{F}}_p\)-Selberg integral of type \(A_n\). Lett. Math. Phys. 111, 71 (2021). https://doi.org/10.1007/s11005-021-01417-x

  22. Selberg, A.: Bemerkninger om et multipelt integral. Norsk Mat. Tidsskr. 26, 71–78 (1944)

    MathSciNet  Google Scholar 

  23. Selberg, A.: Collected Papers I, p. 212. Springer, Heidelberg (1989)

    MATH  Google Scholar 

  24. Spiridonov, V.: On the elliptic beta function. (Russian) Uspekhi Mat. Nauk 5 6(1), 181–182 (2001) [Translation in Russian Math. Surveys 56(1), 185–186 (2001)]

  25. Schechtman, V., Varchenko, A.: Arrangements of hyperplanes and Lie algebra homology. Invent. Math. 106, 139–194 (1991)

    Article  MathSciNet  Google Scholar 

  26. Schechtman, V., Varchenko, A.: Solutions of KZ differential equations modulo \(p\). Ramanujan J. 48(3), 655–683 (2019)

    Article  MathSciNet  Google Scholar 

  27. Slinkin, A., Varchenko, A.: Hypergeometric integrals modulo \(p\) and Hasse–Witt matrices. Arnold Math. J. 7, 267–311 (2021). https://doi.org/10.1007/s40598-020-00168-2

  28. Tarasov, V., Varchenko, A.: Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras. Invent. Math. 128, 501–588 (1997)

    Article  MathSciNet  Google Scholar 

  29. Tarasov, V., Varchenko, A.: Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups. Asterisque 246, 1–135 (1997)

    MATH  Google Scholar 

  30. Tarasov, V., Varchenko, A.: Selberg-type integrals associated with \(\mathfrak{sl}_3\). Lett. Math. Phys. 65, 173–185 (2003)

    Article  MathSciNet  Google Scholar 

  31. van Diejen, J.F., Spiridonov, V.P.: Elliptic Selberg integrals. IMRN 2001(20), 1083–1110 (2001). https://doi.org/10.1155/S1073792801000526

    Article  MathSciNet  MATH  Google Scholar 

  32. Varchenko, A.: Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups. Advanced Series in Mathematical Physics, vol. 21. World Scientific, Singapore (1995)

    Book  Google Scholar 

  33. Varchenko, A.: Special functions, KZ type equations, and representation theory. In: CBMS Regional Conference Series in Mathematics, Vol 98, pp. 118. ISBN: 978-0-8218-2867-0 (2003)

  34. Varchenko, A.: A Selberg integral type formula for an \(\mathfrak{sl}_2\) one-dimensional space of conformal blocks. Mosc. Math. J. 10(2), 469–475 (2010)

    Article  MathSciNet  Google Scholar 

  35. Varchenko, A.: Solutions modulo \(p\) of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz. Mathematics 5(4), 1–18, 52 (2017). https://doi.org/10.3390/math5040052

  36. Varchenko, A.: Hyperelliptic integrals modulo p and Cartier-Manin matrices. Pure Appl. Math. Q. 16(3), 315–336 (2020)

    Article  MathSciNet  Google Scholar 

  37. Varchenko, A.: An invariant subbundle of the KZ connection mod \(p\) and reducibility of \({\widehat{sl}}_2\) Verma modules mod \(p\). Math. Notes. 109(3), 386–397 (2021). https://doi.org/10.1134/S0001434621030068

  38. Varchenko, A.: Determinant of \({\mathbb{F}}_p\)-hypergeometric solutions under ample reduction, 1–22. arXiv:2010.11275

  39. Ole Warnaar, S.: Bisymmetric functions, Macdonald polynomials and \(mathfrak{sl}_3\) basic hypergeometric series. Compos. Math. 144, 271–303 (2008)

    Article  MathSciNet  Google Scholar 

  40. Ole Warnaar, S.: A Selberg integral for the Lie algebra \(A_n\). Acta Math. 203(2), 269–304 (2009)

    Article  MathSciNet  Google Scholar 

  41. Ole Warnaar, S.: The \(\mathfrak{sl}_3\) Selberg integral. Adv. Math. 224(2), 499–524 (2010)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3250, USA

    Richárd Rimányi & Alexander Varchenko

  2. Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, GSP-1, Leninskiye Gory 1, 119991, Moscow, Russia

    Alexander Varchenko

  3. Moscow Center of Fundamental and Applied Mathematics, GSP-1, Leninskiye Gory 1, 119991, Moscow, Russia

    Alexander Varchenko

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  1. Richárd Rimányi
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  2. Alexander Varchenko
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Correspondence to Richárd Rimányi.

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R. Rimányi supported in part by Simons Foundation Grant 523882, A. Varchenko supported in part by NSF Grant DMS-1954266.

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Rimányi, R., Varchenko, A. The \({{\mathbb {F}}}_p\)-Selberg Integral. Arnold Math J. 8, 39–60 (2022). https://doi.org/10.1007/s40598-021-00191-x

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