Skip to main content
SpringerLink
Log in

Supercharacters of Unipotent and Solvable Groups

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The notion of the supercharacter theory was introduced by P. Diaconis and I. M. Isaaks in 2008. In this paper, we present a review of the main notions and facts of the general theory and discuss the construction of the supercharacter theory for algebra groups and the theory of basic characters for unitriangular groups over a finite field. Based on his earlier papers, the author constructs the supercharacter theory for finite groups of triangular type. The structure of the Hopf algebra of supercharacters for triangular groups over finite fields is also characterized.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Aguiar, C. Andrè, C. Benedetti, N. Bergeron, Chen Zhi, P. Diaconis, A. Hendrickson, S. Hsiao, I. M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, Tung Le, S. Lewis, Li Huilan, K. Magaarg, E. Marberg, J.-Ch. Novelli, Pang Amy, F. Saliola, L. Tevlin, J.-Y. Thibon, N. Thiem, V. Venkateswaran, C. R. Vinroot, Yan Ning, and M. Zabricki, “Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras,” Adv. Math., 229, No 4, 2310–2337 (2012).

    Article  MathSciNet  Google Scholar 

  2. C. A. M. André, “Basic characters of the unitriangular group,” J. Algebra, 175, 287–319 (1995).

    Article  MathSciNet  Google Scholar 

  3. C. A. M. André, “Basic sums of coadjoint orbits of the unitriangular group,” J. Algebra, 176, 959–1000 (1995).

    Article  MathSciNet  Google Scholar 

  4. C. A. M. André, “The basic character table of the unitriangular group,” J. Algebra, 241, 437–471 (2001).

    Article  MathSciNet  Google Scholar 

  5. C. A. M. André, “The basic characters of the unitriangular group (for arbitrary primes),” Proc. Am. Math. Soc., 130, 1943–1954 (2002).

    Article  MathSciNet  Google Scholar 

  6. C. A. M. André, “Basic sums of coadjoint orbits of the unitriangular group,” J. Algebra, 176, 959–1000 (1995).

    Article  MathSciNet  Google Scholar 

  7. C. A. M. André, “Hecke algebras for the basic characters of the unitriangular group,” Proc. Am. Math. Soc., 132, 987–996 (2003).

    Article  MathSciNet  Google Scholar 

  8. C. A. M. André, “Supercharacters of unitriangular groups and set-partition combinatorics,” in: CIMPA School: Modern Methods in Combinatorics ECOS 2013, San Luis, Argentina (2013).

  9. C. A. M. André and A. M. Neto, “A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups,” J. Algebra, 322, No. 4, 1273–1294 (2009).

    Article  MathSciNet  Google Scholar 

  10. C. A. M. André and A. Nicolàs, “Supercharacters of the adjoint group of a finite radical ring,” J. Group Theory, 11, 709–746 (2008).

    Article  MathSciNet  Google Scholar 

  11. E. Arias-Castro, P. Diaconis, and R. Stanley, “A super-class walk on upper-triangular matrices,” J. Algebra, 278, 739–765 (2004).

    Article  MathSciNet  Google Scholar 

  12. A. R. Asharafi and F. Koorepazan-Moftakhar, Towards the classification of finite simple groups with exactly three or four supercharacter threories, arXiv:1605.08971

  13. C. Benedett, “Combinatorial Hopf algebra of supercharacters of type D,” J. Alg. Combin., 38, No. 4, 767–783 (2013).

    Article  Google Scholar 

  14. J. L. Brumbaugh, M. Bulkow, P. S. Fleming, L. A. Garcia, S. R. Garcia, G. Karaali, M. Michal, A. P. Turner, H. Suh, “Supercharacters, exponential sums, and the uncertainty principle,” J. Number Theory, 144, 151–175 (2014).

    Article  MathSciNet  Google Scholar 

  15. S. Burkett, J. Lamar, M. L. Lewis, and C. Wynn, Groups with exactly two supercharacter theories, arXiv:1506.00015

  16. C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York–London (1962).

    MATH  Google Scholar 

  17. P. Diaconis and I. M. Isaacs, “Supercharacters and superclasses for algebra groups,” Trans. Am. Math. Soc., 360, 2359–2392 (2008).

    Article  MathSciNet  Google Scholar 

  18. C. F. Fowler, S. R. Garcia, and G. Karaali, “Ramanujan sums as supercharacters,” Ramanujan J., 32, 205–241 (2014).

    Article  MathSciNet  Google Scholar 

  19. D. Grinberg and V. Reiner, Hopf algebras in combinatorics, arXiv:1409.8356

  20. A. O. F. Hendrickson, “Supercharacter theory costructions corresponding to Schur ring products,” Commun. Algebra, 40, No. 12, 4420–4438 (2012).

    Article  Google Scholar 

  21. I. M. Isaacs and D. Karagueuzian, “Involution and characters of upper triangular matrix groups,” Math. Comput., 74, 2027–2033 (2005).

    Article  MathSciNet  Google Scholar 

  22. D. Kazhdan, “Proof of Springer hypothesis,” Isr. J. Math., 28, 272–284 (1977).

    Article  MathSciNet  Google Scholar 

  23. A. A. Kirillov, Lectures on the Orbit Method, Grad. Stud. Math., 64, Springer-Verlag (2004).

  24. A. A. Kirillov, “Variations on the triangular theme,” Am. Math. Soc. Trans., 169, 43–73 (1995).

    MathSciNet  MATH  Google Scholar 

  25. E. Marberg, “Combinatorial methods of character enumeration for the unitriangular group,” J. Algebra, 345, 295–323 (2011).

    Article  MathSciNet  Google Scholar 

  26. E. Marberg and N. Theim, “Superinduction for pattern groups,” J. Algebra, 321, 3681–3703 (2009).

    Article  MathSciNet  Google Scholar 

  27. B. Otto, “Constituents 0f supercharacters and Kirillov functions,” Arch. Math., 94, 319–326 (2010).

    Article  MathSciNet  Google Scholar 

  28. A. N. Panov, “Supercharacter theory for groups of invertible elements of reduced algebras,” St. Petersburg Math. J., 27, 1035–1047 (2016).

    Article  MathSciNet  Google Scholar 

  29. A. N. Panov, Supercharacters for the finite groups of triangular type, arXiv:1508.05767

  30. A. N. Panov, Restriction and induction for supercharacters of finite groups of triangular type, arXiv:1610.04846

  31. A. N. Panov, “Invariants of the coadjoint action on the basic varieties of the unitriangular group,” Transform. Groups, 20, No. 1, 229–246 (2015).

    Article  MathSciNet  Google Scholar 

  32. R. S. Pierce, Associative Algebras, Springer-Verlag, New York (1982).

    Book  Google Scholar 

  33. J.-P. Serr, Linear Representations of Finite Groups, Springer-Verlag, New York (1977).

    Book  Google Scholar 

  34. N. Thiem, “Branching rules in the sing of super functions of unipotent upper-triangular matrices,” J. Alg. Combin., 31, 267–298 (2010).

    Article  Google Scholar 

  35. N. Thiem and V. Venkateswaran, “Restricting supercharacters of the finite unipotent upertriangular matrices,” Electron. J. Combin., 16, No. 23 (2009).

  36. Ning Yan, Representation theory of finite unipotent linear groups, Ph.D. Thesis, Univ. of Pennsylvania (2001); arXiv: 1004.2674

  37. A. Zelevinsky, Representations of Classical Groups. A Hoph Algebra Approach, Lect. Notes Math., 869, Springer-Verlag, Berlin–Heidelberg–New York (1981).

Download references

Author information

Authors and Affiliations

  1. Samara University, Samara, Russia

    A. N. Panov

Authors
  1. A. N. Panov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. N. Panov.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 136, Proceedings of the Seminar on Algebra and Geometry of Samara University, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panov, A.N. Supercharacters of Unipotent and Solvable Groups. J Math Sci 235, 714–739 (2018). https://doi.org/10.1007/s10958-018-4090-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-4090-8

Keywords and phrases

AMS Subject Classification

Search

Navigation