Теория суперхарактеров для борелевской контракции группы GL(n, Fq)

Aleksandr N. Panov

doi:10.21638/11701/spbu01.2020.208

Authors

Aleksandr N. Panov Samara National Research University

DOI:

https://doi.org/10.21638/11701/spbu01.2020.208

Abstract

The notion of a supercharacter theory was introduсed by P. Diaconis and I. M. Isaacs in 2008. A supercharacter theory for a given finite group is a pair of a system of certain complex characters of the group and its partition into classes that have properties similar to the irreducible characters and conjugacy classes. In the present paper, we consider the group obtained by a group contraction from the general linear group over a finite field. For this group, we construct a supercharacter theory. In terms of rook placements, we classify supercharacters and superclasses, calculate values of supercharacters on superclasses.

Keywords:

group representations, irreducible characters, supercharacter theory, superclasses, algebra group

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References

1. Diaconis P., Isaacs I. M. Supercharacters and superclasses for algebra group // Trans. Amer. Math. Soc. 2008. Vol. 360. P. 2359-2392.

2. Панов А. Н. Суперхарактеры унипотентных и разрешимых групп // Итоги науки и техн. Сер. Соврем. мат. и ее прил. Темат. обз. 2017. Т. 136. С. 31-55.

3. André C. A. M. Basic characters of the unitriangular group // J. of Algebra. 1995. Vol. 175. P. 287-319.

4. André C. A. M. Basic sums of coadjoint orbits of the unitriangular group // J. of Algebra. 1995. Vol. 176. P. 959-1000.

5. André C. A. M. The basic character table of the unitriangular group // J. of Algebra. 2001. Vol. 241. P. 437-471.

6. Inönü E., Wigner E. P. On the contraction of groups and their representations // Proc. Nat. Acad. Sci. 1953. Vol. 39. P. 510-524.

7. Винберг Э. Б., Онищик А. Л., Горбацевич В. В. Группы и алгебры Ли - 3. В кн.: Cовр. пробл. матем. Фундам. направл. Т. 41. М.: Изд-во ВИНИТИ, 1990.

8. Панов А. Н. Теория суперхарактеров для групп обратимых элементов приведенных алгебр // Алгебра и анализ. 2015. Т. 27, № 6. C. 242-259.

9. André C. A. M. Hecke algebra for the basic representations of the unitriangular group // Proc. Amer. Math. Soc. 2003. Vol. 132. P. 987-996.

10. Aguiar M., André C., Benedetti C., Bergeron N., Zhi Chen, Diaconis P., Hendrickson A., Hsiao S., Isaacs I. M., Jedwab A., Johnson K., Karaali G., Lauve A., Tung Le, Lewis S., Huilan Li, Magaarg K., Marberg E., Novelli J.-Ch., Pang A., Saliola F., Tevlin L., Thibon J.-Y., Thiem N., Venkateswaran V., Vinroot C. R., Ning Yan, Zabricki M. Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras // Advances in Mathematics. 2012. Vol. 229. P. 2310- 2337.

11. Panov A. N. Supercharacters for ﬁnite groups of triangular type // Comm. Algebra. 2018. Vol. 46. P. 1032-1046.