Abstract
The notion of a supercharacter theory was proposed by P. Diaconis and I.M. Isaacs in 2008. A supercharacter theory for a given finite group is a pair of the system of certain complex characters and the partition of group into classes that have properties similar to the system of irreducible characters and the partition into conjugacy classes. In the present paper, we consider the group obtained by the Borel contraction from the general linear group over a finite field. For this group, we construct the supercharacter theory. In terms of rook placements, we classify supercharacters and superclasses, calculate values of supercharacters on superclasses.
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REFERENCES
P. Diaconis and I. M. Isaacs, “Supercharacters and superclasses for algebra group,” Trans. Am. Math. Soc. 360, 2359–2392 (2008).
A. N. Panov, “Supercharacters of unipotent and solvable groups,” J. Math. Sci. (N. Y.) 235, 714–739 (2018).
C. A. M. André, “Basic characters of the unitriangular group,” J. Algebra 175, 287–319 (1995).
C. A. M. André, “Basic sums of coadjoint orbits of the unitriangular group,” J. Algebra 176, 959–1000 (1995).
C. A. M. André, “The basic character table of the unitriangular group,” J. Algebra 241, 437–471 (2001).
E. Inönü and E. P. Wigner, “On the contraction of groups and their representations,” in Proc. Natl. Acad. Sci. U. S. A. 39, 510–524 (1953).
V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg, Lie Groups and Lie Algebras III (VINITI, Moscow, 1990; Springer-Verlag, Berlin, 1994), in Ser.: Encyclopaedia of Mathematical Sciences, Vol. 41.
A. N. Panov, “Supercharacter theory for groups of invertible elements of reduced algebras,” St. Petersburg Math. J. 27, 1035–1047 (2016).
C. A. M. André, “Hecke algebra for the basic representations of the unitriangular group,” in Proc. Am. Math. Soc. 132, 987–996 (2003).
M. Aguiar, C. Andrè, C. Benedetti, N. Bergeron, Zhi Chen, P. Diaconis, A. Hendrickson, S. Hsiao, I. M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, Tung Le, S. Lewis, Huilan Li, K. Magaarg, E. Marberg, J.‑Ch. Novelli, A. Pang, F. Saliola, L. Tevlin, J.-Y. Thibon, N. Thiem, V. Venkateswaran, C. R. Vinroot, Ning Yan, and M. Zabricki, “Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras,” Adv. Math. 229, 2310–2337 (2012).
A. N. Panov, “Supercharacters for finite groups of triangular type,” Commun. Algebra 46, 1032–1046 (2018).
Funding
This work was supported by the Russian Foundation for Basic Research, project no. 20-01-00091a.
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To Aleksandr Ivanovich Generalov on occasion of the seventieth anniversary
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Panov, A.N. Supercharacter Theory for the Borel Contraction of the Group GL(n, \({{\mathbb{F}}_{q}}\)). Vestnik St.Petersb. Univ.Math. 53, 162–173 (2020). https://doi.org/10.1134/S1063454120020132
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DOI: https://doi.org/10.1134/S1063454120020132