Characters of 2-layered Heisenberg groups

Abstract

We give a classification of irreducible representations of generalized Heisenberg groups $K_n\left({\mathbb{F}}_q\right),n\ge 5,$ which is the pattern group associated to the closed set {(1, i), (2, j), (s, n − 1), (t, n) | 2 ≤ i ≤ n, 3 ≤ j ≤ n, 3 ≤ s < n − 1, 3 ≤ t < n}. In light of the conjectures of Higman, Lehrer and Isaacs for unitriangular groups, this result shows that the number of irreducible characters of $K_n\left({\mathbb{F}}_q\right)$ with a fixed degree is a polynomial in q−1 with non-negative integer coefficients.

Keywords:

• Coadjoint orbits
• unitriangular groups
• unipotent groups
• pattern groups
• Higman conjecture

Acknowledgments

The author is very grateful to the reviewer for his careful reading and suggestions for improving the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author is supported by National Natural Science Foundation of China (Grant no. 11971162) and the Construct Program of the Key Discipline in Hunan Province.