Abstract
Let \(\lambda \) be a partition of an integer n and \({\mathbb F}_q\) be a finite field of order q. Let \(P_\lambda (q)\) be the number of strictly upper triangular \(n\times n\) matrices of the Jordan type \(\lambda \). It is known that the polynomial \(P_\lambda \) has a tendency to be divisible by high powers of q and \(Q=q-1\), and we put \(P_\lambda (q)=q^{d(\lambda )}Q^{e(\lambda )}R_\lambda (q)\), where \(R_\lambda (0)\ne 0\) and \(R_\lambda (1)\ne 0\). In this article, we study the polynomials \(P_\lambda (q)\) and \(R_\lambda (q)\). Our main results: an explicit formula for \(d(\lambda )\) (an explicit formula for \(e(\lambda )\) is known, see Proposition 3.3 below), a recursive formula for \(R_\lambda (q)\) (a similar formula for \(P_\lambda (q)\) is known, see Proposition 3.1 below), the stabilization of \(R_\lambda \) with respect to extending \(\lambda \) by adding strings of 1’s, and an explicit formula for the limit series \(R_{\lambda 1^\infty }\). Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.
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Acknowledgements
We are grateful to the anonymous reviewer for careful reading and useful remarks. We thank D. Golubenko for his help with a table for \(P_\lambda \).
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To the memory of Louiza Kirillova (1937–2021).
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Appendix: Table for \(P_\lambda ,\, n(\lambda )\le 10\).
Appendix: Table for \(P_\lambda ,\, n(\lambda )\le 10\).
The table below is presented in two columns on the next page and in one column in subsequent pages. In every line, the first entry is the notation for the partition \(\lambda \), the second entry is the part \(q^{d(\lambda )}Q^{e(\lambda )}\) of the polynomial \(P_\lambda \) and the sequence in square brackets is the sequence of coefficients of the polynomial \(R(\lambda )\) (starting with the constant term. For example, the line
means \(P_{31}=q^2Q^2(1+3q)\), and the line
means \(P_{51^3}=q^{15}Q^4(1+5q+15q^2+35q^3).\)
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Fuchs, D., Kirillov, A. Jordan Types of Triangular Matrices over a Finite Field. Arnold Math J. 8, 543–559 (2022). https://doi.org/10.1007/s40598-022-00214-1
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DOI: https://doi.org/10.1007/s40598-022-00214-1