On cyclotomic matrices involving Gauss sums over finite fields
Hai-Liang Wu, Jie Li, Li-Yuan Wang, Chi Hoi Yip
TL;DR
This work studies cyclotomic-type matrices built from Gauss sums over finite fields, $A_q(k)$ and $B_q(k)$, as finite-field analogues of Gamma-function determinant problems. It proves integrality and generator-independence of $\det A_q(k)$, establishes a congruence modulo $p$, and provides explicit evaluations for $\det A_q(1)$ and $\det A_q(2)$, along with a general eigenvalue description and a nonsingularity criterion for $\det B_q(k)$ in terms of the order $o_k(p)$. Special cases yield exact determinants: $\det A_q(1)=(-1)^{\frac{(q-2)(q-3)}{2}}(q-1)^{q-1}$ and $\det A_q(2)=(-1)^{\alpha_n}\left(\frac{q-1}{2}\right)^{\frac{q-1}{2}}2^{\frac{p^{n-1}-1}{2}}$, with $\alpha_n=1$ if $n$ is odd and $(p^2+7)/8$ if $n$ is even. For $B_q(k)$, nonsingularity occurs exactly when $o_k(p)=n$, and explicit formulas are given for $k=1$ with $q=p$ and for $k=2$ when $p$ is odd. The results illuminate a finite-field analogue of Gamma-determinant theory, connecting Gauss sums, cyclotomic matrices, and determinant identities in a unified framework.
Abstract
Inspired by the works of L. Carlitz and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function. For example, let $q=p^n$ be an odd prime power with $p$ prime and $n\in\mathbb{Z}^+$. Let $ζ_p=e^{2π{\bf i}/p}$ and let $χ$ be a generator of the group of all mutiplicative characters of the finite field $\mathbb{F}_q$. For the Gauss sum $$G_q(χ^{r})=\sum_{x\in\mathbb{F}_q}χ^{r}(x)ζ_p^{{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(x)},$$ we prove that $$\det \left[G_q(χ^{2i+2j})\right]_{0\le i,j\le (q-3)/2}=(-1)^{α_p}\left(\frac{q-1}{2}\right)^{\frac{q-1}{2}}2^{\frac{p^{n-1}-1}{2}},$$ where $$α_p= \begin{cases} 1 & \mbox{if}\ n\equiv 1\pmod 2, (p^2+7)/8 & \mbox{if}\ n\equiv 0\pmod 2. \end{cases}$$