TY - JOUR

T1 - Traces of powers of matrices over finite fields

AU - Gorodetsky, Ofir

AU - Rodgers, Brad

N1 - Publisher Copyright:
© 2021 American Mathematical Society. All rights reserved.

PY - 2021

Y1 - 2021

N2 - LetM be a random matrix chosen according to Haar measure from the unitary group U(n,C). Diaconis and Shahshahani proved that the traces of M,M2, . . . ,Mk converge in distribution to independent normal variables as n→∞, and Johansson proved that the rate of convergence is superexponential in n. We prove a finite field analogue of these results. Fixing a prime power q = pr, we choose a matrix M uniformly from the finite unitary group U(n, q) ⊆ GL(n, q2) and show that the traces of {Mi} 1≤i≤k, pi converge to independent uniform variables in Fq2 as n. Moreover we show the rate of convergence is exponential in n2. We also consider the closely related problem of the rate at which characteristic polynomial of M equidistributes in 'short intervals' of Fq2 [T]. Analogous results are also proved for the general linear, special linear, symplectic and orthogonal groups over a finite field. In the two latter families we restrict to odd characteristic. The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial.

AB - LetM be a random matrix chosen according to Haar measure from the unitary group U(n,C). Diaconis and Shahshahani proved that the traces of M,M2, . . . ,Mk converge in distribution to independent normal variables as n→∞, and Johansson proved that the rate of convergence is superexponential in n. We prove a finite field analogue of these results. Fixing a prime power q = pr, we choose a matrix M uniformly from the finite unitary group U(n, q) ⊆ GL(n, q2) and show that the traces of {Mi} 1≤i≤k, pi converge to independent uniform variables in Fq2 as n. Moreover we show the rate of convergence is exponential in n2. We also consider the closely related problem of the rate at which characteristic polynomial of M equidistributes in 'short intervals' of Fq2 [T]. Analogous results are also proved for the general linear, special linear, symplectic and orthogonal groups over a finite field. In the two latter families we restrict to odd characteristic. The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial.

UR - http://www.scopus.com/inward/record.url?scp=85107856495&partnerID=8YFLogxK

U2 - 10.1090/tran/8337

DO - 10.1090/tran/8337

M3 - מאמר

AN - SCOPUS:85107856495

VL - 374

SP - 4579

EP - 4638

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -