The Characters of linear permutations

Abstract

Let m > 1 and k > 0 be integers. Let A be a k×k matrix of integers whose determinant dis prime to m. Let V be the space of all mk vectors x =(x1,…,xk) of integers with 0 > xi > m. Let P be the permutation induced on V by the linear transformation x'= Ax+b. We ask whether P is even or odd. In special cases this question was answered by Zolotarev (K = 1,mprime), Lerch(K = 1) and Schur (m odd) in their respective considerations of the quadratic reciprocity law, certain cyclotomic determinants and Gauss sums. The complete answer is sensitive when m is even to the three cases k = 1, k = 2 and k > 2. The second case being the difficult one since the answer depends on A rather than just d.