A Generalization of Circulant Matrices for Non-Abelian Groups

A circulant matrix of order n is the matrix of convolution by a fixed element of the group algebra of the cyclic group Zn. Replacing Zn by an arbitrary finite group G gives the class of matrices that we call G-circulant. We determine the eigenvalues of such matrices with the tools of representation theory and the non-abelian Fourier transform. Definition 1 An n by n matrix C is circulant if there exist c0, . . . , cn−1 such that the i, j entry of C is ci−j mod n, where the rows and columns are numbered from 0 to n − 1 and k mod n means the number between 0 and n− 1 that is congruent to k modulo n. For n = 5 a circulant matrix looks like  c0 c4 c3 c2 c1 c1 c0 c4 c3 c2 c2 c1 c0 c4 c3 c3 c2 c1 c0 c4 c4 c3 c2 c1 c0  Definition 2 Let G = {σ1, . . . , σn} be a finite group of order n. An n by n matrix C is G-circulant (with respect to the ordering of G) if the entry in row i and column j is a function of σiσ j . A circulant matrix is a Zn-circulant matrix with the ordering Zn = {0, 1, . . . , n − 1}. We call a matrix group-circulant if it is G-circulant for some group G and an ordering of the elements of G. Group-circulant matrices naturally arise as the transition matrices of Markov chains on finite groups. The state space is G and the probability of moving from τ to στ is pσ.