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By Charles R. Johnson

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Ryser, Combinatorial Mathematics, Cams Math. Mon. No. 14, Math. Assoc, of America (Washington), 1963. [Ry3] H. J. Ryser, An extension of a theorem of de Bruijn and Erdos on combinatorial designs, J. Algebra 10 (1968), 246-261. [SaSc] B. D. Saunders and H. Schneider, Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices, Discrete Math. 24 (1978), 205-220. [Se] A. Seress, On ^-designs with X = 2p, preprint. [Sh] J. Shao, On a conjecture about the exponent set of primitive matrices, Lin.

Note that the matrix X is not unitary in non-Abelian examples. From this representation one sees that all the usual properties of circulants hold. One way to understand these circulants is to look back at Theorem 1. The matrix Dk has a lot of redundancy, with repeated blocks down the diagonal. For class functions, one can reduce this redundancy by looking only at one block for each representation. One can also consider / acting on A by convolution. This gives a linear map which also leads to simultaneously diagonalizable matrices.

R. Ford, Jr. and D. R. Fulkerson, Flows in Networks, Princeton Univ. Press (Princeton), 1962. [Fr] F. G. Frobenius, tJber die Primfactoren der Gruppendeterminante, Sitzungsberichte der Koniglich Preussischen Akodemie der Wissenschaften zu Berlin (1896), 1343-1382. [GraPo] R. L. Graham and H. O. st. Tech. J. 50 (1971), 2495-2519. [GrLoSc] M. Grfttschel, L. Loyasz, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 169-197. [GrLoSc] M.

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