Normal cyclotomic schemes over a finite commutative ring

Evdokimov, S.; Ponomarenko, I.

doi:10.1090/S1061-0022-08-01027-3

Normal cyclotomic schemes over a finite commutative ring
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by S. Evdokimov and I. Ponomarenko
Translated by: the authors

St. Petersburg Math. J. 19 (2008), 911-929

DOI: https://doi.org/10.1090/S1061-0022-08-01027-3

Published electronically: August 21, 2008

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Abstract:

Cyclotomic association schemes over a finite commutative ring $R$ with identity are studied. The main goal is to identify the normal cyclotomic schemes $\mathcal {C}$, i.e., those for which $\operatorname {Aut}(\mathcal {C})\le A\Gamma L_1(R)$. The problem reduces to the case where the ring $R$ is local, and in this case a necessary condition of normality in terms of the subgroup of $R^\times$ that determines $\mathcal {C}$ is given. This condition is proved to be sufficient for a large class of local rings including the Galois rings of odd characteristic.

References

A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989. MR 1002568, DOI 10.1007/978-3-642-74341-2
John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. MR 1409812, DOI 10.1007/978-1-4612-0731-3
S. A. Evdokimov, Schurity and separability of association schemes, Thesis for a Doctor’s Degree, S.-Peterburg. Gos. Univ., St. Petersburg, 2004. (Russian)
S. A. Evdokimov and I. N. Ponomarenko, Characterization of cyclotomic schemes and normal Schur rings over a cyclic group, Algebra i Analiz 14 (2002), no. 2, 11–55 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 14 (2003), no. 2, 189–221. MR 1925880
R. W. Goldbach and H. L. Claasen, Cyclotomic schemes over finite rings, Indag. Math. (N.S.) 3 (1992), no. 3, 301–312. MR 1186739, DOI 10.1016/0019-3577(92)90037-L
H. Hatalová and T. Šalát, Remarks on two results in the elementary theory of numbers, Acta Fac. Rerum Natur. Univ. Comenian. Math. 20 (1970), 113–117 (1970) (English, with Russian and Slovak summaries). MR 268115
Tatsuro Ito, Akihiro Munemasa, and Mieko Yamada, Amorphous association schemes over the Galois rings of characteristic $4$, European J. Combin. 12 (1991), no. 6, 513–526. MR 1136393, DOI 10.1016/S0195-6698(13)80102-3
Ka Hin Leung and Shing Hing Man, On Schur rings over cyclic groups. II, J. Algebra 183 (1996), no. 2, 273–285. MR 1399027, DOI 10.1006/jabr.1996.0220
R. McConnel, Pseudo-ordered polynomials over a finite field, Acta Arith. 8 (1962/63), 127–151. MR 164953, DOI 10.4064/aa-8-2-127-151
Bernard R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Vol. 28, Marcel Dekker, Inc., New York, 1974. MR 0354768
Carlos J. Moreno and Oscar Moreno, Exponential sums and Goppa codes. I, Proc. Amer. Math. Soc. 111 (1991), no. 2, 523–531. MR 1028291, DOI 10.1090/S0002-9939-1991-1028291-1
J.-L. Nicolas and G. Robin, Majorations explicites pour le nombre de diviseurs de $N$, Canad. Math. Bull. 26 (1983), no. 4, 485–492 (French, with English summary). MR 716590, DOI 10.4153/CMB-1983-078-5
Zhe-Xian Wan, Lectures on finite fields and Galois rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 2008834, DOI 10.1142/5350
Helmut Wielandt, Finite permutation groups, Academic Press, New York-London, 1964. Translated from the German by R. Bercov. MR 0183775
—, Permutation groups through invariant relations and invariant functions, Lecture Notes, Ohio State Univ. Columbus, Dept. Math., Ohio, 1969.