Fredman's reciprocity, invariants of abelian groups, and the permanent of the Cayley table
Dmitri I. Panyushev
DOI: 10.1007/s10801-010-0236-6
Abstract
Let C n {\mathcal{C}}_{n} denote the cyclic group of order n. For G= C n G={\mathcal{C}}_{n}, we compute the Poincaré series of all C n {\mathcal{C}}_{n}-isotypic components in 
(the symmetric tensor exterior algebra of 
). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili-Jibladze. Then we consider the Cayley table, 
, of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of 
equals 
, where n is the order of G.
(the symmetric tensor exterior algebra of ). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili-Jibladze. Then we consider the Cayley table, , of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of equals , where n is the order of G.Pages: 111–125
Keywords: keywords molien formula; Poincaré series; permanent; Ramanujan's sum
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References
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4. Elashvili, A.G., Jibladze, M.: Hermite reciprocity for the regular representations of cyclic groups. Indag. Math. 9(2), 233-238 (1998)
5. Elashvili, A.G., Jibladze, M.: “Hermite reciprocity” for semi-invariants in the regular representations of cyclic groups. Proc. Razmadze Math. Inst 119, 21-24 (1999) (Tbilisi)
6. Elashvili, A.G., Jibladze, M., Pataraia, D.: Combinatorics of necklaces and Hermite reciprocity. J. Al- gebraic Comb. 10, 173-188 (1999)
7. Fredman, M.: A symmetry relationship for a class of partitions. J. Comb. Theory, Ser. A 18, 199-202 (1975)
8. Hall, M.: A combinatorial problem on abelian groups. Proc. Am. Math. Soc. 3, 584-587 (1952)
9. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn., pp. xvi+426. Clarendon/Oxford University Press, New York (1979)
10. Johnson, K.W.: On the group determinant. Math. Proc. Camb. Philos. Soc. 109, 299-311 (1991)
11. Lehmer, D.: Some properties of circulants. J. Number Theory 5, 43-54 (1973)
12. Stanley, R.P.: Invariants of finite groups and their applications to combinatorics. Bull. Am. Math. Soc. (N.S.) 1(3), 475-511 (1979)
13. Thomas, H.: The number of terms in the permanent and the determinant of a generic circulant matrix.
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