Group-theoretical generalization of necklace polynomials
Young-Tak Oh
Department of Mathematics, Sogang University, Seoul, 121-742 Korea
DOI: 10.1007/s10801-011-0307-3
Abstract
Let G be a group, U a subgroup of G of finite index, X a finite alphabet and q an indeterminate. In this paper, we study symmetric polynomials M G ( X, U) and M G q( X, U) M_{G}^{q}(X,U) which were introduced as a group-theoretical generalization of necklace polynomials. Main results are to generalize identities satisfied by necklace polynomials due to Metropolis and Rota in a bijective way, and to express M G q( X, U) M_{G}^{q}(X,U) in terms of M G ( X, V)'s, where [ V] ranges over a set of conjugacy classes of subgroups to which U is subconjugate. As a byproduct, we provide the explicit form of the GL m (\Bbb C)-module whose character is M \mathbb Z q( X, n\mathbb Z) M_{\mathbb{Z}}^{q}(X,n\mathbb{Z}), where m is the cardinality of X.
Pages: 389–420
Keywords: necklace polynomial; $G$-set and $G$-orbit; character; free Lie algebra; symmetric polynomial
Full Text: PDF
References
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2. Brandt, A.: The free Lie ring and Lie representations of the full linear group. Trans. Am. Math. Soc. 352, 528-536 (1944)
3. Dress, A., Siebeneicher, C.: The Burnside ring of profinite groups and the Witt vectors construction. Adv. Math. 70, 87-132 (1988)
4. Dress, A., Siebeneicher, C.: The Burnside ring of the infinite cyclic group and its relation to the necklace algebra, λ-ring and the Universal ring of the Witt vectors. Adv. Math. 78, 1-41 (1989)
5. Gauss, C.F.: Werke Band II, pp. 219-222. Königliche Gesellschaft der Wissenschaften, Göttingen (1863)
6. Hazewinkel, M.: Formal Groups and Applications. Academic Press, New York (1978)
7. Knutson, D.: λ-rings and the representation theory of the symmetric group. In: Lecture Notes in Math., vol.
308. Springer, Berlin (1973)
8. Labelle, G., Leroux, P.: An extension of the exponential formula in enumerative combinatorics. Electron. J. Comb. 3(2), 1-14 (1996)
9. Lenart, C.: Formal group-theoretic generalization of the necklace algebra, including a q-deformation. J. Algebra 199, 703-732 (1998)
10. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995)
11. Metropolis, N., Rota, G.-C.: Witt vectors and the algebra of necklaces. Adv. Math. 50, 95-125 (1983)
12. Moree, P.: The formal series Witt transform. Discrete Math. 295, 143-160 (2005)
13. Nikolov, N., Segal, D.: Finite index subgroups in profinite groups. C. R. Acad. Sci. Paris, Ser. I Math. 83, 303-308 (2003)
14. Oh, Y.-T.: R-analogue of the Burnside ring of profinite groups and free Lie algebras. Adv. Math. 190, 1-46 (2005) J Algebr Comb (2012) 35:389-420
15. Oh, Y.-T.: Necklace rings and logarithmic functions. Adv. Math. 205(2), 434-486 (2006)
16. Oh, Y.-T.: q-deformation of Witt-Burnside rings. Math. Z. 207, 151-191 (2007)
17. Oh, Y.-T.: q-analog of the Möbius function and the cyclotomic identity associated to a profinite group.
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