R-analogue of the Burnside ring of profinite groups and free Lie algebras

Oh, Young-Tak

doi:10.1016/j.aim.2003.10.004

Cited by 10 publications

(35 citation statements)

References 11 publications

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“…On the other hand, a group-theoretical generalization of chL n (V ) first appeared implicitly in [14] to reveal the structure of Witt-Burnside ring of a profinite group G over an arbitrary special λ-ring. To be more precise, to each open subgroup U of G and a finite alphabet X = {x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Group-theoretical generalization of necklace polynomials

2011

J Algebr Comb

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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 q G (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 q G (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 (C)-module whose character is M q Z (X, nZ), where m is the cardinality of X.

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Section: Introductionmentioning

confidence: 99%

Group-theoretical generalization of necklace polynomials

2011

J Algebr Comb

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“…For more information see [13,16]. Unless otherwise stated, the rings we consider will be commutative, but not necessarily unital.…”

Section: The Witt-burnside Ring and The Necklace Ring Of A Profinite mentioning

confidence: 99%

“…called Teichmüller map, for every special λ-ring A ( [13,16]). In the following, we will introduce its q-deformation…”

Section: For Every [U ] ∈ O(g) We Letmentioning

confidence: 99%

“…First, we recall the classical case, i.e., the case where q = 1. References are [13,14]. Given a special λ-ring A, restriction, Res…”

Section: Q-restrictionmentioning

confidence: 99%

“…Recently, in [13,16], it was shown that given any profinite group G, there exists a unique covariant functor N r G from the category of special λ-rings to the category of commutative rings with identity, satisfying…”

Section: Introductionmentioning

confidence: 99%

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q-deformation of Witt-Burnside rings

2007

Math. Z.

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Abstract. In this paper, we construct a q-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where q ranges over the set of integers. When q = 1, it coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. 70 (1988), 87-132). To achieve our goal we first show that there exists a q-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case q = 1, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We also study their properties. As a byproduct, we prove a conjecture due to Lenart (J. Algebra. 199 (1998), 703-732). Finally, we classify W q G up to strict natural isomorphism in case where G is an abelian profinite group.

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Necklace rings and logarithmic functions

2006

Advances in Mathematics

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In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor N r from the category of special λ-rings into the category of special λ-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper (J. Algebra. 253 (2002); no.1, 167-188) to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincaré principle.

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R-analogue of the Burnside ring of profinite groups and free Lie algebras

Cited by 10 publications

References 11 publications

Group-theoretical generalization of necklace polynomials

Group-theoretical generalization of necklace polynomials

q-deformation of Witt-Burnside rings

Necklace rings and logarithmic functions

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