Zeta functions of integral group rings of abelian (p,p)-groups

Takegahara, Yugen

doi:10.1080/00927878708823553

Cited by 8 publications

(8 citation statements)

References 4 publications

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“…In particular this applies to the Z-order ZB. For integral group rings of finite groups, this line of research was initiated by Solomon [42] and extended by work of Reiner [45], Hironaka [37], and Takegahara [46] between 1977 and 1987. Hanaki and Hirasaka generalized the idea to small ASs in 2015.…”

Section: The Algebrasmentioning

confidence: 99%

A survey of semisimple algebras in algebraic combinatorics

Herman

2021

Indian J Pure Appl Math

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This is a survey of semisimple algebras of current interest in algebraic combinatorics, with a focus on questions which we feel will be new and interesting to experts in group algebras, integral representation theory, and computational algebra. The algebras arise primarily in two families: coherent algebras and subconstituent (aka. Terwilliger) algebras. Coherent algebras are subalgebras of full matrix algebras having a basis of 01-matrices satisfying the conditions that it be transpose-closed, sum to the all 1's matrix, and contain a subset ∆ that sums to the identity matrix. The special case when ∆ is a singleton is the important case of an adjacency algebra of a finite association scheme. A Terwilliger algebra is a semisimple extension of a coherent algebra by a set of diagonal 01-matrices determined canonically from its basis elements and a choice of row. We will survey the current state of knowledge of the complex, real, rational, modular, and integral representation theory of these semisimple algebras, indicate their connections with other areas of mathematics, and present several open questions.

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Section: The Algebrasmentioning

confidence: 99%

A survey of semisimple algebras in algebraic combinatorics

Herman

2021

Indian J Pure Appl Math

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“…The explicit formulas for the zeta function of C 2 × C 2 and C 3 × C 3 presented by Takagehara [8] indicate that it will be considerably more difficult to calculate the zeta function of the tensor product of two integral adjacency algebras that are not locally coprime.…”

Section: Examples Of Explicit Zeta Functionsmentioning

confidence: 99%

Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes

Herman

Hirasaka

2017

Communications in Algebra

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The zeta function of an integral lattice Λ is the generating function ζ Λ (s) = ∞ n=0 a n n −s , whose coefficients count the number of left ideals of Λ of index n. We derive a formula for the zeta function of Λ 1 ⊗ Λ 2 , where Λ 1 and Λ 2 are Z-orders contained in finite-dimensional semisimple Q-algebras that satisfy a "locally coprime" condition. We apply the formula obtained above to ZS ⊗ ZT and obtain the zeta function of the adjacency algebra of the direct product of two finite association schemes (X, S) and (Y, T ) in several cases where the Z-orders ZS and ZT are locally coprime and their zeta functions are known.

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“…• G cyclic of prime order ( [1,11,12]); • G cyclic of order p 2 for p a prime ( [11]); • G cyclic of order p 3 for p a prime ( [14]); • G ∈ C 2 × C 2 C 3 × C 3 ( [13]); • G dihedral of order 2p for p an odd prime ( [2]); • G metacyclic ( [7,8]). …”

Section: Introductionmentioning

confidence: 99%

Zeta Functions of Lattices of the Symmetric Group

Hofmann

2016

Communications in Algebra

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The symmetric group $\mathfrak S_{n+1}$ of degree $n+1$ admits an $n$-dimensional irreducible $\mathbf Q \mathfrak S_n$-module $V$ corresponding to the hook partition $(2,1^{n-1})$. By the work of Craig and Plesken we know that there are $\sigma(n+1)$ many isomorphism classes of $\mathbf Z \mathfrak S_{n+1}$-lattices which are rationally equivalent to $V$, where $\sigma$ denotes the divisor counting function. In the present paper we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the $\mathbf Z \mathfrak S_{n+1}$-lattice defined by the Specht basis

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Zeta functions of integral group rings of abelian (p,p)-groups

Cited by 8 publications

References 4 publications

A survey of semisimple algebras in algebraic combinatorics

A survey of semisimple algebras in algebraic combinatorics

Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes

Zeta Functions of Lattices of the Symmetric Group

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