Research on properties of generating functions of the number of special permutation representations and their applications.

特殊排列表示数生成函数的性质及其应用研究。

基本信息

批准号：
15540002
负责人：
TAKEGAHARA Yugen
金额：
$ 2.37万
依托单位：
MURORAN INSTITUTE OF TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540002/
关键词：
finite group homomorphism exponential generation function symmetric group complex reflection group Weyl group irreducible character p-adic property 交代群置換表現母関数 p進解析関数 Frobeniusの定理 Frobenius予想

项目摘要

a)We study the properties of two p-adic power series exp(f(x)) and exp(g(x)), which appear in the exponential generating function for the number of solutions in the alternating group on n-letters to the equation x^d=1. Moreover, when exp(f(x)) and exp(g(x)) are considered as the exponential generating functions for the sequences {h_n} and {r_n}, respectively, we express h_n and r_n by p-adic analytic functions.b)Let H be a finite simple group, and identify H with the group consisting of all inner automorphisms of H. Let G be a group of automorphisms of H, and suppose that an element a of G is not an element of H and that a^2 is an element of H. The conjecture : "if a divisor e of the order of H is a multiple of the largest power of 2 dividing the order of H and if the number of solutions in aH to the equation x^<2e>=1 is e, then every element of aH is a solution to the equation x^<2e>=1" is jure if H is the alternating group on n-letters.c)We consider G_n to be the kernel of a homomorphism from the wreath product of a finite group G with the symmetric group on n-letters which is determined by a homomorphism from G to the cyclic group C_m generated by a primitive m-th root of the unity in the complex numbers. Using the first orthogonality relation, we prove that the exponential generating function for the number of homomorphisms from a finitely generated group A to G_n is described as a sum of exponential functions of the form exp(f(x)) which are determined by elements cφ_m(A) of the factor group A/φ_m(A) of A by the intersection φ_m(A) of all kernels of homomorphisms from A to C_m.d)We obtain, p-adic properties of the exponential generating function of the number of homomorphisms from a finite cyclic p-group to the wreath product of a finite cyclic group of order p by the symmetric group of n-letters.

a) 我们研究两个 p 进幂级数 exp(f(x)) 和 exp(g(x)) 的性质，它们出现在 n 个字母的交替群中解的数量的指数生成函数中此外，当 exp(f(x)) 和 exp(g(x)) 分别被视为序列 {h_n} 和 {r_n} 的指数生成函数时，我们表示 h_n 和r_n b) 设 H 是一个有限单群，并将 H 等同于由 H 的所有内自同构组成的群。设 G 是 H 的一组自同构，并假设 G 的元素 a 是不是 H 的元素，并且 a^2 是 H 的元素。猜想：“如果 H 阶的除数 e 是除 H 阶的 2 的最大幂的倍数，并且如果解的数量aH 代入方程x^<2e>=1 是 e，那么 aH 的每个元素都是方程 x^<2e>=1 的解”，如果 H 是 n 个字母上的交替群，则成立。c）我们认为 G_n 是来自有限群 G 与 n 个字母上的对称群的花环乘积的同态核，该同态由从 G 到循环群 C_m 的同态确定，循环群 C_m 是由单位的本原 m 次根生成的使用第一个正交关系，我们证明从有限生成群 A 到 G_n 的同态数的指数生成函数被描述为 exp(f(x)) 形式的指数函数之和，这些指数函数被确定通过 A 的因子群 A 的元素 cφ_m(A)/A 的 φ_m(A) 通过从 A 到 C_m 的所有同态核的交集 φ_m(A)，我们得到，从有限循环 p 群到 p 阶有限循环群与 n 字母对称群的花环乘积的同态数的指数生成函数的 p 进属性。