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）我们研究了两个p-adic功率序列exp（f（x））和exp（g（x））的性质，这些属性出现在n字母上的替代组中的溶液数量中，n字母x^d = 1。此外，当分别将exp（f（x））和exp（g（x））视为序列{h_n}和{r_n}序列的指数生成函数时，我们通过p-adiC Analytic函数表达了H_n和R_N，h_n和r_n由h组成的组，并与所有内部自动组成的组一起识别h。并假设G的元素A的元素不是H的元素，并且A^2是H的一个元素。 n-字母c）我们认为g_n是来自有限组G的花圈产物与N-字母上的对称组的同构的核，该组由从G到环状基团C_M的同型c_m确定，该组由统一的统一统一的原始M-在复杂数字中产生。使用第一个正交关系，我们证明，来自有限生成的A组到G_N的同构数量的指数生成函数被描述为形式exp（f（x））的指数函数的总和，这些函数由e/φ_______________________的元素（a）确定。在C_M.D上，我们获得了从有限循环P组到对称的N字母组有限循环组的有限循环组的花圈产物的指数生成函数的P-ADIC特性。