p-adic estimates of the number of permutation representations
p-adic estimates of the number of permutation representations
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排列表示数量的 p-adic 估计
DOI：
10.1016/j.aim.2019.04.008
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发表时间：
2019
影响因子：
1.7
通讯作者：
Yugen Takegahara
中科院分区：
文献类型：
--
作者：
A. Takahashi;K. Takahashi;S. Settepanella;Yugen Takegahara
设p为素数，设G为类型为λ=（λ 1, λ 2，…）且λ 1≥λ 2≥⋯∑λ i= s的有限阿贝p群。设u= max (λ 1,[(s+ 1)/2]}, v= s - u。对于每一个非负整数n，设h n (G)为G到对称群sn在n个字母上的同态个数。除了情况p = 2和u +δv 0≤v + 1,克罗内克符号δ,或p = 3 u = v≥1,存在p进分析函数f (X) r = 0, 1,…,p u + 1−1和一个多项式η(X)与整数系数,对于任何非负整数y, y h p u + 1 + r (G) = p{∑j = 1 u p−−(u v)} y f r (y)∏j = 1 yη(j)和奥德p y (h p u + 1 + r (G)) ={∑j = 1 u p−−(u v)} y +奥德p (f r (y))。若p= 2, λ 3= 0，且u= v≥1，或p= 3, u= v≥1，则h n (G)具有类似性质。在λ 3= 0的假设下，给出了关于p阶循环群的环积中G的置换表示个数的一些结果。
Let p be a prime, and let G be a finite abelian p-group of type λ=(λ 1, λ 2,…) with λ 1≥ λ 2≥⋯ and∑ λ i= s. Set u= max{λ 1,[(s+ 1)/2]} and v= s− u. For each nonnegative integer n, let h n (G) be the number of homomorphisms from G to the symmetric group S n on n letters. Except for the case where p= 2 and u+ δ v 0≤ v+ 1, δ the Kronecker delta, or p= 3 and u= v≥ 1, there exist p-adic analytic functions f r (X) for r= 0, 1,…, p u+ 1− 1 and a polynomial η (X) with integer coefficients such that for any nonnegative integer y, h p u+ 1 y+ r (G)= p {∑ j= 1 u p j−(u− v)} y f r (y)∏ j= 1 y η (j) and ord p (h p u+ 1 y+ r (G))={∑ j= 1 u p j−(u− v)} y+ ord p (f r (y)). If p= 2, λ 3= 0, and u= v≥ 1 or if p= 3 and u= v≥ 1, then h n (G) has analogous properties. Under the assumption that λ 3= 0, some results for the number of permutation representations of G in the wreath product of a cyclic group of order p with S n are also presented.