Clifford群的不变量
The invariants of the Clifford groups
摘要：
2 ^m维Barnes-Wall格L_m（m不是3）的自同构群是某个"Clifford群“C_m（一个由正交群扩张的2^（1+ 2 m）阶超特殊群）中的指数为2的子群。这个群及其复类似物CC_m是近年来在正交展开、Kerdock集、Grassmannian空间中的填充、量子码、Siegel模形式和球面设计的构造中出现的。本文给出了Runge 1996年结果的一个较简单的证明：2k次C_m的不变量空间被长度为2k的二元自对偶码与域GF（2^m）张量得到的码的完全重量计数器所张成;当m >= k-1时，这些重量计数器是基。我们还给出了L_m和C_m的新构造：设M是具有Gram矩阵[2，sqrt（2）; sqrt（2），2]的Z[sqrt（2）]-格。则L_m是M的第m个张量幂幂的有理数部分，C_m是这个张量幂幂的自同构群。此外，如果C是一个二元自对偶码，不是由权为2的向量生成的，则C_m恰好是C和GF（2 ^m）的张量积的完备权枚举数的自同构群。对于复群CC_m，所有这些结果都有类似的结果，只是用"双偶自对偶码“代替"自对偶码”.
英文摘要：
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not 3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an extraspecial group of order 2^(1+2m) extended by an orthogonal group). This group and its complex analogue CC_m have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge's 1996 result that the space of invariants for C_m of degree 2k is spanned by the complete weight enumerators of the codes obtained by tensoring binary self-dual codes of length 2k with the field GF(2^m); these are a basis if m >= k-1. We also give new constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix [2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power of M, and C_m is the automorphism group of this tensor power. Also, if C is a binary self-dual code not generated by vectors of weight 2, then C_m is precisely the automorphism group of the complete weight enumerator of the tensor product of C and GF(2^m). There are analogues of all these results for the complex group CC_m, with ``doubly-even self-dual code'' instead of ``self-dual code''.
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