The wavelet group and wavelet representation associated with shifts coming from a two dimensional crystal symmetry group \(\Gamma \) and dilations by powers of 3, are defined and studied. The main result is an explicit decomposition of this \(3\Gamma \)-wavelet representation into irreducible representations of the wavelet group. Because we prove that the \(3\Gamma \)-wavelet representation is multiplicity free, this direct integral decomposition is essentially unique.

中文翻译：

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按墙纸组分解小波表示的平移

定义和研究了与来自二维晶体对称群\（\ Gamma \）的位移相关联的小波组和小波表示以及以3的幂进行的扩张。主要结果是将此\（3 \ Gamma）-小波表示形式显式分解为小波组的不可约表示形式。因为我们证明了\（3 \ Gamma \）小波表示是无多重性的，所以这种直接积分分解本质上是唯一的。