We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huang's sensitivity theorem using "pseudo-characters", which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size $t$-intersecting families in the symmetric group and the perfect matching scheme.

中文翻译：

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对称群及其后的复杂度测度

我们将函数复杂性度量的定义扩展到对称群等域。我们考虑的复杂度度量包括度、近似度、决策树复杂度、敏感性、块敏感性和其他一些。我们表明这些复杂性度量对于对称群和许多其他域是多项式相关的。为了表明除敏感性之外的所有度量都是多项式相关的，我们概括了 Nisan 和其他人的经典论证。为了增加混合的敏感性，我们使用“伪字符”简化为 Huang 的敏感性定理，它见证了函数的程度。使用类似的想法，我们将 Ellis、Friedgut 和 Pilpel 对称群上的布尔 1 次函数的表征扩展到完美匹配方案。作为我们想法的另一种应用，