The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class of functions in the study of Boolean functions. A function $f:\{0,1\}^n \to \{0,1\}$ is called transitive (or weakly-symmetric) if there exists a transitive group $G$ of $S_n$ such that $f$ is invariant under the action of $G$ - that is the function value remains unchanged even after the bits of the input of $f$ are moved around according to some permutation $\sigma \in G$. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades. In this work, we study transitive functions in light of several combinatorial measures. We look at the maximum separation between various pairs of measures for transitive functions. Such study for general Boolean functions has been going on for past many years. The best-known results for general Boolean functions have been nicely compiled by Aaronson et. al (STOC, 2021). The separation between a pair of combinatorial measures is shown by constructing interesting functions that demonstrate the separation. But many of the celebrated separation results are via the construction of functions (like "pointer functions" from Ambainis et al. (JACM, 2017) and "cheat-sheet functions" Aaronson et al. (STOC, 2016)) that are not transitive. Hence, we don't have such separation between the pairs of measures for transitive functions. In this paper we show how to modify some of these functions to construct transitive functions that demonstrate similar separations between pairs of combinatorial measures.

翻译：在布尔函数$f:\{0,1\}^n \to \{0,1\}$中，对称性的作用已在复杂度理论中得到广泛研究。例如，对称函数（即在$S_n$作用下保持不变的函数）是布尔函数研究中的重要函数类别。若存在$S_n$的传递群$G$使得函数$f$在$G$作用下保持不变——即当输入比特按照$\sigma \in G$进行置换后函数值不变——则称函数$f:\{0,1\}^n \to \{0,1\}$为传递函数（或弱对称函数）。过去数十年间，理解传递函数的各种复杂度度量已成为一个丰富的研究领域。本文从若干组合度量的角度研究传递函数。我们关注传递函数中不同度量对之间的最大分离程度。针对一般布尔函数的此类研究已持续多年，其最著名结果由Aaronson等人（STOC 2021）系统整理。组合度量对之间的分离通常通过构造能体现该分离的特殊函数来证明。然而许多经典的分离结果依赖于非传递函数的构造（如Ambainis等人（JACM 2017）的"指针函数"和Aaronson等人（STOC 2016）的"作弊函数"），因此目前尚无传递函数中度量对之间此类分离的结果。本文展示了如何修改这些函数以构造能体现类似组合度量对分离的传递函数。