The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant ${\gamma }_G(M)$ for a finite dimensional module M of a finite group G which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.

中文翻译：

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Benson - 普通和有符号置换模块的 Symonds 不变量

有符号置换模块是对称群的普通置换模块和扭曲置换模块的同时推广。在最近的一篇论文中，Dave Benson 和 Peter Symonds 定义了一个新的不变量${\gamma }_G(米)$对于有限群G的有限维模块M，它试图量化模块与投影的接近程度。在本文中，我们使用表示论和组合学的工具为对称群的所有有符号置换模块确定了这个不变量。