Maximal Length Projections in Group Algebras with Applications to Linear Rank Tests of Uniformity

Let \(G\) be a finite group, let \(\mathbb{C}G\) be the complex group algebra of \(G\), and let \(p \in \mathbb{C}G\). In this paper, we show how to construct submodules\(S\) of \(\mathbb{C}G\) of a fixed dimension with the property that the orthogonal projection of \(p\) onto \(S\) has maximal length. We then provide an example of how such submodules for the symmetric group \(S_n\) can be used to create new linear rank tests of uniformity in statistics for survey data that arises when respondents are asked to give a complete ranking of \(n\) items.