Reflection groups and a convolution theorem
A converse to a theorem of \textit{M. L. Eaton} and \textit{M. D. Perlman} [Pac. J. Math. 73, 73-80 (1977; Zbl 0365.15007) and Ann. Probab. 5, 829-860 (1977; Zbl 0401.51006)] on convolutions of \(G\)-decreasing functions is proved. The original theorem states that the Convolution Theorem is satisfied by reflection subgroups of the orthogonal group. In this paper it is shown that if \(G\) is a finite linear group that satisfies the Convolution Theorem, then \(G\) is a reflection group. Furthermore, it is shown that if \(\rho\colon G\hookrightarrow\text{GL}(V)\) is a faithful representation that satisfies the Convolution Theorem, then \(\rho\) is a direct sum of the canonical Coxeter representation of \(G\) and a trivial representation. Both of these results concern a connection between the theory of reflection groups and a class of probability inequalities that is of interest to statisticians.
scientific article; zbMATH DE number 1136631