Reflection groups and a convolution theorem
type: Item
instance of: scholarly article
title: Reflection groups and a convolution theorem
zbMATH Open document ID: 0903.20016
author: Suren L. Fernando
published in: Pacific Journal of Mathematics
publication date: 1999-01-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 1136631
zbMATH Keywords: \(G\)-decreasing functions
zbMATH Keywords: convolution theorem
zbMATH Keywords: orthogonal groups
zbMATH Keywords: finite linear groups
zbMATH Keywords: faithful representations
zbMATH Keywords: Coxeter representations
zbMATH Keywords: reflection groups
reviewed by: Milica Stojanović
MaRDI profile type: Publication
full work available at URL: pjm.1997.180.229
OpenAlex ID: W2018096291
DOI: 10.2140/PJM.1997.180.229
description: scientific article; zbMATH DE number 1136631