Rational Representations of Finite Groups: The Story of &#947

Abstract

Mathematics is full of examples that illustrate the principle of extend and conquer. Put simply, mathematical results become clearer and more complete when the framework of the discussion is enlarged. The fundamental theorem of algebra is a good example. For it is usually difficult, if not impossible, to describe the number of integers or rational numbers that are solutions to a polynomial equation; but by enlarging the scope of the discussion to include complex numbers and by agreeing to count multiple solutions separately, we obtain the simple and elegant statement that every polynomial equation of positive degree n has exactly n complex solutions. This article discusses an example of the extend and conquer principle that arises in the representation theory of finite groupsthe Artin induction theorem. This theorem shows that every linear representation of a finite group over the field of rational numbers extends to a permutation representation of the group and, as such, provides a link between representing the elements of a finite group as linear transformations of a vector space and representing them as permutations of a set. But more importantly, when the Artin theorem is stated in terms of group characters, it provides a precise way to measure, numerically, how close the rational representations of the group are to being permutation representations. We denote this numerical measure by the symbol y. The purpose of this article is to discuss the history of y, its existence and known values, and to indicate briefly the local algebraic methods for determining this invariant.