Positive definite functions of finitary isometry groups over fields of odd characteristic.
A group is said to be finitary linear if it acts faithfully on a vector space such that the fixed-point space of every element of the group has finite codimension in the vector space. Let \(G\) be a group. A normalized positive definite \(\mathbb{C}\)-linear class function is a function \(f\colon\mathbb{C} G\to\mathbb{C}\) such that \(f(1)=1\) and \(f(x^*x)\geq 0\) for all \(x\in\mathbb{C} G\), where * denotes a certain anti-automorphism of \(\mathbb{C} G\). Extremal positive definite class functions can be considered as generalized irreducible characters. The authors investigate finitary classical groups over finite fields of odd characteristic. They give a classification of extremal positive definite class functions for the orthogonal group, the unitary group, and the symplectic group.
scientific article; zbMATH DE number 5083177
scientific article