class function
Given a field K, a K–valued class function on a group G is a function f:G⟶K such that f(g)=f(h) whenever g and h are elements of the same conjugacy class of G.
An important example of a class function is the character of a group representation. Over the complex numbers, the set of characters of the irreducible representations of G form a basis for the vector space of all ℂ–valued class functions, when G is a compact Lie group.
Relation to the convolution algebra
Class functions are also known as central functions, because they correspond to functions f in the convolution algebra C*(G) that have the property f*g=g*f for all g∈C*(G) (i.e., they commute with everything under the convolution operation). More precisely, the set of measurable complex valued class functions f is equal to the set of central elements of the convolution algebra C*(G), for G a locally compact group admitting a Haar measure.
Title | class function |
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Canonical name | ClassFunction |
Date of creation | 2013-03-22 12:18:06 |
Last modified on | 2013-03-22 12:18:06 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 8 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 20A05 |
Synonym | central function |