What are the characters of a group? | Representation theory episode 4

Exclusive content on Patreon: https://www.patreon.com/user?u=86649007 #grouptheory #linearalgebra #matrices #representationtheory #charactertheory A character is simply a homomorphism that maps each group element to a complex number. We look at the trivial charactrer, character tables, complex roots of unity, and the Schur orthogonality relations. There are many interesting connections between character theory and matrix representations, finite Fourier analysis, the Riemann zeta function, and more. Many thanks to professor Karel Dekimpe from the university of Leuven for helping me understand representations and characters. Don't forget to check out these links: [3B1B 1]    • Pi hiding in prime regularities   A video in which a multiplicative character function plays a central role. It helps count the number of dots inside a large circle, in order to find a value for pi. [KH 1]    • Introduction to analytic number theory 1: ...   In the first 2 videos of this playlist, Masoud Khalkhali talks about group characters. When we cycle through them, we can turn them into periodic functions. He then goes on to show how they are used as the coefficients of Dirichlet series. The multiplicative nature of the characters is the crucial ingredient that makes everything work. [WIKI 1] https://en.wikipedia.org/wiki/Divisor... Wikipedia article about the sigma function that counts the number of divisors of any positive integer. [WIKI 2] https://en.wikipedia.org/wiki/Multipl... Explanation about the properties of multiplicative functions. [UM 1]    • Video   Constructing the character table for Z4 (addition modulo 4, which I call C4 in the video). Shows how to decompose a 2D character into 1D parts. 0:00 Introduction 1:13 What are characters? 3:59 Characters and 1-dimensional representations 5:22 The character table 6:37 Characters for Abelian groups 8:26 Trivial characters 9:03 Irreducible characters are orthogonal 11:31 Finite Fourier analysis 12:40 The norm of a character 14:31 Character tables are orthogonal matrices 15:28 Number theory, Riemann Zeta function 19:15 Wrapping up This video is published under a CC Attribution license ( https://creativecommons.org/licenses/... )