The S3 character table - a (somewhat) new meaning | Diffusion Symmetry 2 | N J Wildberger

With diffusion symmetry, we explore mathematical objects or physical systems by spreading or diffusing from an initial point. The algebraic objects that result are hypergroups, or fusion algebra, or one of many similar and almost equivalent systems found in combinatorics, group theory, number theory, von Neumann algebras and conformal field theories.

Here we show how this point of view allows us to reconsider the meaning of the character table for S3, the smallest non-abelian finite group. In fact we go back and look not just at Frobenius' final take on characters as traces of irreducible representations of a group, but also on his earlier understanding of a character as a particular function on conjugacy classes which respects multiplicative structure.

In particular we will look at the hypergroup version of the S3 character table, and finally give also an new combinatorial interpretation of it.

Video Contents (thanks to Lucas Lofaro)

0:00 Intro
0:07 Overview
1:36 Character Table of The Symmetric Group S3
4:55 Irreducible Representations of S3
13:01 Characters of Commutative Groups
17:21 Characters of Non-Commutative Groups via Representations
19:36 Definition of the Character Table of a Group
21:40 Example of Multiplying Conjugacy Classes
25:59 Fusion Rule Algebra for Conjugacy Classes of S3
27:58 The Class Hypergroup of S3
31:02 The Hypergroup Character Table for S3
36:40 A geometric Interpretation

The Playlist for this series is at    • Плейлист  

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This is closely related to the Exceptional Structures in Mathematics and Physics lectures which are part of the Explore Research Level Mathematics video lectures. These are designed for motivated amateur, or professional, mathematicians wanting to delve into pure maths research without the abstraction and complexity that currently besets much of the subject.

To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at    / @wildeggmathematicscourses  , or become a Patron at   / njwildberger   . There are already dozens of videos there, and many more will appear over the next months and years. One additional advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.

Here are the Insights into Mathematics Playlists:

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Here are the Wild Egg Maths Playlists (some available only to Members!)

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