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    • Playlist   ************************ 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. ************** Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Insights into Mathematics YT channel:    / njwildberger   Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos) Wild Egg Maths Playlists:    • Box Arithmetic Book: A finite multiset fou...   (NEW!!)    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)    • The Hexagrammum Mysticum: a Geometric Gem ...   (14 videos)    • Algebraic Calculus Two   (8 videos)    • Special (Polynomial) Functions and Maxel N...   (25 videos)    • Dynamics on Graphs: The ADE Phenomenon & B...   (30 videos)