In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry. Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram, D4. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin(8). This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spin representations. These automorphisms do not project to automorphisms of SO(8). The vector representation—the natural action of SO(8) (hence Spin(8)) on F8—consists over the real numbers of Euclidean 8-vectors and is generally known as the "defining module", while the chiral spin representations are also known as "half-spin representations", and all three of these are fundamental representations. No other connected Dynkin diagram has an automorphism group of order greater than 2; for other Dn (corresponding to other even Spin groups, Spin(2n)), there is still the automorphism corresponding to switching the two half-spin representations, but these are not isomorphic to the vector representation. Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the Tits building associated with the group. For special linear groups, one obtains projective duality. For Spin(8), one finds a curious phenomenon involving 1-, 2-, and 4-dimensional subspaces of 8-dimensional space, historically known as "geometric triality". The exceptional 3-fold symmetry of the D4 diagram also gives rise to the Steinberg group 3D4. (Wikipedia).
Testing and Online Experimentation
Join Data Science Dojo and Statsig for a conversation on experimentation and testing. Learn how leading companies like Facebook use experimentation to build better products and accelerate their growth with 10x as much testing. Web experimentation can range from simple projects like design
From playlist A/B Testing & Beyond
Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this pheno
From playlist Algebra
What is the difference between theoretical and experimental physics?
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From playlist Science Unplugged: Physics
Building A Product From The Ground Up
For most seasoned business owners and aspiring entrepreneurs, the product development process often carries a mystical aura. Product development refers to the complete process of taking a product to market. It also covers renewing an existing product and introducing an old product to a new
From playlist Product Development
A short a cappella tribute to experimentalists. It is sung while performing three simple experiments with household items: Mentos dropped in diet Coke, a tea bag emptied and burned, and a ping pong ball floating in the air stream of a hair dryer.
From playlist Science Experiments!
Statistics Lesson #3: Randomized Experiments & Observational Studies
This video is for my College Algebra and Statistics students (and anyone else who may find it helpful). I define a randomized experiment, show a couple of examples, and define some important vocabulary related to experiments. Then I define an observational study, give an example, and discu
From playlist Statistics
What is a hypothesis test? The meaning of the null and alternate hypothesis, with examples. Overview of test statistics and confidence levels.
From playlist Hypothesis Tests and Critical Values
Monster simple group(Monster) 2 by Alexander Ivanov
DATE & TIME 05 November 2016 to 14 November 2016 VENUE Ramanujan Lecture Hall, ICTS Bangalore Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstra
From playlist Group Theory and Computational Methods
Relativistic dot products and complex numbers II 40b | Wild Linear Algebra B | NJ Wildberger
We continue discussing the triality of blue, red and green dot products, and introduce associated complex numbers. These are nicely organized inside the algebra of 2x2 matrices. Chromogeometry is the study of the interactions between these three geometries. ************************ Screen
From playlist WildLinAlg: A geometric course in Linear Algebra
Yaping Yang: The perverse coherent sheaves on toric Calabi-Yau 3-folds
30 September 2021 Yaping Yang: The perverse coherent sheaves on toric Calabi-Yau 3-folds and Cohomological Hall algebras Abstract: Let X be a smooth local toric Calabi-Yau 3-fold. On the cohomology of the moduli spaces of certain sheaves on X, there is an action of the cohomological Hall
From playlist Representation theory's hidden motives (SMRI & Uni of Münster)
Reliability 1: External reliability and rater reliability and agreement
In this video, I discuss external reliability, inter- and intra-rater reliability, and rater agreement.
From playlist Reliability analysis
Kostiantyn Drach: Box renormalization as a 'black box'
HYBRID EVENT Recorded during the meeting "Advancing Bridges in Complex Dynamics" the September 23, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Luca Récanzone Find this video and other talks given by worldwide mathematicians on CIRM's Audi
From playlist Dynamical Systems and Ordinary Differential Equations
Monoidal Structures on GL(2)-Modules and Abstractly Automorphic Representations - Gal Dor
Joint IAS/Princeton University Number Theory Seminar Topic: Monoidal Structures on GL(2)-Modules and Abstractly Automorphic Representations Speaker: Gal Dor Affiliation: Tel Aviv University Date: March 04, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Symmetries in QFT and their Relationship with Category Theory (Lecture 2) by Lakshya Bhardwaj
INFOSYS-ICTS STRING THEORY LECTURES SYMMETRIES IN QFT AND THEIR RELATIONSHIP WITH CATEGORY THEORY SPEAKER: Lakshya Bhardwaj (Mathematical Institute, University of Oxford) DATE : 10 October 2022 to 12 October 2022 VENUE: Madhava Lecture Hall (Hybrid) Lecture 1: 10 October 2022 at 3:30 pm
From playlist Infosys-ICTS String Theory Lectures
Research Methodology for Beginners || Research Methodology Lecture
Research methodology is the specific procedures or techniques used to identify, select, process, and analyze information about a topic. In a research paper, the methodology section allows the reader to critically evaluate a study's overall validity and reliability. In this course you w
From playlist Research Method
The Large N Limit of Chern Simons Vector Models: Past and Present by Shiroman Prakash
11 January 2017 to 13 January 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru String theory has come a long way, from its origin in 1970's as a possible model of strong interactions, to the present day where it sheds light not only on the original problem of strong interactions, but
From playlist String Theory: Past and Present
A02 Independence of the solution set
The independence of a linear system. How to make sure that a set of solutions are not constant multiples of each other.
From playlist A Second Course in Differential Equations
In Wildness: Walking Tasmania's Overland Track - Liam Carroll
Liam Carroll takes us on a virtual hike across a landscape of bits, and at the same time across his thoughts about his experiences leading hikes through the Tasmanian wilderness. He introduces us to the landscape, flora and fauna of the place, and explains what its Wildness means to him. W
From playlist metauni festival 2023
Twisted generating functions and the nearby Lagrangian conjecture - Sylvain Courte
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Topic: Twisted generating functions and the nearby Lagrangian conjecture Speaker: Sylvain Courte Affiliation: Université Grenoble Alpes Date: February 26, 2021 For more video please visit http://video.ias.edu Courte-2021-02
From playlist Mathematics
Validity, reliability and accuracy explained
What doe validity, reliability and accuracy mean in experiments? Watch and find out. Support me on Patreon - https://www.patreon.com/HighSchoolPhysicsExplained Find me on facebook - www.facebook.com/HighSchoolPhysicsExplained credit Pendulum animation - PhET Interactive Simulations Unive
From playlist general