In mathematical representation theory, a minuscule representation of a semisimple Lie algebra or group is an irreducible representation such that the Weyl group acts transitively on the weights. Some authors exclude the trivial representation. A quasi-minuscule representation (also called a basic representation) is an irreducible representation such that all non-zero weights are in the same orbit under the Weyl group; each simple Lie algebra has a unique quasi-minuscule representation that is not minuscule, and the multiplicity of the zero weight is the number of short nodes of the Dynkin diagram. The minuscule representations are indexed by the weight lattice modulo the root lattice, or equivalently by irreducible representations of the center of the simply connected compact group. For the simple Lie algebras, the dimensions of the minuscule representations are given as follows. * An (n+1k) for 0 ≤ k ≤ n (exterior powers of vector representation). Quasi-minuscule: n2+2n (adjoint) * Bn 1 (trivial), 2n (spin). Quasi-minuscule: 2n+1 (vector) * Cn 1 (trivial), 2n (vector). Quasi-minuscule: 2n2–n–1 if n>1 * Dn 1 (trivial), 2n (vector), 2n−1 (half spin), 2n−1 (half spin). Quasi-minuscule: 2n2–n (adjoint) * E6 1, 27, 27. Quasi-minuscule: 78 (adjoint) * E7 1, 56. Quasi-minuscule: 133 (adjoint) * E8 1. Quasi-minuscule: 248 (adjoint) * F4 1. Quasi-minuscule: 26 * G2 1. Quasi-minuscule: 7 (Wikipedia).
From playlist Week 0 2015 Shorts
Representation theory: Introduction
This lecture is an introduction to representation theory of finite groups. We define linear and permutation representations, and give some examples for the icosahedral group. We then discuss the problem of writing a representation as a sum of smaller ones, which leads to the concept of irr
From playlist Representation theory
How do we represent the scalar of a vector
Learn the basics of vector operations. Vectors can be added, subtracted and multiplied. To add or subtract two or more vectors, we add each of the corresponding components of the vectors. To multiply a scalar to a vector, we simply multiply the scalar to each of the components of the vecto
From playlist Vectors
Ex: Write Numbers as Roman Numerals
This video explains how to write numbers when using Roman numerals. Site: http://mathispower4u.com
From playlist Roman Numerals
Visualization of tensors - part 1
This video visualizes tensors. It shows some introduction to tensor theory and demonstrates it with the Cauchy stress tensor. Future parts of this series will show more theory and more examples. It talks about the term 'tensor' as used in physics and math. In the field of AI the term 'te
From playlist Animated Physics Simulations
Reverse Plane Partitions and Modules for the Preprojective Algebra - Anne Dranowski
Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: Reverse Plane Partitions and Modules for the Preprojective Algebra Speaker: Anne Dranowski Affiliation: Member, School of Mathematics Date: November 19, 2020 For more video please visit http://video.ias.edu
From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory
Ex: Write the Number for Roman Numerals
This video explains how to determine the number when it is written using Roman numerals. Site: http://mathispower4u.com
From playlist Roman Numerals
Knot Categorification From Mirror Symmetry (Lecture- 1) by Mina Aganagic
PROGRAM QUANTUM FIELDS, GEOMETRY AND REPRESENTATION THEORY 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pan
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Robert Cass: Perverse mod p sheaves on the affine Grassmannian
28 September 2021 Abstract: The geometric Satake equivalence relates representations of a reductive group to perverse sheaves on an affine Grassmannian. Depending on the intended application, there are several versions of this equivalence for different sheaf theories and versions of the a
From playlist Representation theory's hidden motives (SMRI & Uni of Münster)
Rigidity and Flexibility of Schubert classes - Colleen Robles
Colleen Robles Texas A & M University; Member, School of Mathematics January 27, 2014 Consider a rational homogeneous variety X. The Schubert classes of X form a free additive basis of the integral homology of X. Given a Schubert class S in X, Borel and Haefliger asked: aside from the Schu
From playlist Mathematics
Represent a Discrete Function Using Ordered Pairs, a Table, and Function Notation
This video explains how to represent a discrete function given as points as ordered pairs, a table, and using function notation. http://mathispower4u.com
From playlist Introduction to Functions: Function Basics
How do we represent subtracting vectors graphically and algebraically
Learn the basics of vector operations. Vectors can be added, subtracted and multiplied. To add or subtract two or more vectors, we add each of the corresponding components of the vectors. To multiply a scalar to a vector, we simply multiply the scalar to each of the components of the vecto
From playlist Vectors
Knot Categorification From Mirror Symmetry (Lecture- 2) by Mina Aganagic
PROGRAM QUANTUM FIELDS, GEOMETRY AND REPRESENTATION THEORY 2021 (ONLINE) ORGANIZERS: Aswin Balasubramanian (Rutgers University, USA), Indranil Biswas (TIFR, india), Jacques Distler (The University of Texas at Austin, USA), Chris Elliott (University of Massachusetts, USA) and Pranav Pan
From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)
Joel Kamnitzer - Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry 3/5
In the 21st century, there has been a great interest in the study of symplectic resolutions, such as cotangent bundles of flag varieties, hypertoric varieties, quiver varieties, and affine Grassmannian slices. Mathematicians, especially Braden-Licata-Proudfoot-Webster, and physicists obser
From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory
The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties - Will Sawin
Joint IAS/Princeton University Number Theory Seminar Topic: The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties Speaker: Will Sawin Affiliation: Columbia University Date: March 18, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Vectors: Addition and Scalar Multiplication
This is the first video of a series from the Worldwide Center of Mathematics explaining the basics of vectors. This video deals with vector notation, vector addition, and scalar multiplication. For more math videos, visit our channel or go to www.centerofmath.org
From playlist Basics: Vectors
Bea Schumann - String Cones and Cluster Varieties
We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence. This is based on joint work with Gle
From playlist Combinatorics and Arithmetic for Physics: special days
Cong Xue - Cohomology of stacks of shtukas
Correction: The affiliation of Lei Fu is Tsinghua University. Let X be a geometrically connected smooth projective curve over Fq and G a reductive group over the function field of X. For any finite set I we have the stacks of shtukas over X^I, and the Satake sheaves over the stacks of sht
From playlist Conférence « Géométrie arithmétique en l’honneur de Luc Illusie » - 5 mai 2021
In this video, we'll learn how to view a complex number as a 2x2 matrix with a special form. We'll also see that there is a matrix version for the number 1 and a matrix representation for the imaginary unit, i. Furthermore, the matrix representation for i has the defining feature of the im
From playlist Complex Numbers
Models for Galois deformation rings - Brandon Levin
Workshop on Motives, Galois Representations and Cohomology Around the Langlands Program Topic: Models for Galois deformation rings Speaker: Brandon Levin Affiliation: University of Chicago Date: November 9, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics