Representation theory of finite groups

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

RT1: Representation Theory Basics

Representation Theory: We present basic concepts about the representation theory of finite groups. Representations are defined, as are notions of invariant subspace, irreducibility and full reducibility. For a general finite group G, we suggest an analogue to the finite abelian case, whe

From playlist *** The Good Stuff ***

Representation theory: Abelian groups

This lecture discusses the complex representations of finite abelian groups. We show that any group is iomorphic to its dual (the group of 1-dimensional representations, and isomorphic to its double dual in a canonical way (Pontryagin duality). We check the orthogonality relations for the

From playlist Representation theory

RT6. Representations on Function Spaces

Representation Theory: We note how to transfer a group action of a group G on a set X to a group action on F(X), the functions on X. Because F(X) is a vector space, we obtain a representation of the group, and we can apply previous techniques. In particular, the group acts on itself in

From playlist Representation Theory

RT7.3. Finite Abelian Groups: Convolution

Representation Theory: We define convolution of two functions on L^2(G) and note general properties. Three themes: convolution as an analogue of matrix multiplication, convolution with character as an orthogonal projection on L^2(G), and using using convolution to project onto irreduci

From playlist Representation Theory

RT8.2. Finite Groups: Classification of Irreducibles

Representation Theory: Using the Schur orthogonality relations, we obtain an orthonormal basis of L^2(G) using matrix coefficients of irreducible representations. This shows the sum of squares of dimensions of irreducibles equals |G|. We also obtain an orthonormal basis of Class(G) usin

From playlist Representation Theory

RT2: Unitary Representations

Representation Theory: We explain unitarity and invariant inner products for representations of finite groups. Full reducibility of such representations is derived. Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html

From playlist Representation Theory

RT7.2. Finite Abelian Groups: Fourier Analysis

Representation Theory: With orthogonality of characters, we have an orthonormal basis of L^2(G). We note the basic philosophy behind the Fourier transform and apply it to the character basis. From this comes the definition of convolution, explored in 7.3. Course materials, including pro

From playlist Representation Theory

Representation theory: Induced representations

We define induced representations of finite groups in two ways as either left or right adjoints of the restriction functor. We calculate the character of an induced representation, and give an example of an induced representation of S3.

From playlist Representation theory

Representations of p-adic reductive groups by Tasho Kaletha

PROGRAM ZARISKI-DENSE SUBGROUPS AND NUMBER-THEORETIC TECHNIQUES IN LIE GROUPS AND GEOMETRY (ONLINE) ORGANIZERS: Gopal Prasad, Andrei Rapinchuk, B. Sury and Aleksy Tralle DATE: 30 July 2020 VENUE: Online Unfortunately, the program was cancelled due to the COVID-19 situation but it will

From playlist Zariski-dense Subgroups and Number-theoretic Techniques in Lie Groups and Geometry (Online)

Geordie Williamson 6 August 2020

Topic: Modular Representation Theory and Geometry Abstract: This will be a broad survey talk on interactions between geometry and representation theory, with a focus on representations in positive characteristic (“modular representation theory”). I will outline several basic questions (e.

From playlist Geordie Williamson external seminars

Parahoric Subgroups and Supercuspidal Representations of p-Adic groups - Dick Gross

Dick Gross Harvard University December 9, 2010 This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is th

From playlist Mathematics

On a Hecke algebra isomorphism of Kazhdan by Radhika Ganapathy

PROGRAM GROUP ALGEBRAS, REPRESENTATIONS AND COMPUTATION ORGANIZERS: Gurmeet Kaur Bakshi, Manoj Kumar and Pooja Singla DATE: 14 October 2019 to 23 October 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Determining explicit algebraic structures of semisimple group algebras is a fund

From playlist Group Algebras, Representations And Computation

13 - Deformations of Galois representations and applications

Orateur(s) : M. Emerton Public : Tous Date : jeudi 27 octobre Lieu : Institut Henri Poincaré

From playlist Colloque Evariste Galois

Moduli Stacks of Galois Representations by Mathew Emerton

Program Recent developments around p-adic modular forms (ONLINE) ORGANIZERS: Debargha Banerjee (IISER Pune, India) and Denis Benois (University of Bordeaux, France) DATE: 30 November 2020 to 04 December 2020 VENUE: Online This is a follow up of the conference organized last year arou

From playlist Recent Developments Around P-adic Modular Forms (Online)

Nigel Higson: Isomorphism conjectures for non discrete groups

The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: "The Farrell-Jones conjecture" I shall discuss aspects of the C*-algebraic version of the Farrell-Jones conjecture (namely the Baum-Connes conjecture) for Lie groups and p-adic groups. The conj

From playlist HIM Lectures: Junior Trimester Program "Topology"

Representation theory and geometry – Geordie Williamson – ICM2018

Plenary Lecture 17 Representation theory and geometry Geordie Williamson Abstract: One of the most fundamental questions in representation theory asks for a description of the simple representations. I will give an introduction to this problem with an emphasis on the representation theor

From playlist Plenary Lectures

Profinite rigidity – Alan Reid – ICM2018

Topology Invited Lecture 6.7 Profinite rigidity Alan Reid Abstract: We survey recent work on profinite rigidity of residually finite groups. © International Congress of Mathematicians – ICM www.icm2018.org     Os direitos sobre todo o material deste canal pertencem ao Instituto de Mat

From playlist Topology

RT7.1: Finite Abelian Groups: Character Orthogonality

We establish an analogue of Fourier analysis for a finite abelian group G. A decomposition of L^2(G) is given in terms of characters. Versions of Schur Orthogonality Relations and the Peter-Weyl Theorem are given. Course materials, including problem sets and solutions, available at htt

From playlist Representation Theory