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
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
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
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 ***
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
RT4.2. Schur's Lemma (Expanded)
Representation Theory: We introduce Schur's Lemma for irreducible representations and apply it to our previous constructions. In particular, we identify Hom(V,V) with invariant sesquilinear forms on V when (pi, V) is unitary. Course materials, including problem sets and solutions, availa
From playlist Representation Theory
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
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
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
Holomorphic Curves in Compact Quotients of SL(2,C) by Sorin Dumitrescu
DISCUSSION MEETING TOPICS IN HODGE THEORY (HYBRID) ORGANIZERS: Indranil Biswas (TIFR, Mumbai, India) and Mahan Mj (TIFR, Mumbai, India) DATE: 20 February 2023 to 25 February 2023 VENUE: Ramanujan Lecture Hall and Online This is a followup discussion meeting on complex and algebraic ge
From playlist Topics in Hodge Theory - 2023
The congruence subgroup property for SL(2,Z) - William Yun Chen
Arithmetic Groups Topic: The congruence subgroup property for SL(2,Z) Speaker: William Yun Chen Affiliation: Member, School of Mathematics Date: November 10, 2021 Somehow, despite the title, SL(2,Z) is the poster child for arithmetic groups not satisfying the congruence subgroup property
From playlist Mathematics
Richard Hain - 4/4 Universal mixed elliptic motives
Prof. Richard HAIN (Duke University, Durham, USA) Universal mixed elliptic motives are certain local systems over a modular curve that are endowed with additional structure, such as that of a variation of mixed Hodge structure. They form a tannakian category. The coordinate ring of its fu
From playlist Richard Hain - Universal mixed elliptic motives
A mini-course on vertex operator algebras of N= 2 Superconformal... (Lecture 3) by Madalena Lemos
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)
Talk by Charlotte Chan (MIT, USA)
Flag Varieties and Representations of p-adic Groups
From playlist Seminars: Representation Theory and Number Theory
Why do we care about characters of tilting modules? - Shotaro Makisumi
SL2 Seminar Topic: Why do we care about characters of tilting modules? Speaker: Shotaro Makisumi Affiliation: Columbia University; Member, School of Mathematics Date: January 26, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Modular forms of half-integral weight on exceptional groups
Joint IAS/Princeton University Number Theory Seminar Topic: Modular forms of half-integral weight on exceptional groups Speaker: Spencer Leslie Affiliation: Duke University Half-integral weight modular forms are classical objects with many important arithmetic applications. In terms of
From playlist Joint IAS/PU Number Theory Seminar
Jayce Getz: New avenues for the circle method, Lecture IV
Motivated by research arising from automorphic representation theory, I will present some ideas that should open up new avenues of research in the circle method. In the first half of the lectures I will discuss an adelic version of the delta-method of Duke, Friedlander, Iwaniec and Heath-B
From playlist Hausdorff School "The Circle Method"
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
How to diagonalize a functor - Benjamin Elias
Members' Seminar Topic: How to diagonalize a functor Speaker: Benjamin Elias Affiliation: University of Oregon; von Neumann Fellow, School of Mathematics Date: October 5, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics