- published: 23 Jul 2023
- views: 85839
Group of Lie type
In mathematics, a group of Lie type is a group closely related to the group, G(k), of rational points of a reductive linear algebraic group, G, with values in the field, k. Finite groups of Lie type give the bulk of non-abelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
All Lie groups are groups of Lie type, but not vice-versa.
Dieudonné (1971) and Carter (1989) are standard references for groups of Lie type.
Classical groups
An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other fields by Jordan (1870). These groups were studied by L. E. Dickson and Jean Dieudonné. Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence.
A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series An, Bn, Cn, Dn,2An, 2Dn of Chevalley and Steinberg groups.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Group_of_Lie_type
Podcasts:
-
Why study Lie theory? | Lie groups, algebras, brackets #1
Next video: https://youtu.be/erA0jb9dSm0 Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets. Files for download: Go to https://www.mathemaniac.co.uk/download and enter the following password: lievideoseries Sources: History of group theory: https://en.wikipedia.org/wiki/History_of_group_theory Lie’s theory for differential equations (actually developed further after Lie died): https://arxiv.org/pdf/1901.01543.pdf https://www.physics.drexel.edu/~bob/LieGroups/LG_16.pdf https://www.researchgate.net/profile/Michael-Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differe...
published: 23 Jul 2023 -
Group of Lie type
If you find our videos helpful you can support us by buying something from amazon. https://www.amazon.com/?tag=wiki-audio-20 Group of Lie type In mathematics, a group of Lie type is a group closely related to the group, G(k), of rational points of a reductive linear algebraic group, G, with values in the field, k.Finite groups of Lie type give the bulk of non-abelian finite simple groups. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=FjVKDxTpBsQ
published: 22 Jan 2016 -
Exceptional Finite groups of Lie type and their Primitive actions, E.Pan (University of Warwick)
Young Group Theorists workshop: exploring new connections
published: 20 Oct 2022 -
What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3
Part 4: https://youtu.be/9CBS5CAynBE A bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular. Basically, Lie groups are groups and manifolds, and thinking about them as manifolds, we know that we want to understand Lie algebras; and thinking about them as groups, we know what additional structure we want on the Lie algebras - the Lie bracket. YouTube, please do not demonetise this video for me saying “Tits group”. This is an actual mathematical object named after a French mathematician Jacques Tits. Files for download: Go to https://www.mathemaniac.co.uk/download and enter the following password: so3embeddedin5dim SO(3) embedded in R^5: http://at.yorku.ca/b/ask-an-algebraic-topologist/2020/2618.htm https://en.wikipedia.org/wiki/W...
published: 10 Aug 2023 -
Representations of finite groups of Lie type (Lecture 1) by Dipendra Prasad
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 fundamental problem, which has played a central role in the development of the representation theory of finite groups. The tools of the representation theory of finite groups extend in various ways to profinite groups such as compact linear groups over the ring of integers of a local field (for example GL_n(Z_p)). However, the continuous representations or even representation growth of profinite groups is not well understood and is one of the current exciting areas of research. The importance of computationa...
published: 31 Dec 2019 -
Character theory of finite groups of Lie type (Meinolf Geck) 1
In these lectures we provide an introduction to Lusztig's classification of the irreducible characters of a finite group of Lie type. This essentially relies on structural properties of the underlying algebraic group, which will be surveyed in the first lecture. We then go on to discuss the partition of the set of characters into series and the Jordan decomposition of characters. Finally, we address the problem of computing character values, in the framework of Lusztig's theory of character sheaves
published: 04 Aug 2020 -
Gerhard Hiss: Representation theory for groups of Lie type I
The lecture was held within the framework of the Hausdorff Trimester Program: Logic and Algorithms in Group Theory. Abstract: The lectures begin by introducing finite groups of Lie type, in particular their structural properties relevant for their representation theory. Then some of the most fundamental goals and the state of the art will be discussed. Harish-Chandra theory, one of the major tools, will be presented. The focus will be on Deligne-Lusztig theory, which yields a classification of the irreducible complex representations of the groups of Lie type.
published: 26 Sep 2018 -
Jon Carlson: Endotrivial modules for finite groups of Lie type
Abstract: This is a report on joint work with Jesper Grodal, Nadia Mazza and Dan Nakano. Suppose that G is a finite group and that k is a field of characteristic p greater than 0. A kG-module M is an endotrivial module if Homk(M, M) = M∗ ⊗ M = k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modu...
published: 29 May 2024 -
Jesper Grodal: String topology of finite groups of Lie type
Abstract: Finite groups of Lie type, such as SL_n(F_q), Sp_n(F_q)..., are ubiquitous in mathematics, and calculating their cohomology has been a central theme over the years, and serves as an input to many other calculations, ranging from K-theory to stratification results. It has calculationally been observed that (when calculable) their mod ell cohomology agree with the mod ell cohomology of LBG(C), the free loop space on BG(C), the classifying space of the corresponding complex algebraic group G(C), as long as q is congruent to 1 mod ell. In recent joint work with Anssi Lahtinen, that combines ell-compact groups with string topology à la Chas-Sullivan, we attempt to provide a reason behind this "coincidence". I'll give an introduction to this line of thought.
published: 14 Dec 2022 -
Spinors for Beginners 16: Lie Groups and Lie Algebras
Full spinors playlist: https://www.youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs Leave me a tip: https://ko-fi.com/eigenchris Powerpoint slide files + Exercise answers: https://github.com/eigenchris/MathNotes/tree/master/SpinorsForBeginners Videos on momentum operator generating translations in quantum mechanics: Physics with Elliot: https://www.youtube.com/watch?v=_lz1VfI6Wxk&t=596s&pp=ygUecGh5c2ljcyB3aXRoIGVsbGlvdCBjb21tdXRhdG9y Professor M Does Science's video: https://www.youtube.com/watch?v=978mMgGYs1M&t=392s&pp=ygUkcHJvZmVzc29yIG0gZG9lcyBzY2llbmNlIHRyYW5zbGF0aW9u Quantum Sense': https://www.youtube.com/watch?v=A7yDvA8VQC8&t=305s&pp=ygUZcXVhbnR1bSBzZW5zZSB0cmFuc2xhdGlvbg%3D%3D 0:00 - Introduction 2:45 - Groups & Lie Groups 4:00 - Exponent of a so(3) Matrix 7:40 - Cal...
published: 10 Dec 2023
developed with YouTube
4:26
Why study Lie theory? | Lie groups, algebras, brackets #1
- Order: Reorder
- Duration: 4:26
- Uploaded Date: 23 Jul 2023
- views: 85839
Next video: https://youtu.be/erA0jb9dSm0
Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for p...
Next video: https://youtu.be/erA0jb9dSm0
Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets.
Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: lievideoseries
Sources:
History of group theory: https://en.wikipedia.org/wiki/History_of_group_theory
Lie’s theory for differential equations (actually developed further after Lie died):
https://arxiv.org/pdf/1901.01543.pdf
https://www.physics.drexel.edu/~bob/LieGroups/LG_16.pdf
https://www.researchgate.net/profile/Michael-Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf
https://en.wikipedia.org/wiki/Lie_point_symmetry
Lie theory applications to mathematics: https://math.stackexchange.com/questions/1322206/what-are-applications-of-lie-groups-algebras-in-mathematics
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: https://mathemaniac.myspreadshop.co.uk
Ko-fi: https://ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
https://wn.com/Why_Study_Lie_Theory_|_Lie_Groups,_Algebras,_Brackets_1
Next video: https://youtu.be/erA0jb9dSm0
Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets.
Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: lievideoseries
Sources:
History of group theory: https://en.wikipedia.org/wiki/History_of_group_theory
Lie’s theory for differential equations (actually developed further after Lie died):
https://arxiv.org/pdf/1901.01543.pdf
https://www.physics.drexel.edu/~bob/LieGroups/LG_16.pdf
https://www.researchgate.net/profile/Michael-Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf
https://en.wikipedia.org/wiki/Lie_point_symmetry
Lie theory applications to mathematics: https://math.stackexchange.com/questions/1322206/what-are-applications-of-lie-groups-algebras-in-mathematics
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: https://mathemaniac.myspreadshop.co.uk
Ko-fi: https://ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
16:02
Group of Lie type
- Order: Reorder
- Duration: 16:02
- Uploaded Date: 22 Jan 2016
- views: 198
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Group of Lie type
In mathematic...
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Group of Lie type
In mathematics, a group of Lie type is a group closely related to the group, G(k), of rational points of a reductive linear algebraic group, G, with values in the field, k.Finite groups of Lie type give the bulk of non-abelian finite simple groups.
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=FjVKDxTpBsQ
https://wn.com/Group_Of_Lie_Type
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Group of Lie type
In mathematics, a group of Lie type is a group closely related to the group, G(k), of rational points of a reductive linear algebraic group, G, with values in the field, k.Finite groups of Lie type give the bulk of non-abelian finite simple groups.
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=FjVKDxTpBsQ
- published: 22 Jan 2016
- views: 198
15:45
Exceptional Finite groups of Lie type and their Primitive actions, E.Pan (University of Warwick)
- Order: Reorder
- Duration: 15:45
- Uploaded Date: 20 Oct 2022
- views: 70
Young Group Theorists workshop: exploring new connections
Young Group Theorists workshop: exploring new connections
https://wn.com/Exceptional_Finite_Groups_Of_Lie_Type_And_Their_Primitive_Actions,_E.Pan_(University_Of_Warwick)
Young Group Theorists workshop: exploring new connections
- published: 20 Oct 2022
- views: 70
21:57
What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3
- Order: Reorder
- Duration: 21:57
- Uploaded Date: 10 Aug 2023
- views: 311898
Part 4: https://youtu.be/9CBS5CAynBE
A bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular.
Basically...
Part 4: https://youtu.be/9CBS5CAynBE
A bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular.
Basically, Lie groups are groups and manifolds, and thinking about them as manifolds, we know that we want to understand Lie algebras; and thinking about them as groups, we know what additional structure we want on the Lie algebras - the Lie bracket.
YouTube, please do not demonetise this video for me saying “Tits group”. This is an actual mathematical object named after a French mathematician Jacques Tits.
Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: so3embeddedin5dim
SO(3) embedded in R^5: http://at.yorku.ca/b/ask-an-algebraic-topologist/2020/2618.htm
https://en.wikipedia.org/wiki/Whitney_embedding_theorem (n-dim manifold can be properly embedded in R^(2n): if you only want “the overall picture”, but perhaps distances are distorted)
https://en.wikipedia.org/wiki/Nash_embedding_theorems (n-dim Riemannian manifold can be isometrically embedded in n(3n+11)/2 dim if compact, n(n+1)(3n+11)/2 dim if not compact: if you want everything to remain intact, i.e. distances are preserved)
BCH formula (why Lie brackets are useful): https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula
Finite simple groups as building blocks: https://en.wikipedia.org/wiki/Composition_series
Classification of finite simple groups: https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
Levi decomposition (the more precise meaning of “building blocks” in Lie algebra): https://en.wikipedia.org/wiki/Levi_decomposition
E8 (the monster group of Lie algebras):
https://aimath.org/E8/e8.html
https://en.wikipedia.org/wiki/E8_(mathematics)
https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything
Video chapters:
00:00 Introduction
01:26 Lie groups - groups
05:41 Lie groups - manifolds
10:23 Lie algebras
14:16 Lie brackets
18:03 The "Lie theory picture"
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: https://mathemaniac.myspreadshop.co.uk
Ko-fi: https://ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
https://wn.com/What_Is_Lie_Theory_Here_Is_The_Big_Picture._|_Lie_Groups,_Algebras,_Brackets_3
Part 4: https://youtu.be/9CBS5CAynBE
A bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular.
Basically, Lie groups are groups and manifolds, and thinking about them as manifolds, we know that we want to understand Lie algebras; and thinking about them as groups, we know what additional structure we want on the Lie algebras - the Lie bracket.
YouTube, please do not demonetise this video for me saying “Tits group”. This is an actual mathematical object named after a French mathematician Jacques Tits.
Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: so3embeddedin5dim
SO(3) embedded in R^5: http://at.yorku.ca/b/ask-an-algebraic-topologist/2020/2618.htm
https://en.wikipedia.org/wiki/Whitney_embedding_theorem (n-dim manifold can be properly embedded in R^(2n): if you only want “the overall picture”, but perhaps distances are distorted)
https://en.wikipedia.org/wiki/Nash_embedding_theorems (n-dim Riemannian manifold can be isometrically embedded in n(3n+11)/2 dim if compact, n(n+1)(3n+11)/2 dim if not compact: if you want everything to remain intact, i.e. distances are preserved)
BCH formula (why Lie brackets are useful): https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula
Finite simple groups as building blocks: https://en.wikipedia.org/wiki/Composition_series
Classification of finite simple groups: https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
Levi decomposition (the more precise meaning of “building blocks” in Lie algebra): https://en.wikipedia.org/wiki/Levi_decomposition
E8 (the monster group of Lie algebras):
https://aimath.org/E8/e8.html
https://en.wikipedia.org/wiki/E8_(mathematics)
https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything
Video chapters:
00:00 Introduction
01:26 Lie groups - groups
05:41 Lie groups - manifolds
10:23 Lie algebras
14:16 Lie brackets
18:03 The "Lie theory picture"
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: https://mathemaniac.myspreadshop.co.uk
Ko-fi: https://ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
- published: 10 Aug 2023
- views: 311898
49:11
Representations of finite groups of Lie type (Lecture 1) by Dipendra Prasad
- Order: Reorder
- Duration: 49:11
- Uploaded Date: 31 Dec 2019
- views: 2393
PROGRAM : GROUP ALGEBRAS, REPRESENTATIONS AND COMPUTATION
ORGANIZERS: Gurmeet Kaur Bakshi, Manoj Kumar and Pooja Singla
DATE: 14 October 2019 to 23 October 20...
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 fundamental problem, which has played a central role in the development of the representation theory of finite groups. The tools of the representation theory of finite groups extend in various ways to profinite groups such as compact linear groups over the ring of integers of a local field (for example GL_n(Z_p)). However, the continuous representations or even representation growth of profinite groups is not well understood and is one of the current exciting areas of research. The importance of computational methods in all pursuits of pure mathematics is no more obscure, and the subject has established itself as a powerful tool, aiding quick maturing of intuition about concrete mathematical structures. The focus of this program is on theoretical aspects of group algebras and representation theory of finite and profinite groups complemented by computational techniques using discrete algebra system GAP.
The first part of this program, a 5-day instructional workshop during October 14 - 18, 2019, is planned to touch upon various aspects of group algebras and representation theory, including Wedderburn decomposition as well as idempotents of group algebra and their applications, representations of finite simple groups of Lie type and p-adic analytic groups, representation growth and computational methods.
It will consist of three mini-courses:
Representations of finite and profinite groups
Structure of group algebras
Computations with finite groups
The plan is to have five hours of teaching and five hours of tutorials and guided practice sessions in each mini-course.
The second part of the program, a 4-day discussion meeting during October 20-23, 2019, is aimed at exploring some of the recent developments and problems of current mathematical interest in the above mentioned and related areas. Eminent speakers from India as well as abroad will present their work on a variety of topics. To encourage discussions among the participants, plenty of time will be provided between the talks. It is planned to have poster presentations by young researchers to manifest their mathematical thoughts and to foster interactions with the senior mathematicians. We'll dedicate October 21, 2019, to group rings, "Group Ring Day", in honor of the 80th birthday of Professor I.B.S. Passi.
The participants, Ph.D. students, post-doctoral fellows, and young faculty members, are expected to have exposure to advanced group theory and to have a good grasp on basic algebra and representation theory.
Eligibility criteria: The program is mainly aimed at Ph.D. students, post-doctoral fellows, and young faculty members. However, a couple of highly motivated final year master students shall also be given a chance to participate in the first part of the program.
CONTACT US: [email protected]
PROGRAM LINK: https://www.icts.res.in/program/garc2019
Table of Contents (powered by https://videoken.com)
0:00:00 Representations of finite groups of Lie type (Lecture 1)
0:01:16 Introduction
0:01:50 Recall one of the most basic representation considered for groups
0:06:16 Trivial and dimensional representation
0:08:08 Steinberg representation
0:09:25 Principle series
0:09:42 Boreal subgroup
0:10:53 Induced representation
0:14:10 Basic building blocks of representation theory
0:17:27 Group theory
0:21:46 Theory for Gl(Fq) and very analogous theory for Gln(Fq)
0:27:02 Linguistic theory
0:27:40 Philosophy of cusp forms (Harish Chandra)
0:28:53 Glossary of terms
0:31:38 Appealing and reductive groups
0:32:00 Borel subgroups
0:32:45 Parabolic subgroups
0:35:45 Symmetric groups
0:36:29 Twisted groups
0:42:50 The representation of theory of the group G is reduce to understand cuspital representation of all smaller groups
0:45:52 Q&A
https://wn.com/Representations_Of_Finite_Groups_Of_Lie_Type_(Lecture_1)_By_Dipendra_Prasad
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 fundamental problem, which has played a central role in the development of the representation theory of finite groups. The tools of the representation theory of finite groups extend in various ways to profinite groups such as compact linear groups over the ring of integers of a local field (for example GL_n(Z_p)). However, the continuous representations or even representation growth of profinite groups is not well understood and is one of the current exciting areas of research. The importance of computational methods in all pursuits of pure mathematics is no more obscure, and the subject has established itself as a powerful tool, aiding quick maturing of intuition about concrete mathematical structures. The focus of this program is on theoretical aspects of group algebras and representation theory of finite and profinite groups complemented by computational techniques using discrete algebra system GAP.
The first part of this program, a 5-day instructional workshop during October 14 - 18, 2019, is planned to touch upon various aspects of group algebras and representation theory, including Wedderburn decomposition as well as idempotents of group algebra and their applications, representations of finite simple groups of Lie type and p-adic analytic groups, representation growth and computational methods.
It will consist of three mini-courses:
Representations of finite and profinite groups
Structure of group algebras
Computations with finite groups
The plan is to have five hours of teaching and five hours of tutorials and guided practice sessions in each mini-course.
The second part of the program, a 4-day discussion meeting during October 20-23, 2019, is aimed at exploring some of the recent developments and problems of current mathematical interest in the above mentioned and related areas. Eminent speakers from India as well as abroad will present their work on a variety of topics. To encourage discussions among the participants, plenty of time will be provided between the talks. It is planned to have poster presentations by young researchers to manifest their mathematical thoughts and to foster interactions with the senior mathematicians. We'll dedicate October 21, 2019, to group rings, "Group Ring Day", in honor of the 80th birthday of Professor I.B.S. Passi.
The participants, Ph.D. students, post-doctoral fellows, and young faculty members, are expected to have exposure to advanced group theory and to have a good grasp on basic algebra and representation theory.
Eligibility criteria: The program is mainly aimed at Ph.D. students, post-doctoral fellows, and young faculty members. However, a couple of highly motivated final year master students shall also be given a chance to participate in the first part of the program.
CONTACT US: [email protected]
PROGRAM LINK: https://www.icts.res.in/program/garc2019
Table of Contents (powered by https://videoken.com)
0:00:00 Representations of finite groups of Lie type (Lecture 1)
0:01:16 Introduction
0:01:50 Recall one of the most basic representation considered for groups
0:06:16 Trivial and dimensional representation
0:08:08 Steinberg representation
0:09:25 Principle series
0:09:42 Boreal subgroup
0:10:53 Induced representation
0:14:10 Basic building blocks of representation theory
0:17:27 Group theory
0:21:46 Theory for Gl(Fq) and very analogous theory for Gln(Fq)
0:27:02 Linguistic theory
0:27:40 Philosophy of cusp forms (Harish Chandra)
0:28:53 Glossary of terms
0:31:38 Appealing and reductive groups
0:32:00 Borel subgroups
0:32:45 Parabolic subgroups
0:35:45 Symmetric groups
0:36:29 Twisted groups
0:42:50 The representation of theory of the group G is reduce to understand cuspital representation of all smaller groups
0:45:52 Q&A
- published: 31 Dec 2019
- views: 2393
59:32
Character theory of finite groups of Lie type (Meinolf Geck) 1
- Order: Reorder
- Duration: 59:32
- Uploaded Date: 04 Aug 2020
- views: 662
In these lectures we provide an introduction to Lusztig's classification
of the irreducible characters of a finite group of Lie type. This essentially relies on...
In these lectures we provide an introduction to Lusztig's classification
of the irreducible characters of a finite group of Lie type. This essentially relies on
structural properties of the underlying algebraic group, which will be surveyed
in the first lecture. We then go on to discuss the partition of the set of characters
into series and the Jordan decomposition of characters. Finally, we address the
problem of computing character values, in the framework of Lusztig's theory of
character sheaves
https://wn.com/Character_Theory_Of_Finite_Groups_Of_Lie_Type_(Meinolf_Geck)_1
In these lectures we provide an introduction to Lusztig's classification
of the irreducible characters of a finite group of Lie type. This essentially relies on
structural properties of the underlying algebraic group, which will be surveyed
in the first lecture. We then go on to discuss the partition of the set of characters
into series and the Jordan decomposition of characters. Finally, we address the
problem of computing character values, in the framework of Lusztig's theory of
character sheaves
- published: 04 Aug 2020
- views: 662
51:56
Gerhard Hiss: Representation theory for groups of Lie type I
- Order: Reorder
- Duration: 51:56
- Uploaded Date: 26 Sep 2018
- views: 803
The lecture was held within the framework of the Hausdorff Trimester Program: Logic and Algorithms in Group Theory.
Abstract:
The lectures begin by introducing...
The lecture was held within the framework of the Hausdorff Trimester Program: Logic and Algorithms in Group Theory.
Abstract:
The lectures begin by introducing finite groups of Lie type, in particular their structural properties relevant for their representation theory. Then some of the most fundamental goals and the state of the art will be discussed. Harish-Chandra theory, one of the major tools, will be presented. The focus will be on Deligne-Lusztig theory, which yields a classification of the irreducible complex representations of the groups of Lie type.
https://wn.com/Gerhard_Hiss_Representation_Theory_For_Groups_Of_Lie_Type_I
The lecture was held within the framework of the Hausdorff Trimester Program: Logic and Algorithms in Group Theory.
Abstract:
The lectures begin by introducing finite groups of Lie type, in particular their structural properties relevant for their representation theory. Then some of the most fundamental goals and the state of the art will be discussed. Harish-Chandra theory, one of the major tools, will be presented. The focus will be on Deligne-Lusztig theory, which yields a classification of the irreducible complex representations of the groups of Lie type.
- published: 26 Sep 2018
- views: 803
54:19
Jon Carlson: Endotrivial modules for finite groups of Lie type
- Order: Reorder
- Duration: 54:19
- Uploaded Date: 29 May 2024
- views: 10
Abstract: This is a report on joint work with Jesper Grodal, Nadia Mazza and Dan Nakano. Suppose that G is a finite group and that k is a field of characteristi...
Abstract: This is a report on joint work with Jesper Grodal, Nadia Mazza and Dan Nakano. Suppose that G is a finite group and that k is a field of characteristic p greater than 0. A kG-module M is an endotrivial module if Homk(M, M) = M∗ ⊗ M = k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modules have dimension one, but the exceptions are notable.
https://wn.com/Jon_Carlson_Endotrivial_Modules_For_Finite_Groups_Of_Lie_Type
Abstract: This is a report on joint work with Jesper Grodal, Nadia Mazza and Dan Nakano. Suppose that G is a finite group and that k is a field of characteristic p greater than 0. A kG-module M is an endotrivial module if Homk(M, M) = M∗ ⊗ M = k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modules have dimension one, but the exceptions are notable.
- published: 29 May 2024
- views: 10
1:00:35
Jesper Grodal: String topology of finite groups of Lie type
- Order: Reorder
- Duration: 1:00:35
- Uploaded Date: 14 Dec 2022
- views: 129
Abstract: Finite groups of Lie type, such as SL_n(F_q), Sp_n(F_q)..., are ubiquitous in mathematics, and calculating their cohomology has been a central theme o...
Abstract: Finite groups of Lie type, such as SL_n(F_q), Sp_n(F_q)..., are ubiquitous in mathematics, and calculating their cohomology has been a central theme over the years, and serves as an input to many other calculations, ranging from K-theory to stratification results. It has calculationally been observed that (when calculable) their mod ell cohomology agree with the mod ell cohomology of LBG(C), the free loop space on BG(C), the classifying space of the corresponding complex algebraic group G(C), as long as q is congruent to 1 mod ell. In recent joint work with Anssi Lahtinen, that combines ell-compact groups with string topology à la Chas-Sullivan, we attempt to provide a reason behind this "coincidence". I'll give an introduction to this line of thought.
https://wn.com/Jesper_Grodal_String_Topology_Of_Finite_Groups_Of_Lie_Type
Abstract: Finite groups of Lie type, such as SL_n(F_q), Sp_n(F_q)..., are ubiquitous in mathematics, and calculating their cohomology has been a central theme over the years, and serves as an input to many other calculations, ranging from K-theory to stratification results. It has calculationally been observed that (when calculable) their mod ell cohomology agree with the mod ell cohomology of LBG(C), the free loop space on BG(C), the classifying space of the corresponding complex algebraic group G(C), as long as q is congruent to 1 mod ell. In recent joint work with Anssi Lahtinen, that combines ell-compact groups with string topology à la Chas-Sullivan, we attempt to provide a reason behind this "coincidence". I'll give an introduction to this line of thought.
- published: 14 Dec 2022
- views: 129
36:23
Spinors for Beginners 16: Lie Groups and Lie Algebras
- Order: Reorder
- Duration: 36:23
- Uploaded Date: 10 Dec 2023
- views: 28545
Full spinors playlist: https://www.youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs
Leave me a tip: https://ko-fi.com/eigenchris
Powerpoint slide f...
Full spinors playlist: https://www.youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs
Leave me a tip: https://ko-fi.com/eigenchris
Powerpoint slide files + Exercise answers: https://github.com/eigenchris/MathNotes/tree/master/SpinorsForBeginners
Videos on momentum operator generating translations in quantum mechanics:
Physics with Elliot: https://www.youtube.com/watch?v=_lz1VfI6Wxk&t=596s&pp=ygUecGh5c2ljcyB3aXRoIGVsbGlvdCBjb21tdXRhdG9y
Professor M Does Science's video: https://www.youtube.com/watch?v=978mMgGYs1M&t=392s&pp=ygUkcHJvZmVzc29yIG0gZG9lcyBzY2llbmNlIHRyYW5zbGF0aW9u
Quantum Sense': https://www.youtube.com/watch?v=A7yDvA8VQC8&t=305s&pp=ygUZcXVhbnR1bSBzZW5zZSB0cmFuc2xhdGlvbg%3D%3D
0:00 - Introduction
2:45 - Groups & Lie Groups
4:00 - Exponent of a so(3) Matrix
7:40 - Calculating so(3) generators
9:50 - Momentum generators translations
10:58 - so(3) traceless proof
13:35 - so(3) anti-symmetric proof
15:11 - Warning about matrix exponentials
16:43 - Lie Algebra Bracket
20:24 - Structure coefficients
21:48 - Lie Algebras as Tangent Spaces
23:30 - Lie Algebra Property Proofs
30:15 - Summary of so(3)
31:05 - Overview of so+(1,3)
32:50 - Spin-1 and Spin-1/2 representations
34:40 - Math vs Physics conventions
https://wn.com/Spinors_For_Beginners_16_Lie_Groups_And_Lie_Algebras
Full spinors playlist: https://www.youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs
Leave me a tip: https://ko-fi.com/eigenchris
Powerpoint slide files + Exercise answers: https://github.com/eigenchris/MathNotes/tree/master/SpinorsForBeginners
Videos on momentum operator generating translations in quantum mechanics:
Physics with Elliot: https://www.youtube.com/watch?v=_lz1VfI6Wxk&t=596s&pp=ygUecGh5c2ljcyB3aXRoIGVsbGlvdCBjb21tdXRhdG9y
Professor M Does Science's video: https://www.youtube.com/watch?v=978mMgGYs1M&t=392s&pp=ygUkcHJvZmVzc29yIG0gZG9lcyBzY2llbmNlIHRyYW5zbGF0aW9u
Quantum Sense': https://www.youtube.com/watch?v=A7yDvA8VQC8&t=305s&pp=ygUZcXVhbnR1bSBzZW5zZSB0cmFuc2xhdGlvbg%3D%3D
0:00 - Introduction
2:45 - Groups & Lie Groups
4:00 - Exponent of a so(3) Matrix
7:40 - Calculating so(3) generators
9:50 - Momentum generators translations
10:58 - so(3) traceless proof
13:35 - so(3) anti-symmetric proof
15:11 - Warning about matrix exponentials
16:43 - Lie Algebra Bracket
20:24 - Structure coefficients
21:48 - Lie Algebras as Tangent Spaces
23:30 - Lie Algebra Property Proofs
30:15 - Summary of so(3)
31:05 - Overview of so+(1,3)
32:50 - Spin-1 and Spin-1/2 representations
34:40 - Math vs Physics conventions
- published: 10 Dec 2023
- views: 28545
4:26
Why study Lie theory? | Lie groups, algebras, brackets #1
Next video: https://youtu.be/erA0jb9dSm0
Lie’s theory of continuous symmetries was origin...
published: 23 Jul 2023
Why study Lie theory? | Lie groups, algebras, brackets #1
Why study Lie theory? | Lie groups, algebras, brackets #1
- Report rights infringement
- published: 23 Jul 2023
- views: 85839
Next video: https://youtu.be/erA0jb9dSm0
Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets.
Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: lievideoseries
Sources:
History of group theory: https://en.wikipedia.org/wiki/History_of_group_theory
Lie’s theory for differential equations (actually developed further after Lie died):
https://arxiv.org/pdf/1901.01543.pdf
https://www.physics.drexel.edu/~bob/LieGroups/LG_16.pdf
https://www.researchgate.net/profile/Michael-Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf
https://en.wikipedia.org/wiki/Lie_point_symmetry
Lie theory applications to mathematics: https://math.stackexchange.com/questions/1322206/what-are-applications-of-lie-groups-algebras-in-mathematics
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: https://mathemaniac.myspreadshop.co.uk
Ko-fi: https://ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
16:02
Group of Lie type
If you find our videos helpful you can support us by buying something from amazon.
https:/...
published: 22 Jan 2016
Group of Lie type
Group of Lie type
- Report rights infringement
- published: 22 Jan 2016
- views: 198
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Group of Lie type
In mathematics, a group of Lie type is a group closely related to the group, G(k), of rational points of a reductive linear algebraic group, G, with values in the field, k.Finite groups of Lie type give the bulk of non-abelian finite simple groups.
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=FjVKDxTpBsQ
15:45
Exceptional Finite groups of Lie type and their Primitive actions, E.Pan (University of Warwick)
Young Group Theorists workshop: exploring new connections
published: 20 Oct 2022
Exceptional Finite groups of Lie type and their Primitive actions, E.Pan (University of Warwick)
Exceptional Finite groups of Lie type and their Primitive actions, E.Pan (University of Warwick)
- Report rights infringement
- published: 20 Oct 2022
- views: 70
Young Group Theorists workshop: exploring new connections
21:57
What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3
Part 4: https://youtu.be/9CBS5CAynBE
A bird's eye view on Lie theory, providing motivatio...
published: 10 Aug 2023
What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3
What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3
- Report rights infringement
- published: 10 Aug 2023
- views: 311898
Part 4: https://youtu.be/9CBS5CAynBE
A bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular.
Basically, Lie groups are groups and manifolds, and thinking about them as manifolds, we know that we want to understand Lie algebras; and thinking about them as groups, we know what additional structure we want on the Lie algebras - the Lie bracket.
YouTube, please do not demonetise this video for me saying “Tits group”. This is an actual mathematical object named after a French mathematician Jacques Tits.
Files for download:
Go to https://www.mathemaniac.co.uk/download and enter the following password: so3embeddedin5dim
SO(3) embedded in R^5: http://at.yorku.ca/b/ask-an-algebraic-topologist/2020/2618.htm
https://en.wikipedia.org/wiki/Whitney_embedding_theorem (n-dim manifold can be properly embedded in R^(2n): if you only want “the overall picture”, but perhaps distances are distorted)
https://en.wikipedia.org/wiki/Nash_embedding_theorems (n-dim Riemannian manifold can be isometrically embedded in n(3n+11)/2 dim if compact, n(n+1)(3n+11)/2 dim if not compact: if you want everything to remain intact, i.e. distances are preserved)
BCH formula (why Lie brackets are useful): https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula
Finite simple groups as building blocks: https://en.wikipedia.org/wiki/Composition_series
Classification of finite simple groups: https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
Levi decomposition (the more precise meaning of “building blocks” in Lie algebra): https://en.wikipedia.org/wiki/Levi_decomposition
E8 (the monster group of Lie algebras):
https://aimath.org/E8/e8.html
https://en.wikipedia.org/wiki/E8_(mathematics)
https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything
Video chapters:
00:00 Introduction
01:26 Lie groups - groups
05:41 Lie groups - manifolds
10:23 Lie algebras
14:16 Lie brackets
18:03 The "Lie theory picture"
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: https://www.facebook.com/mathemaniacyt
Instagram: https://www.instagram.com/_mathemaniac_/
Twitter: https://twitter.com/mathemaniacyt
Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: https://mathemaniac.myspreadshop.co.uk
Ko-fi: https://ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
49:11
Representations of finite groups of Lie type (Lecture 1) by Dipendra Prasad
PROGRAM : GROUP ALGEBRAS, REPRESENTATIONS AND COMPUTATION
ORGANIZERS: Gurmeet Kaur Bakshi...
published: 31 Dec 2019
Representations of finite groups of Lie type (Lecture 1) by Dipendra Prasad
Representations of finite groups of Lie type (Lecture 1) by Dipendra Prasad
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- published: 31 Dec 2019
- views: 2393
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 fundamental problem, which has played a central role in the development of the representation theory of finite groups. The tools of the representation theory of finite groups extend in various ways to profinite groups such as compact linear groups over the ring of integers of a local field (for example GL_n(Z_p)). However, the continuous representations or even representation growth of profinite groups is not well understood and is one of the current exciting areas of research. The importance of computational methods in all pursuits of pure mathematics is no more obscure, and the subject has established itself as a powerful tool, aiding quick maturing of intuition about concrete mathematical structures. The focus of this program is on theoretical aspects of group algebras and representation theory of finite and profinite groups complemented by computational techniques using discrete algebra system GAP.
The first part of this program, a 5-day instructional workshop during October 14 - 18, 2019, is planned to touch upon various aspects of group algebras and representation theory, including Wedderburn decomposition as well as idempotents of group algebra and their applications, representations of finite simple groups of Lie type and p-adic analytic groups, representation growth and computational methods.
It will consist of three mini-courses:
Representations of finite and profinite groups
Structure of group algebras
Computations with finite groups
The plan is to have five hours of teaching and five hours of tutorials and guided practice sessions in each mini-course.
The second part of the program, a 4-day discussion meeting during October 20-23, 2019, is aimed at exploring some of the recent developments and problems of current mathematical interest in the above mentioned and related areas. Eminent speakers from India as well as abroad will present their work on a variety of topics. To encourage discussions among the participants, plenty of time will be provided between the talks. It is planned to have poster presentations by young researchers to manifest their mathematical thoughts and to foster interactions with the senior mathematicians. We'll dedicate October 21, 2019, to group rings, "Group Ring Day", in honor of the 80th birthday of Professor I.B.S. Passi.
The participants, Ph.D. students, post-doctoral fellows, and young faculty members, are expected to have exposure to advanced group theory and to have a good grasp on basic algebra and representation theory.
Eligibility criteria: The program is mainly aimed at Ph.D. students, post-doctoral fellows, and young faculty members. However, a couple of highly motivated final year master students shall also be given a chance to participate in the first part of the program.
CONTACT US: [email protected]
PROGRAM LINK: https://www.icts.res.in/program/garc2019
Table of Contents (powered by https://videoken.com)
0:00:00 Representations of finite groups of Lie type (Lecture 1)
0:01:16 Introduction
0:01:50 Recall one of the most basic representation considered for groups
0:06:16 Trivial and dimensional representation
0:08:08 Steinberg representation
0:09:25 Principle series
0:09:42 Boreal subgroup
0:10:53 Induced representation
0:14:10 Basic building blocks of representation theory
0:17:27 Group theory
0:21:46 Theory for Gl(Fq) and very analogous theory for Gln(Fq)
0:27:02 Linguistic theory
0:27:40 Philosophy of cusp forms (Harish Chandra)
0:28:53 Glossary of terms
0:31:38 Appealing and reductive groups
0:32:00 Borel subgroups
0:32:45 Parabolic subgroups
0:35:45 Symmetric groups
0:36:29 Twisted groups
0:42:50 The representation of theory of the group G is reduce to understand cuspital representation of all smaller groups
0:45:52 Q&A
59:32
Character theory of finite groups of Lie type (Meinolf Geck) 1
In these lectures we provide an introduction to Lusztig's classification
of the irreducibl...
published: 04 Aug 2020
Character theory of finite groups of Lie type (Meinolf Geck) 1
Character theory of finite groups of Lie type (Meinolf Geck) 1
- Report rights infringement
- published: 04 Aug 2020
- views: 662
In these lectures we provide an introduction to Lusztig's classification
of the irreducible characters of a finite group of Lie type. This essentially relies on
structural properties of the underlying algebraic group, which will be surveyed
in the first lecture. We then go on to discuss the partition of the set of characters
into series and the Jordan decomposition of characters. Finally, we address the
problem of computing character values, in the framework of Lusztig's theory of
character sheaves
51:56
Gerhard Hiss: Representation theory for groups of Lie type I
The lecture was held within the framework of the Hausdorff Trimester Program: Logic and Al...
published: 26 Sep 2018
Gerhard Hiss: Representation theory for groups of Lie type I
Gerhard Hiss: Representation theory for groups of Lie type I
- Report rights infringement
- published: 26 Sep 2018
- views: 803
The lecture was held within the framework of the Hausdorff Trimester Program: Logic and Algorithms in Group Theory.
Abstract:
The lectures begin by introducing finite groups of Lie type, in particular their structural properties relevant for their representation theory. Then some of the most fundamental goals and the state of the art will be discussed. Harish-Chandra theory, one of the major tools, will be presented. The focus will be on Deligne-Lusztig theory, which yields a classification of the irreducible complex representations of the groups of Lie type.
54:19
Jon Carlson: Endotrivial modules for finite groups of Lie type
Abstract: This is a report on joint work with Jesper Grodal, Nadia Mazza and Dan Nakano. S...
published: 29 May 2024
Jon Carlson: Endotrivial modules for finite groups of Lie type
Jon Carlson: Endotrivial modules for finite groups of Lie type
- Report rights infringement
- published: 29 May 2024
- views: 10
Abstract: This is a report on joint work with Jesper Grodal, Nadia Mazza and Dan Nakano. Suppose that G is a finite group and that k is a field of characteristic p greater than 0. A kG-module M is an endotrivial module if Homk(M, M) = M∗ ⊗ M = k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modules have dimension one, but the exceptions are notable.
1:00:35
Jesper Grodal: String topology of finite groups of Lie type
Abstract: Finite groups of Lie type, such as SL_n(F_q), Sp_n(F_q)..., are ubiquitous in ma...
published: 14 Dec 2022
Jesper Grodal: String topology of finite groups of Lie type
Jesper Grodal: String topology of finite groups of Lie type
- Report rights infringement
- published: 14 Dec 2022
- views: 129
Abstract: Finite groups of Lie type, such as SL_n(F_q), Sp_n(F_q)..., are ubiquitous in mathematics, and calculating their cohomology has been a central theme over the years, and serves as an input to many other calculations, ranging from K-theory to stratification results. It has calculationally been observed that (when calculable) their mod ell cohomology agree with the mod ell cohomology of LBG(C), the free loop space on BG(C), the classifying space of the corresponding complex algebraic group G(C), as long as q is congruent to 1 mod ell. In recent joint work with Anssi Lahtinen, that combines ell-compact groups with string topology à la Chas-Sullivan, we attempt to provide a reason behind this "coincidence". I'll give an introduction to this line of thought.
36:23
Spinors for Beginners 16: Lie Groups and Lie Algebras
Full spinors playlist: https://www.youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_A...
published: 10 Dec 2023
Spinors for Beginners 16: Lie Groups and Lie Algebras
Spinors for Beginners 16: Lie Groups and Lie Algebras
- Report rights infringement
- published: 10 Dec 2023
- views: 28545
Full spinors playlist: https://www.youtube.com/playlist?list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs
Leave me a tip: https://ko-fi.com/eigenchris
Powerpoint slide files + Exercise answers: https://github.com/eigenchris/MathNotes/tree/master/SpinorsForBeginners
Videos on momentum operator generating translations in quantum mechanics:
Physics with Elliot: https://www.youtube.com/watch?v=_lz1VfI6Wxk&t=596s&pp=ygUecGh5c2ljcyB3aXRoIGVsbGlvdCBjb21tdXRhdG9y
Professor M Does Science's video: https://www.youtube.com/watch?v=978mMgGYs1M&t=392s&pp=ygUkcHJvZmVzc29yIG0gZG9lcyBzY2llbmNlIHRyYW5zbGF0aW9u
Quantum Sense': https://www.youtube.com/watch?v=A7yDvA8VQC8&t=305s&pp=ygUZcXVhbnR1bSBzZW5zZSB0cmFuc2xhdGlvbg%3D%3D
0:00 - Introduction
2:45 - Groups & Lie Groups
4:00 - Exponent of a so(3) Matrix
7:40 - Calculating so(3) generators
9:50 - Momentum generators translations
10:58 - so(3) traceless proof
13:35 - so(3) anti-symmetric proof
15:11 - Warning about matrix exponentials
16:43 - Lie Algebra Bracket
20:24 - Structure coefficients
21:48 - Lie Algebras as Tangent Spaces
23:30 - Lie Algebra Property Proofs
30:15 - Summary of so(3)
31:05 - Overview of so+(1,3)
32:50 - Spin-1 and Spin-1/2 representations
34:40 - Math vs Physics conventions
Group of Lie type
In mathematics, a group of Lie type is a group closely related to the group, G(k), of rational points of a reductive linear algebraic group, G, with values in the field, k. Finite groups of Lie type give the bulk of non-abelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
All Lie groups are groups of Lie type, but not vice-versa.
Dieudonné (1971) and Carter (1989) are standard references for groups of Lie type.
Classical groups
An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other fields by Jordan (1870). These groups were studied by L. E. Dickson and Jean Dieudonné. Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence.
A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series An, Bn, Cn, Dn,2An, 2Dn of Chevalley and Steinberg groups.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Group_of_Lie_type
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