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Title: the 2-character theory for finite 2-groups.
Abstract: In this work, we generalize the notion of character for 2-representations of finite 2-groups. The properties of 2-characters bear strong similarities to those classical characters of finite groups, including conjugation invariance, additivity, multiplicativity and orthogonality. With a careful analysis using homotopy fixed points and quotients for categories with 2-group actions, we prove that the category of class functors on a 2-group $\mathcal G$ is equivalent to the Drinfeld center of the 2-group algebra $\mathrm{Vec}_{\mathcal G}$, which categorifies the Fourier transform on finite abelian groups. After transferring the canonical nondegenerate braided monoidal structure from $\mathfrak Z_1(\mathrm{Vec}_{\mathcal G})$, we discover that irreducible 2-characters of $\mathcal G$ coincide with full centers of the corresponding 2-representations, which are in a one-to-one correspondence with Lagrangian algebras in the category of class functors on $\mathcal G$. In particular, the fusion rule of $2\mathrm{Rep}(\mathcal G)$ can be calculated from the pointwise product of Lagrangian algebras as class functors. From a topological quantum field theory (TQFT) point of view, the commutative Frobenius algebra structure on a 2-character is induced from a 2D topological sigma-model with target space $\lvert \mathrm{B} \mathcal G \rvert$.
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MATH Seminar: “Endotrivial modules for finite groups of Lie type”, Nadia Mazza, 1:30PM April 3 2024 (EN)
You are cordially invited to the Algebra Seminar organized by the Department of Mathematics.
Speaker: Nadia Mazza (Lancaster University)
“Endotrivial modules for finite groups of Lie type.”
Abstract: Let G be a finite group and k a field of positive characteristic p diving the order of G. An endotrivial kG-module is a finitely generated kG-module which is “invertible” in some suitable sense. Since the late 70s, these modules have been intensely studied in modular representation theory. In this talk, we review the essential background on endotrivial modules, and present some results about endotrivial modules for finite groups of Lie type, obtained jointly with Carlson, Grodal and Nakano.
Date: Wednesday, April 3, 2024 Time: 13:30 Place: ZOOM
To request the event link, please send a message to [email protected]
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Representation Theory
A First Course
- William Fulton 0 ,
- Joe Harris 1
Department of Mathematics, University of Michigan, Ann Arbor, USA
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Department of Mathematics, Harvard University, Cambridge, USA
Part of the book series: Graduate Texts in Mathematics (GTM, volume 129)
Part of the book sub series: Readings in Mathematics (READMATH)
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Table of contents (26 chapters)
Front matter, finite groups, representations of finite groups.
- William Fulton, Joe Harris
Examples; Induced Representations; Group Algebras; Real Representations
Representations of \({\mathfrak{s}_{_d}}\) : young diagrams and frobenius’s character formula, representations of \({\mathfrak{u}_d}\) and \(g{l_2}\left( {{\mathbb{f}_q}} \right)\), weyl’s construction, lie groups and lie algebras, lie algebras and lie groups, initial classification of lie algebras, lie algebras in dimensions one, two, and three, representations of sl2ℂ, representations of sl3ℂ, part i, representations ofsl3ℂ, part ii: mainly lots of examples, the classical lie algebras and their representations, the general setup: analyzing the structure and representations of an arbitrary semisimple lie algebra, sl4ℂ and slnℂ, symplectic lie algebras.
- Abelian group
- cohomology group
- finite group
- group action
- Lie algebra
- representation theory
- Vector space
William Fulton
Book Title : Representation Theory
Book Subtitle : A First Course
Authors : William Fulton, Joe Harris
Series Title : Graduate Texts in Mathematics
DOI : https://doi.org/10.1007/978-1-4612-0979-9
Publisher : Springer New York, NY
eBook Packages : Springer Book Archive
Copyright Information : Springer Science+Business Media New York 2004
Hardcover ISBN : 978-0-387-97527-6 Published: 22 October 1991
Softcover ISBN : 978-0-387-97495-8 Published: 22 October 1991
eBook ISBN : 978-1-4612-0979-9 Published: 01 December 2013
Series ISSN : 0072-5285
Series E-ISSN : 2197-5612
Edition Number : 1
Number of Pages : XV, 551
Topics : Topological Groups, Lie Groups
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The Science
The ways atomic nuclei and their protons and neutrons can combine are extremely complex and often difficult to study experimentally. As one interesting way to make theoretical predictions, physicists use methods called finite-volume simulations with periodic boundary conditions. These simulations create an imaginary box around a group of protons and neutrons. The periodic boundary conditions mean that particles leave the volume on one side of the box and reenter on the other. This box has an effect on large-scale computations of the atomic nuclei formed by the protons and neutrons.
This work solves a long-standing and fundamental problem for electrically charged systems in “periodic boxes” in finite-volume simulations. The study derives the mathematical equation that describes how the properties of electrically charged systems depend on the size of the simulation volume. This “volume dependence” makes it possible for physicists to learn the real-world physical properties of a system. This will aid in studies of how elements form in stars .
This work solves a long-standing and fundamental problem regarding charged systems in a finite-volume with periodic boundary conditions. It derives the expected volume dependence for the binding energies of nuclei with two or more protons. In an atomic nucleus, the "strong nuclear force" binds protons and neutrons together. However, accurate descriptions of a nucleus must also consider the electromagnetic repulsion between protons. This force is particularly strong at the lowest energies, where many important processes take place that create the elements that make up the world we know. This study means that nuclear binding energies can now be more accurately predicted from lattice simulations that use finite-volume boxes and must be extrapolated to infinite box size. Among other implications, the study allows researchers to extract parameters that play an important role in low-energy astrophysical reactions where one nucleus is captured by another to produce a new element.
Sebastian Koenig North Carolina State University [email protected] Dean Lee Facility for Rare Isotope Beams, Michigan State University [email protected]
Two of the researchers were supported in part by the National Science Foundation. A third researcher was supported by the Department of Energy (DOE) and its Nuclear Computational Low-Energy Initiative (NUCLEI) SciDAC-5 project. This material is based upon work supported by the DOE Office of Science, Office of Nuclear Physics, under the FRIB Theory Alliance.
Publications
Yu, H., Koenig, S., and Lee, D., Charged-Particle Bound States in Periodic Boxes . Physical Review Letters 131 , 212502 (2023). [DOI: 10.1103/physrevlett.131.212502]
Related Links
Understanding Charged Particles Helps Physicists Simulate Element Creation in Stars , North Carolina State University press release
IMAGES
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The representation theory of groups is a part of mathematics which examines how groups act on given structures.. Here the focus is in particular on operations of groups on vector spaces.Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations.. Other than a few marked exceptions, only finite groups ...
What is Representation Theory? Group representations describe elements of a group in terms of invertible linear transformations. Representation theory, then, allows questions regarding abstract ... REPRESENTATION THEORY FOR FINITE GROUPS 5 Proposition 3.3. If ˆ: G!GL(V) is a representation then the dual representa-
We can now define a group representation. Definition 1.6. Let G be a group. A representation of G (also called a G-representation, or just a representation) is a pair (p,V) where V is a vector space and p: G !Homvect(V,V) is a group action. I.e., an action on the set V so that for each g 2G, p(g) : V !V is a linear map. Remark 1.7.
6 I. REPRESENTATION THEORY OF FINITE GROUPS The role of σ−1 in the right hand side is that it guarantees us the right order in composi-tions: we then have (στ) ∗ = σ ∗τ ∗. Any group homomorphism s: G→ Aut(X) defines the linear representation ρ s of Gin k[X] by the formula: ρ s(σ) = s(σ) ∗. This ρ
The second chapter contains the core of the representation theory covered in the course. The third chapter contains several constructions of representations (for instance, tensor product and induced representations). Finally, the fourth chapter contains applications of the theory in Chapters 2 and 3 to group theory and also
Steinberg is an algebraist interested in a broad range of areas including semigroups, geometric group theory and representation theory. Other research interests include automata theory, finite state Markov chains and algebras associated to etale groupoids. Steinberg is the co-author of a 2009 Springer publication in the SMM series entitled "The ...
Representations of Algebras and Finite Groups 7 Preface These notes describe the basic ideas of the theory of representations of nite groups. Most of the essential structural results of the theory follow imme-diately from the structure theory of semisimple algebras, and so this topic occupies a long chapter.
Let G be a finite group of order |G|. Prove that any representation of G contains an invariant subspace of dimension less than or equal to |G|. 3.2 Maschke's theorem. Theorem 3.3. (Maschke) Let G be a finite group such that \( {\text {char}}\, k \) does not divide |G|.
Representation Theory of Finite Groups. Anupam Singh. The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the theory over complex numbers which is Character Theory.
This book provides an accessible introduction to the state of the art of representation theory of finite groups. Starting from a basic level that is summarized at the start, the book proceeds to cover topics of current research interest, including open problems and conjectures. The central themes of the book are block theory and module theory ...
Resources Online textbooks: P.Webb, Representation Theory Book We need the first 5 sections (pages 1-62). A.Baker, Representations of finite groups A.N.Sengupta, Notes on representations of algebras and finite groups D.M.Jackson, Notes on the representation theory of finite groups P.Etingof et al. Introduction to representation theory also discusses category theory, Dynkin diagrams, and ...
This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics.
reading and reference will be Martin Isaacs' Character Theory of Finite Groups. We will cover about half of the book over the course of this semester. It is (according to Professor Hermann) a readable book, so it would be appropriate for this (planned-to-be) reading course. Representation Theory of Finite Groups Professor: Dr. Peter Hermann
Representation theory is the study of groups acting on vector spaces. As such, we have the following de nition: De nition 1. A representation of a group Gis a pair (V;ˆ) where V is a vector space over C and ˆis a homomorphism ˆ: G!GL(V). We will often refer to representations by their vector space and assume that the morphism ˆis clear from ...
A Course in Finite Group Representation Theory Peter Webb February 23, 2016. Preface The representation theory of nite groups has a long history, going back to the 19th ... Most students who attend an advanced course in group representation theory do not go on to be specialists in the subject, for otherwise the class would be much smaller. ...
g∈G. This representation is called the trivial representation. Example 2.3. Let Gbe a finite group acting on a finite setX, and let V be the vector space with basis elements {e x} x∈X, indexed by X. Then we define ρ: G→GL(V) where ρ(g) is the map such that, e x →e gx. This representation is referred to as the permutation representation.
Representation of a Group 11 2.1. Commutator Subgroup and One Dimensional Representations 14 Chapter 3. Maschke's Theorem 17 Chapter 4. Schur's Lemma 23 Chapter 5. Representation Theory of Finite Abelian Groups over C 25 5.1. Example of Representation over Q 27 Chapter 6. The Group Algebra k[G] 29 Chapter 7. Constructing New Representations 31
Abstract. Although much of the theory of finite-dimensional algebras had its origins in the theory of group representations, it seems simpler nowadays to develop the theory of algebras first and then use it to give an account of group representations. This theory has been a powerful tool in the study of groups, especially the modular theory ...
1.2.1 Representations & modules F will denote an arbitrary field andGa finite group. All modules considered in this course will be finite-dimensional left modules. A (finite-dimensional)representation of Gover F is a group homomorphism ρ: G→ GL(V), where V is a (finite-dimensional) vector space overF. We write g·vfor ρ(g)(v).
Monday, Chapter 1. Representations of Finite Groups Representation theory of finite groups is originally concerned with the ways of writing a finite group G as a group of matrices, that is using group homomorphisms from Gto the general linear group GL npKq of invertible n n-matrices with coefficients in a field K for some positive integer n ...
2.3. New representations from old 21 2.4. Permutation representations 23 2.5. Properties of permutation representations 25 2.6. Calculating in permutation representations 27 2.7. Generalized permutation representations 28 Exercises on Chapter 2 30 Chapter 3. Character theory 33 3.1. Characters and class functions on a flnite group 33 3.2.
In this work, we generalize the notion of character for 2-representations of finite 2-groups. The properties of 2-characters bear strong similarities to those classical characters of finite groups, including conjugation invariance, additivity, multiplicativity and orthogonality. With a careful analysis using homotopy fixed points and quotients for categories with 2-group actions, we prove that ...
Abstract: Let G be a finite group and k a field of positive characteristic p diving the order of G. An endotrivial kG-module is a finitely generated kG-module which is "invertible" in some suitable sense. Since the late 70s, these modules have been intensely studied in modular representation theory.
Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the ...
Printed Dec. 12, 2007 Finite Group Representations 4 representation is an example of a permutation representation, namely one in which every group element acts by a permutation matrix. Regarding representations of Gas RG-modules has the advantage that many def-initions we wish to make may be borrowed from module theory. Thus we may study
As one interesting way to make theoretical predictions, physicists use methods called finite-volume simulations with periodic boundary conditions. These simulations create an imaginary box around a group of protons and neutrons. The periodic boundary conditions mean that particles leave the volume on one side of the box and reenter on the other.