Irreducible Characters of Slk(ℤ/P ℤ)
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REPRESENTATION THEORY. WEEK 4 1. Induced Modules Let B ⊂ a Be
REPRESENTATION THEORY. WEEK 4 VERA SERGANOVA 1. Induced modules Let B ⊂ A be rings and M be a B-module. Then one can construct induced A module IndB M = A ⊗B M as the quotient of a free abelian group with generators from A × M by relations (a1 + a2) × m − a1 × m − a2 × m, a × (m1 + m2) − a × m1 − a × m2, ab × m − a × bm, and A acts on A ⊗B M by left multiplication. Note that j : M → A ⊗B M defined by j (m) = 1 ⊗ m is a homomorphism of B-modules. Lemma 1.1. Let N be an A-module, then for ϕ ∈ HomB (M, N) there exists a unique ψ ∈ HomA (A ⊗B M, N) such that ψ ◦ j = ϕ. Proof. Clearly, ψ must satisfy the relation ψ (a ⊗ m)= aψ (1 ⊗ m)= aϕ (m) . It is trivial to check that ψ is well defined. Exercise. Prove that for any B-module M there exists a unique A-module satisfying the conditions of Lemma 1.1. Corollary 1.2. (Frobenius reciprocity.) For any B-module M and A-module N there is an isomorphism of abelian groups ∼ HomB (M, N) = HomA (A ⊗B M, N) . F Example. Let k ⊂ F be a field extension. Then induction Indk is an exact functor from the category of vector spaces over k to the category of vector spaces over F , in the sense that the short exact sequence 0 → V1 → V2 → V3 → 0 becomes an exact sequence 0 → F ⊗k V1⊗→ F ⊗k V2 → F ⊗k V3 → 0. Date: September 27, 2005. -
Frobenius Recip-Rocity and Extensions of Nilpotent Lie Groups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 298, Number I, November 1986 FROBENIUS RECIP-ROCITY AND EXTENSIONS OF NILPOTENT LIE GROUPS JEFFREY FOX ABSTRACT. In §1 we use COO-vector methods, essentially Frobenius reciprocity, to derive the Howe-Richardson multiplicity formula for compact nilmanifolds. In §2 we use Frobenius reciprocity to generalize and considerably simplify a reduction procedure developed by Howe for solvable groups to general extensions of nilpotent Lie groups. In §3 we give an application of the previous results to obtain a reduction formula for solvable Lie groups. Introduction. In the representation theory of rings the fact that the tensor product functor and Hom functor are adjoints is of fundamental importance and at the time very simple and elegant. When this adjointness property is translated into a statement about representations of finite groups via the group ring, the classical Frobenius reciprocity theorem emerges. From this point of view, Frobenius reciproc- ity, aside from being a useful tool, provides a very natural explanation for many phenomena in representation theory. When G is a Lie group and r is a discrete cocompact subgroup, a main problem in harmonic analysis is to describe the decomposition of ind~(1)-the quasiregular representation. When G is semisimple the problem is enormously difficult, and little is known in general. However, if G is nilpotent, a rather detailed description of ind¥(l) is available, thanks to the work of Moore, Howe, Richardson, and Corwin and Greenleaf [M, H-I, R, C-G]. If G is solvable Howe has developed an inductive method of describing ind~(l) [H-2]. -
Representations of Finite Groups’ Iordan Ganev 13 August 2012 Eugene, Oregon
Notes for ‘Representations of Finite Groups’ Iordan Ganev 13 August 2012 Eugene, Oregon Contents 1 Introduction 2 2 Functions on finite sets 2 3 The group algebra C[G] 4 4 Induced representations 5 5 The Hecke algebra H(G; K) 7 6 Characters and the Frobenius character formula 9 7 Exercises 12 1 1 Introduction The following notes were written in preparation for the first talk of a week-long workshop on categorical representation theory. We focus on basic constructions in the representation theory of finite groups. The participants are likely familiar with much of the material in this talk; we hope that this review provides perspectives that will precipitate a better understanding of later talks of the workshop. 2 Functions on finite sets Let X be a finite set of size n. Let C[X] denote the vector space of complex-valued functions on X. In what follows, C[X] will be endowed with various algebra structures, depending on the nature of X. The simplest algebra structure is pointwise multiplication, and in this case we can identify C[X] with the algebra C × C × · · · × C (n times). To emphasize pointwise multiplication, we write (C[X], ptwise). A C[X]-module is the same as X-graded vector space, or a vector bundle on X. To see this, let V be a C[X]-module and let δx 2 C[X] denote the delta function at x. Observe that δ if x = y δ · δ = x x y 0 if x 6= y It follows that each δx acts as a projection onto a subspace Vx of V and Vx \ Vy = 0 if x 6= y. -
Representation Theory with a Perspective from Category Theory
Representation Theory with a Perspective from Category Theory Joshua Wong Mentor: Saad Slaoui 1 Contents 1 Introduction3 2 Representations of Finite Groups4 2.1 Basic Definitions.................................4 2.2 Character Theory.................................7 3 Frobenius Reciprocity8 4 A View from Category Theory 10 4.1 A Note on Tensor Products........................... 10 4.2 Adjunction.................................... 10 4.3 Restriction and extension of scalars....................... 12 5 Acknowledgements 14 2 1 Introduction Oftentimes, it is better to understand an algebraic structure by representing its elements as maps on another space. For example, Cayley's Theorem tells us that every finite group is isomorphic to a subgroup of some symmetric group. In particular, representing groups as linear maps on some vector space allows us to translate group theory problems to linear algebra problems. In this paper, we will go over some introductory representation theory, which will allow us to reach an interesting result known as Frobenius Reciprocity. Afterwards, we will examine Frobenius Reciprocity from the perspective of category theory. 3 2 Representations of Finite Groups 2.1 Basic Definitions Definition 2.1.1 (Representation). A representation of a group G on a finite-dimensional vector space is a homomorphism φ : G ! GL(V ) where GL(V ) is the group of invertible linear endomorphisms on V . The degree of the representation is defined to be the dimension of the underlying vector space. Note that some people refer to V as the representation of G if it is clear what the underlying homomorphism is. Furthermore, if it is clear what the representation is from context, we will use g instead of φ(g). -
Induction and Restriction As Adjoint Functors on Representations of Locally Compact Groups
Internat. J. Math. & Math. Scl. 61 VOL. 16 NO. (1993) 61-66 INDUCTION AND RESTRICTION AS ADJOINT FUNCTORS ON REPRESENTATIONS OF LOCALLY COMPACT GROUPS ROBERT A. BEKES and PETER J. HILTON Department of Mathematics Department of Mathematical Sciences Santa Clara University State University of New York at Binghamton Santa Clara, CA 95053 Bingharnton, NY 13902-6000 (Received April 16, 1992) ABSTRACT. In this paper the Frobenius Reciprocity Theorem for locally compact groups is looked at from a category theoretic point of view. KEY WORDS AND PHRASES. Locally compact group, Frobenius Reciprocity, category theory. 1991 AMS SUBJECT CLASSIFICATION CODES. 22D30, 18A23 I. INTRODUCTION. In [I] C. C. Moore proves a global version of the Frobenius Reciprocity Theorem for locally compact groups in the case that the coset space has an invariant measure. This result, for arbitrary closed subgroups, was obtained by A. Kleppner [2], using different methods. We show how a slight modification of Moore's original proof yields the general result. It is this global version of the reciprocity theorem that is the basis for our categorical approach. (See [3] for a description of the necessary category-theoretical concepts.) We begin by setting up the machinery necessary to discuss the reciprocity theorem. Next we show how, using the global version of the theorem, the functors of induction and restriction are adjoint. The proofs of these results are at the end of the paper. 2. THE MAIN RFULTS. Throughout G is a separable locally compact group and K is a closed subgroup. Let G/K denote the space of right cosets of K in G and for s E G, we write for the coset Ks. -
Part II — Representation Theory Theorems
Part II | Representation Theory Theorems Based on lectures by S. Martin Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Linear Algebra and Groups, Rings and Modules are essential Representations of finite groups Representations of groups on vector spaces, matrix representations. Equivalence of representations. Invariant subspaces and submodules. Irreducibility and Schur's Lemma. Complete reducibility for finite groups. Irreducible representations of Abelian groups. Character theory Determination of a representation by its character. The group algebra, conjugacy classes, and orthogonality relations. Regular representation. Permutation representations and their characters. Induced representations and the Frobenius reciprocity theorem. Mackey's theorem. Frobenius's Theorem. [12] Arithmetic properties of characters Divisibility of the order of the group by the degrees of its irreducible characters. Burnside's paqb theorem. [2] Tensor products Tensor products of representations and products of characters. The character ring. Tensor, symmetric and exterior algebras. [3] 1 Representations of S and SU2 1 The groups S , SU2 and SO(3), their irreducible representations, complete reducibility. The Clebsch-Gordan formula. *Compact groups.* [4] Further worked examples The characters of one of GL2(Fq);Sn or the Heisenberg group. [3] 1 Contents II Representation Theory (Theorems) Contents 0 Introduction 3 1 Group actions 4 2 Basic definitions 5 3 Complete reducibility and Maschke's theorem 6 4 Schur's lemma 7 5 Character theory 8 6 Proof of orthogonality 10 7 Permutation representations 11 8 Normal subgroups and lifting 12 9 Dual spaces and tensor products of representations 13 9.1 Dual spaces . -
Introduction to the Representation Theory of Finite Groups
Basic Theory of Finite Group Representations over C Adam B Block 1 Introduction We introduce the basics of the representation theory of finite groups in characteristic zero. In the sequel, all groups G will be finite and all vector spaces V will be finite dimensional over C. We first define representa- tions and give some basic examples. Then we discuss morphisms and subrepresentations, followed by basic operations on representations and character theory. We conclude with induced and restricted representa- tions and mention Frobenius Reciprocity. These notes are intended as the background to the UMS summer seminar on the representation theory of symmetric groups from [AMV04]. The author learned this material originally from [FH] and recommends this source for more detail in the following; many of the proofs in the sequel likely come from this source. 2 Definitions and Examples Representation theory is the study of groups acting on vector spaces. As such, we have the following definition: Definition 1. A representation of a group G is a pair (V; ρ) where V is a vector space over C and ρ is a homomorphism ρ : G ! GL(V ). We will often refer to representations by their vector space and assume that the morphism ρ is clear from context. Every representation defines a unique C[G]-module, with the action of G on V being g · v = ρ(g)(v) and vice versa. We refer to the dimension of the representation, defined to be dim V . With the definition in mind, we begin with a few examples. Example 2. Let G be any group and V any vector space. -
Introduction to Representation Theory
Introduction to representation theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina February 2, 2011 Contents 1 Basic notions of representation theory 5 1.1 What is representation theory? . ......... 5 1.2 Algebras........................................ ... 7 1.3 Representations................................. ...... 7 1.4 Ideals .......................................... 10 1.5 Quotients ....................................... 11 1.6 Algebras defined by generators and relations . ............. 11 1.7 Examplesofalgebras.............................. ...... 11 1.8 Quivers ......................................... 13 1.9 Liealgebras..................................... 15 1.10 Tensorproducts................................. ...... 17 1.11 Thetensoralgebra ............................... ...... 19 1.12 Hilbert’sthirdproblem. ......... 19 arXiv:0901.0827v5 [math.RT] 1 Feb 2011 1.13 Tensor products and duals of representations of Lie algebras.............. 20 1.14 Representations of sl(2) .................................. 20 1.15 ProblemsonLiealgebras . ....... 21 2 General results of representation theory 23 2.1 Subrepresentations in semisimple representations . ................. 23 2.2 Thedensitytheorem ............................... ..... 24 2.3 Representations of direct sums of matrix algebras . ............... 24 2.4 Filtrations..................................... ..... 25 2.5 Finitedimensionalalgebras . ......... 26 1 2.6 Characters of representations . ......... -
Homogeneous Vector Bundles and Induced Representations
Topics in Representation Theory: Homogeneous Vector Bundles and Induced Representations 1 Homogeneous Vector Bundles We will begin by going over some terminology for those students who haven't seen some of these concepts before. The geometrical structures that appear in representation theory are very simple examples of basic geometrical structures that are normally studied in geometry and topology courses. They are as good a place as any to start learning some of this terminology if you haven't seen it before. Given a Lie group G with a Lie subgroup H, one can form the space of right cosets G=H. This is a manifold and it has a transitive G action, an element g acts by left multiplication: g0H ! gg0H and any two points on G=H are related by such a mapping. Spaces like this are called \homogeneous spaces". At each point in G=H there is a subgroup of G such that left multiplication by elements in it leaves the point invariant, this is called the \isotropy group" of the point. At the identity eH 2 G=H this is just the subgroup H, at other points it will be a translation of H. The space G=H is best thought of not just as a manifold, but as the \base- space" for a bundle. The total space of the bundle is G and the projection map from the bundle to the base-space is just the standard projection π : g 2 G ! gH 2 G=H The inverse image π−1(gH) of a point in G=H is the “fiber" above that point, it is a copy of H and one can think of G, the total space of the bundle, as a family of copies of H, parametrized by point in G=H. -
Arxiv:1901.00730V1 [Math.RT]
SCHWARTZ HOMOLOGIES OF REPRESENTATIONS OF ALMOST LINEAR NASH GROUPS YANGYANG CHEN AND BINYONG SUN Abstract. Let G be an almost linear Nash group, namely, a Nash group which admits a Nash homomorphism with finite kernel to some GLk(R). A homology theory (the Schwartz homology) is established for the category of smooth Fr´echet representations of G of moderate growth. Frobenius reciprocity and Shapiro’s lemma are proved in this category. As an application, we give a criterion for automatic extensions of Schwartz homologies of Schwartz sections of a tempered G-vector bundle. Contents 1. Introduction 2 1.1. Smooth representations 2 1.2. Smooth cohomologies and smooth homologies 3 1.3. Smooth Fr´echet representations of moderate growth 5 1.4. Schwartz homologies 6 1.5. Automatic extensions of Schwartz homologies 9 1.6. Structure of this article 11 2. Preliminaries on some function spaces 11 2.1. Schwartz functions 11 2.2. Linear families of moderate growth 12 2.3. Tempered linear families 14 3. Preliminaries on representations 16 3.1. The action of compactly supported distributions 16 3.2. The category SmodG 16 4. A proof of Theorem 1.4 17 5. Schwartz homologies 19 5.1. Relatively projective representations 19 arXiv:1901.00730v1 [math.RT] 3 Jan 2019 5.2. The coinvariants of relatively projective representations 22 5.3. Schwartz homologies 24 6. Tempered vector bundles and Schwartz inductions 25 6.1. Tempered vector bundles 25 2010 Mathematics Subject Classification. 22E20, 46T30. Key words and phrases. Schwartz homologies, Hausdorffness, tempered vector bundle, Schwartz induction, Shapiro lemma, Lie algebra homology, automatic extensions. -
Arxiv:Math/9910135V1 [Math.OA] 26 Oct 1999 Introduction Ihrrltv Commutants Relative Higher Nterltv Oiino H Nlso N a Enago Oreof Source Good a Been Has Subfactors
C*-TENSOR CATEGORIES AND FREE PRODUCT BIMODULES SHIGERU YAMAGAMI DEPARTMENT OF MATHEMATICAL SCIENCES IBARAKI UNIVERSITY MITO, 310–8512, JAPAN Abstract. A C*-tensor category with simple unit object is realized by von Neumann algebra bimodules of finite Jones index if and only if it is rigid. Introduction Given a subfactor N ⊂ M of finite Jones index, the associated sequence of ′ higher relative commutants {N ∩ Mj}j≥0 with M0 = M contains rich information on the relative position of the inclusion and has been a good source of combinatorial structures behind subfactors. An axiomatization of higher relative commutants is performed by S. Popa in the form of so-called standard lattices, which is based on a generalization of his preceding result on free product construction of irreducible subfactors of arbitrarily given indices. On the other hand, F. Radulescu invented a method to construct free product subfactors from commuting squares satisfying some strong conditions on non-degeneracy and connectedness. Under the background of these results on subfactors, we shall present here an abstract characterization of C*-tensor categories which can be realized by bimodules of finite Jones index. To do this, we need to impose some structure of rigidity in C*-tensor categories: we assume Frobenius duality together with a systematic choice of dual objects in C*-tensor categories. The notion of Frobenius duality is formulated in [21] as an abstraction from the Ocneanu’s Frobenius reciprocity in bimodules of finite Jones index (see [3, Chapter 9]), which produces promptly all the combinatorial features in subfactor theory such as commuting squares, Markov traces, Perron-Frobenius eigenvectors and so on ([24]). -
FROBENIUS RECIPROCITY and the HAAGERUP TENSOR PRODUCT 3 in the Setting of Hilbert Space Representations
FROBENIUS RECIPROCITY AND THE HAAGERUP TENSOR PRODUCT TYRONE CRISP Abstract. In the context of operator-space modules over C∗-algebras, we give a complete characterisation of those C∗-correspondences whose associated Haagerup tensor product functors admit left adjoints. The characterisation, which builds on previous joint work with N. Higson, exhibits a close connec- tion between the notions of adjoint operators and adjoint functors. As an application, we prove a Frobenius reciprocity theorem for representations of locally compact groups on operator spaces: the functor of unitary induction for a closed subgroup H of a locally compact group G admits a left adjoint in this setting if and only if H is cocompact in G. The adjoint functor is given by Haagerup tensor product with the operator-theoretic adjoint of Rieffel’s induction bimodule. 1. Introduction The reciprocity between induced and restricted group characters discovered by Frobenius is a cornerstone in the representation theory of finite groups, establish- ing a useful explicit relationship between the representations of a group and the representations of its subgroups. The computation of the character table of the symmetric groups by Frobenius himself is an early example of the reciprocity law’s many applications. See [Cur99] for an account of Frobenius’s computation. Turn- ing from finite-dimensional representations of finite groups to infinite-dimensional representations of locally compact groups, Frobenius reciprocity was also a central concern of Mackey in his influential work on induced unitary representations, and various forms of the reciprocity law have been established in this context. See [Wei40], [Mac52], [Mau52], [Mac53], [Fel64] and [Ros77], for example.