GEIRELLINGSRUD MA4270—REPRESENTATION THEORY Contents 1 Basics about representations 7 2 Representations of Abelian groups 37 3 Complete reducibilty 49 4 Characters 83 5 Induction and Restriction 131 6 Induction theorems 163 7 Frobenius Groups 181 8 Some algebra 187 9 The symmetric groups 191 Introduction 1 Basics about representations Basics Preliminary version 0.2 as of 14th November, 2017 Changes: 22/8. Have tidied up here and there. Expanded the text on the Burn- side ring. Corrected misprints. Added an example (1.6) and an exercise (1.9) 18/9: Corrected a stupidity in praragraph (1.8) on page 15. The first time I gave this course, back in 2012, was initiated by students interested in physics; some were master- or phd-students in physics. Naturally this influenced the attitude toward the content, certainly with Lie-groups and their complex (or real) representations being center stage. However, this time I was approached by students in topology (with a bias towards number theory) to give the course, and naturally this changes the attitude. Still the complex representations will be the leading stars, but we shall to some extent pursue the theory over general fields (including fields of positive characteristic). I also think it is appropriate to use the language of categories, it is not essen- tial but clarifies things enormously (at least if you speak the language). And for those with intension of pursuing the subject in an algebraic direction, it will be essential. Prerequisite knowledge: • Basics of algebra (commutative and non-commutative) • Basics of group theory • Rudiments of categories and functors • rudiments of topology and manifolds. and of course a good bit of mathematical maturity. (1.1) In this first part of the course groups will will tacitly be assumed to be finite. The order of a group G, that is the number of its elements, will be de- noted by jGj. Groups are, with few exceptions, multiplicatively written, and 8 ma4270—representation theory the group law will be denoted by juxtaposition (or occasionally with the clas- sical dot), i.e., the product of g and h is denoted by gh (or g · h). The unit ele- ment will be denoted by 1. (1.2)—About ground fields. We fix a ground field k. To begin with no re- strictions are imposed on k. We shall however not dig very deep into the the- ory when k is of positive characteristic—our main interest will be in the case when k is of characteristic zero—but we do what easily can be done in general without to much additional machinery. But of course when k is algebraically closed, the theory is particularly elegant and simple. The relation between the field and the group has a strong influence on the theory, basically in two different ways. Firstly, if the characteristic of k is pos- itive and divides the order jGj the situation is markedly more complicated Friendly field then if not. In the good case we call k a field friendly1 for G; that is, when the 1 This is highly non-standard termin- characteristic of k either is zero or relatively prime to the order jGj of G. ology, but a name is needed to avoid repeating the phrase “a field whose Secondly, when k in addition to being friendly is “sufficiently large”, the the- characteristic either is zero or positive ory tends to be uniform and many features are independent of the field. In this and not dividing jGj” case k is called a big friendly field is for G. For the moment, this is admittedly Big friendly field very vague, but in the end (after having proved a relevant theorem of Burn- side), a field will be friendly if it contains n different n-th roots of unity where n is the exponent2 of G; equivalently the polynomial tn − 1 splits into a product 2 Recall:The exponent is smallest ele- of n different linear factors over k. ments killing the whole G; or in less murderous terms, it is the least com- Every algebraically closed field of characteristic zero is big and friendly for mon multiple of the order of elements every group. The all important example to have in mind is of course the field in G. The exponent divides the order jGj, but they are not necessary equal. of complex numbers C. Another field important in number theory, is the field of algebraic numbers Q. (1.3) To give a twist to a famous sentence from Animal Farm: “All groups are equal, but some groups are more equal that others”. Among the “more equal groups”, we shall meet the symmetric groups Sn, the alternating groups An, the dihedral groups Dn, the linear groups Gl(Fq, n) and Sl(Fq, n) over finite fields, the generalized quaternionic groups. And of course the cyclic groups Cn. basics about representations 9 1.1 Definitions In the beginning there will of course be lots of definition to make. The fun- damental one is to tell what a representation is. When that is in place, we in- troduce the basic concepts related to representations, and the operations they can be exposed to. The development follows more or less a unifying pattern: Well known concepts from linear algebra are interpreted in the category of representations. The first and fundamental concept Given a finite group and a field k.A representation of G afforded by the vector Representation of a group space V over k is a group homomorphism r : G ! Aut(V); that is, a map such that r(gh) = r(g) ◦ r(h) and r(1) = idV . Equivalently one may consider a map G × V ! V, temporarily denoted by r, satisfying the three requests • r(g, r(h, v)) = r(gh, v) for all g, h 2 G and v 2 V, • r(e, v) = v for all v 2 V, • r(g, v) is k-linear as a function of v. Or in short, the map r is a k-linear action of G on V. The link between the two notions is the relation r(g) · v = r(g, v). The two first conditions above are equivalent to r being a homomorphism; and the third ensures that r takes ( ) values in Aut V . Frequently and indiscriminately we shall use the term a Linear G-action linear G-action for a representation, and a third term will be a G-module—the G-module reason for which will be clear later. (1.1) In most of this course V will be of finite dimension over k and we say that V is a finite representation. In the few cases when V will be of infinite Finite representations dimension, the field k will either be R or C, and both the vector space V and the group G will be equipped with some (nice) topologies. One says that the action is continuous if the map r is continuous. As a matter of convenience the reference to the map r will most often be skipped and the action written as gv or g · v in stead of r(g)v; also the nota- tion gjV for r(v) will be in frequent use. In the same vein, we say that V is a 10 ma4270—representation theory representation, the action of G being understood, and even the base field will sometimes be understood. (1.2) Any vector space V affords a trivial action; i.e., one such that g · v = v for 2 2 = all g G and all v V. In case V k we call it the trivial representation and The trivial representation denote it by kG (or just by k if there is no imminent danger of confusion). The zero representation is, as the name indicates, the zero vector space with the trivial action. The set of vectors v 2 V that are invariant under all elements in G plays a special role in the story. It is a linear subspace denoted by VG. One has VG = f v 2 V j g · v = v for all g 2 G g. The notation is in concordance with the general notation XG for the set of fixed points of an action of G on a set X. Examples 1.1. Let G be the symmetric group Sn on n-letters. Assume that Vn is a vector SymmStandard space of dimension n over k with a basis feig1≤i≤n. Letting s 2 Sn act on basis elements s · ei = es(i) and extending this by linearity, one obtains a representation of G on Vn. 1.2. Let x1,..., xn be variables and let the symmetric group Sn act on the xi- HomPoly s by permutation; i.e., s(xi) = xs(i) for s 2 Sn. For every natural number r this induces a representation of Sn on the space of homogenous of degree r polynomials in the xi with coefficients in a field k. C∗ 1.3. The (cyclic) subgroup mn of the multiplicative group of non-zero com- RootsOfUnity1 plex numbers consisting of the n-th roots of unity acts by multiplication on C. The corresponding representation is one-dimensional and denoted by L(1). 1.4. Continuing the previous example, let m be an integer. One defines a rep- RootsofUnity2 resentation mn on C by by letting a root of unity h 2 mn act via multiplication by the power hm. This representation is denoted by L(m). 1.5.—P ermutation representations. Let G act on the finite set X, and de- PermReps note by Lk(X) the set of maps from X to k.