Representation Theory - University of California, Berkeley

1 Representation TheoryCT, Lent 20051 What is Representation Theory ?Groups arise in nature as sets of symmetries (of an object), which are closed under compo-sition and under taking inverses . For example, thesymmetric groupSnis the group of allpermutations (symmetries) of{1,..,n}; thealternating groupAnis the set of all symmetriespreserving the parity of the number of ordered pairs (did you really remember that one?); thedihedral groupD2nis the group of symmetries of the regularn-gon in the plane. TheorthogonalgroupO(3) is the group of distance-preserving transformations of Euclidean space which fix theorigin. There is also the group ofalldistance-preserving transformations, which includes thetranslations along with O(3).

2 1 The official definition is of course more abstract, a group is a setGwith a binary operation which is associative, has a unit elementeand for which inverses exist. Associativity allows aconvenient abuse of notation, where we writeghforg h; we haveghk= (gh)k=g(hk) andparentheses are unnecessary. I will often write 1 fore, but this is dangerous on rare occasions,such as when studying the groupZunder addition; in that case,e= abstract definition notwithstanding, the interesting situation involves a group acting on a set. Formally, an action of a groupGon a setXis an action map a:G X Xwhichiscompatible with the group law, in the sense thata(h,a(g,x)) =a(hg,x)anda(e,x) = justifies the abusive notationa(g,x) = evengx, for we haveh(gx) = (hg) this point of view, geometry asks, Given a geometric objectX, what is its group ofsymmetries?

3 Representation Theory reverses the question to Given a groupG, what objectsXdoes it act on? and attempts to answer this question by classifying suchXup to restricting to the linear case, our main concern, let us remember another way todescribe an action ofGonX. Everyg Gdefines a mapa(g) :X Xbyx7 gx. Thismap is a bijection, with inverse mapa(g 1): indeed,a(g 1) a(g)(x) =g 1gx=ex=xfromthe properties of the action. Hencea(g) belongs to the set Perm(X) of bijective self-maps set forms a group under composition, and the properties of an action imply action ofGonX is the same as a group homomorphism :G Perm(X). is a logical abuse here, clearly an action, defined as a mapa:G X Xisnot the same as the homomorphism in the Proposition; you are meant to read that specifyingone is completely equivalent to specifying the other, unambiguously.

4 But the definitions aredesigned to allow such abuse without much danger, and I will frequently indulge in that (in factI denoted byain lecture).21 This group is isomorphic to thesemi-direct productO(3)nR3 but if you do not know what this means,do not respect to abuse, you may wish to err on the side of caution when writing up solutions in your exam!1 The reformulation of Prop. leads to the following observation. For any actiona HonXand group homomorphism :G H, there is defined arestrictedorpulled-backaction aofGonX, as a=a . In the original definition, the action sends (g,x) to (g)(x).( ) Example: Tautological action ofPerm(X)onXThis is the obvious action, call itT, sending (f,x) tof(x), wheref:X Xis a bijectionandx X.

5 Check that it satisfies the properties of an action! In this language, the actionaofGonXis T, with the homomorphism of the proposition the pull-back under of thetautological action.( ) question of classifying all possibleXwith action ofGis hopeless in such generality, butone should recall that, in first approximation, mathematics is linear. So we shall take ourXtoavector spaceover some groundfield, and ask that the action ofGbe linear, as well, in otherwords, that it should preserve the vector space structure. Our interest is mostly confined to thecase when the field of scalars isC, although we shall occasional mention how the picture changeswhen other fields are linear Representation ofGon a complex vector spaceVis a set-theoreticaction onVwhich preserves the linear structure, that is, (g)(v1+v2) = (g)v1+ (g)v2, v1,2 V, (g)(kv) =k (g)v, k C,v VUnless otherwise mentioned,representationwill meanfinite-dimensional complex Representation .

6 ( ) Example: The general linear groupLetVbe a complex vector space of dimensionn < . After choosing a basis, we can identify itwithCn, although we shall avoid doing so without good reason. Recall that theendomorphismalgebraEnd(V) is the set of all linear maps (oroperators)L:V V, with the natural additionof linear maps and the composition as multiplication. (If you do not remember, you shouldverify that the sum and composition of two linear maps is also a linear map.) IfVhas beenidentified withCn, a linear map is uniquely representable by a matrix, and the addition of linearmaps becomes the entry-wise addition, while the composition becomes the matrix multiplication.(Another good fact to review if it seems lost in the mists of time.)

7 Inside End(V) there is contained the group GL(V) ofinvertiblelinear operators (thoseadmitting a multiplicative inverse); the group operation, of course, is composition (matrix mul-tiplication). I leave it to you to check that this is a group, with unit the identity operator following should be obvious enough, from the naturally a Representation ofGL(V).It is called thestandardrepresentation of GL(V). The following corresponds to Prop. ,involving the same abuse of Representation ofGonV is the same as a group homomorphism fromGtoGL(V). that, to give a linear action ofGonV, we must assign to eachg Ga linearself-map (g) End(V). Compatibility of the action with the group law requires (h) ( (g)(v)) = (hg)(v), (1)(v) =v, v V,whence we conclude that (1) = Id, (hg) = (h) (g).

8 Takingh=g 1shows that (g) isinvertible, hence lands in GL(V). The first relation then says that we are dealing with a between two representations ( 1,V1) and ( 2,V2) ofGis alinear isomorphism :V1 V2which intertwines with the action ofG, that is, satisfies ( 1(g)(v)) = 2(g)( (v)).Note that the equality makes sense even if is not invertible, in which case it is just calledanintertwining operatororG-linear map. However, if is invertible, we can write instead 2= 1 1,( )meaning that we have an equality of linear maps after inserting any group elementg. Observethat this relation determines 2, if 1and are known. We can finally formulate theBasic Problem of Representation Theory :Classify all representations of a given groupG,up to arbitraryG, this is very hard!

9 We shall concentrate on finite groups, where a very goodgeneral Theory exists. Later on, we shall study some examples of topological compact groups,such as U(1) and SU(2). The general Theory for compact groups is also completely understood,but requires more difficult close with a simple observation, tying in with Definition Given any Representation ofGon a spaceVof dimensionn, a choice of basis inVidentifies this linearly withCn. Callthe isomorphism . Then, by formula ( ), we can define a new Representation 2ofGonCn, which is isomorphic to ( ,V). So anyn-dimensional Representation ofGis isomorphic to arepresentation onCn. The use of an abstract vector space does not lead to new Representation ,but it does free us from the presence of a distinguished LectureToday we discuss the representations of a cyclic group, and then proceed to define the importantnotions of irreducibility and complete reducibility( ) Concrete realisation of isomorphism classesWe observed last time that everym-dimensional Representation of a groupGwas isomorphicto a Representation onCm.

10 This leads to a concrete realisation of the set ofm-dimensionalisomorphism classes of set ofm-dimensional isomorphism classes ofG-representations is inbijection with the quotientHom (G; GL(m;C))/GL(m;C)of the set of group homomorphism toGL(m)by the overall conjugation action on the by GL(m) sends a homomorphism to the new homomorphismg7 (g) 1. According to Definition , this has exactly the effect of identifying proposition is not as useful (for us) as it looks. It can be helpful in under-standing certain infinite discrete groups such asZbelow in which case the set Hom canhave interesting geometric structures. However, for finite groups, the set of isomorphism classesis finite so its description above is not too ( ) Example: Representations shall classify all representations of the groupZ, with its additive structure.