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Representation rings of semidirect products of tori by finite groups Stykow, Maxim 2015

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Representation Rings of Semidirect Productsof Tori by Finite GroupsbyMaxim StykowA thesis submitted in partial fulfillment of the requirements forthe degree ofDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)The University of British Columbia(Vancouver)August 2015c© Maxim Stykow, 2015AbstractThis dissertation studies semidirect products of a torus by a finite groupfrom the representation theory point of view. The finite group of greatestinterest is the cyclic group of prime order. Such semidirect products occur innature as isotropy groups of Lie groups acting on themselves by conjugationand as normalizers of maximal tori in reductive linear algebraic groups.The main results of this dissertation are• the calculation of the representation ring of such semidirect productsas an algebra over the integers for certain special cases,• the adaptation of an algorithm from invariant theory to find finitepresentations of representation rings,• the computation of the topological K-theory of the classifying spaceof certain semidirect products,• the demonstration that the equivariant K-theory of the projective uni-tary group of degree 2 acting on itself by conjugation is not a freemodule over its representation ring.iiPrefaceThis dissertation is original, unpublished, independent work by me, MaximStykow, under the supervision of Dr. Alejandro A´dem. I also thank Dr. ManChuen Cheng, Dr. Julia Gordon, Dr. Kee Lam, and Dr. Zinovy Reichsteinfor fruitful discussions.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viSome Notation and Conventions . . . . . . . . . . . . . . . . . . viiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Background and Preliminary Results . . . . . . . . . . . . . 41.1 Representations of semidirect products . . . . . . . . . . . . 41.2 The classification of ZG-lattices . . . . . . . . . . . . . . . . 82 Representation Rings of Semidirect Products . . . . . . . . 142.1 From representations to invariant theory . . . . . . . . . . . 142.2 Rings of invariants . . . . . . . . . . . . . . . . . . . . . . . 232.3 Semidirect products by other groups . . . . . . . . . . . . . 313 Applications to Lie Groups and K-Theory . . . . . . . . . . 363.1 Tori of compact connected Lie groups . . . . . . . . . . . . . 36iv3.2 Topological K-theory of the classifying space of semidirectproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Results with mod p coefficients . . . . . . . . . . . . . . . . 424 Adjoint Action and Equivariant K-Theory of Lie Groups 464.1 The projective unitary group of degree 2 . . . . . . . . . . . 484.2 The projective unitary group of degree 3 . . . . . . . . . . . 525 Further Directions . . . . . . . . . . . . . . . . . . . . . . . . . 59Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A Equivariant algebraic topology . . . . . . . . . . . . . . . . . 68B Equivariant K-theory . . . . . . . . . . . . . . . . . . . . . . . 71C K-theory with coefficients . . . . . . . . . . . . . . . . . . . . 73vList of Figures4.1 The action of W on T . . . . . . . . . . . . . . . . . . . . . . 494.2 Translates of the fundamental domain of T {W (top left square). 54viSome Notation andConventionsThe symbol Cn denotes the cyclic group of order n and Zp the p-adic integers.Given an R-module M and an ideal I Ă R MpI :“ limÐÝM{InM denotes theI-adic completion of M .The class number of the ring of algebraic integers in the nth cyclotomicfield will be referred to simply as the class number of n.If S is a set, curly brackets tSu indicate the ring or algebra generated byS while pointy brackets xSy indicate the group generated by S.When occurring between representations, dot ¨ denotes tensor productand plus ` denotes direct sum.The symbol Sn denotes a sphere of dimension n. The following kinds ofequivalences occur:– is an isomorphism of algebraic structures (such as groups, rings, repre-sentations) or Lie groups« is a homeomorphism of topological spaces” is a congruence of integers modulo another fixed integerA square denotes the end of a proof. A triangle 4 denotes the end ofan example.viiDedicationTo Yogi RamsuratkumarviiiIntroductionConsider a split extension of groups1 Ñ T Ñ Γ Ñ GÑ 1or equivalently the semidirect product Γ – T ¸ G with T “ S1 ˆ ¨ ¨ ¨ ˆ S1a torus of rank n and G a finite group. Such semidirect products occur innature as isotropy groups of Lie groups acting on themselves by conjugation,cf. chapter 4, and as normalizers of maximal tori in reductive linear algebraicgroups; cf. [LMMR13].The topic of this dissertation is the calculation of the representation ringof Γ and the topological K-theory of the classifying space of Γ.In the first chapter, algebraic tools necessary to study the representationtheory of semidirect products and the K-theory of their classifying spaceare introduced. Key to our study is the action of G on the character grouppT – Zn, a Z-free ZG-module known as a ZG-lattice.The second chapter studies in detail the case when G “ Cp, the cyclicgroup of prime order p. After finding a presentation of the representationring RpΓq as a Z-algebra, we establish a link to the multiplicative invariantalgebra of G which enables us to find a finite presentation of RpΓq bZ Qpζqwhere ζ is a primitive pth root of unity. Some restrictions apply when theclass number of Zrζs is not trivial. The reason we begin with Cp is that a)T¸Cp was the first such semidirect product encountered as an isotropy groupand b) ZCp-lattices are well understood. We go on to discuss the cases of1G “ D2p, the dihedral group of order 2p, and G “ Sn, the symmetric groupon n letters. The main tools used to describe the various representationrings are Mackey’s method of “little groups”, the theory of ZG-lattices, andseveral theorems of Reiner classifying ZG-lattices.Let TM denote a torus with character ZG-lattice M and MG the sublat-tice of G-invariants in M . The main result of the second chapter isTheorem 2.1.7. Let G “ Cp “ xσ | σp “ 1y, where p ą 1 is any prime, andlet ZG and IG be the group ring respectively augmentation ideal of G. If Mis the ZG-lattice ZGs ‘ IGt and ΓM “ TM ¸G is the associated semidirectproduct, thenRpΓMq – Z xm : m PMG(b Z indΓMTM xm : rms PM{G(b Zrvs{pvp ´ 1qsubject to the relations(i) indΓMTM xm “ xm ¨ Φp for m PMG and Φp “ 1` v ` ¨ ¨ ¨ ` vp´1,(ii) indΓMTM xm ¨ indΓMTM xm1 “p´1ÿi“0indΓMTM xm`σim1,(iii) xm ¨ xm1“ xm`m1,(iv) xm ¨ vj ¨ indΓMTM xm1 “ indΓMTM xm`m1.In the third chapter, we apply our results to the K-theory of the clas-sifying space BΓ of Γ by using the Atiyah-Segal completion theorem. Bypassing to a subset of T known as a p-discrete torus it is possible to cir-cumvent the number theoretic issues encountered in chapter 2 and presenta complete answer for the mod p K-theory of BΓ when G “ Cp:Theorem 3.3.1. Let G “ Cp, where p ą 1 is any prime. If T is a torus ofrank n on which G acts by automorphisms, and Γ “ T ¸G is the associatedsemidirect product, thenK˚pBΓ;Cpq – K˚pBΓM ;Cpq2where ΓM “ TM ¸ G with M “ Zr ‘ ZGs ‘ IGt, for some integers r, s, tobeying r ` sp` tpp´ 1q “ n.The fourth chapter discusses the equivariant K-theory K˚GpGq of a com-pact connected Lie group G acting on itself by conjugation and shows thatK˚GpGq is not a free module over the representation ring of G when G isthe projective unitary group PUp2q of degree 2. It is then discussed how toextend this result to PUp3q.In the last chapter we mention natural ways by which the results of thiswork could be extended and outline the potentials and difficulties to do so.An interesting parallel to our research has been pursued by Adem et al.who computed the cohomology of semidirect products known as crystallo-graphic groups1; cf. [AP06] and [AGPP08]. In this case the torus is replacedby its character lattice pT – Zn to form Zn ¸ Cp. The K-theory of the clas-sifying space of Zn ¸ Cp was determined based on these computations; cf.[DL13]. In our work the situation is reversed: the K-theory of the classifyingspace of T ¸ Cp is computed while its cohomology remains unknown.1Cf. [Cha86, p. 74] for a definition.3Chapter 1Background and PreliminaryResults1.1 Representations of semidirect productsIn this section we assume all groups to be locally compact and Hausdorff.Let G be a subgroup of a group Γ and A a normal subgroup of Γ suchthat AXG “ t1u and Γ “ AG. Then Γ is known as the semidirect productof A by G, denoted A¸G. Conjugation of elements of A by elements of Gyields an action of G on A. Conversely, given any two groups A and G anda group homomorphism ψ : G Ñ AutpAq, we can construct a new groupA¸ψ G with underlying set AˆG and multiplication given bypa1, h1q ¨ pa2, h2q :“ pa1ψh1pa2q, h1h2q. ( A¸ψG is the semidirect product of subgroups isomorphic to A and Gin the sense just established. This dissertation is concerned with computingrepresentation rings of semidirect products Γ “ A¸G when A is abelian.Definition 1.1.1. Given a subgroup H of a group G and a continuous H-representation V we define the induced representation indGH V of G to be the4vector space C0HpG, V q of all continuous H-maps from G to V . A continuousG-action is given by g ¨ fpxq “ fpg´1xq; cf. [BtD85, III.6].Note that since an H-map is determined by its values on a system ofrepresentatives of G{H,dimC0HpG, V q “ |G{H| ¨ dimV. ( that |G{H| need not be finite. There is also a restriction map whichtakes a G-representation U and restricts the action to a subgroup H of G,denoted by resGH U . The following will be a useful formula when H is anormal subgroup of finite index in G; cf., e.g., [BtD85, VI.7.4]:resGH indGH V –àrgsPG{HV g ( the H-representation V g is defined to be V as a vector space with theH-actionh ¨ v “ ghg´1 ¨ v.Often all irreducible representations of Γ “ A ¸ G can be obtained byinducing representations of the form χb ρ where χ is a character of A andρ an irreducible representation of a subgroup of G. For example:Proposition 1.1.2 ([Seg68b, 3.12]). Let A Ñ Γ Ñ G be an extension of anilpotent group G by a topologically cyclic1 group A. Then any irreduciblerepresentation of Γ is monomial, i.e., induced from a one-dimensional rep-resentation of a subgroup of finite index.This scenario applies when Γ is the semidirect product of a torus of rankn by a cyclic group of prime order p.Specifically which representations to induce was worked out by G. W.Mackey. The original idea goes back to physics Nobel prize winner E.1Cf.’s method of “little groups” by which he computed representationsof the Poincare´ group, the group of isometries of Minkowski spacetime; cf.[Wig39]. The Poincare´ group can be seen as a group extension of the Lorentzgroup—a noncompact, nonabelian Lie group which is not connected.Mackey extended Wigner’s ideas to locally compact Hausdorff groups ingeneral. The theorem below is an application of his imprimitivity theoremfor infinite groups. Following [Mac52] we briefly describe the method:Recall that A is abelian and so its irreducible representations are one-dimensional. They are the elements of the character or Pontryagin dualgroup pA :“ HompA,C˚q. The action of G on A induces an action on pA.Definition 1.1.3 ([Sun96]). Recall that a Borel set is any set in a topolog-ical space that can be formed from open or equivalently closed sets throughthe operations of countable union, countable intersection, and relative com-plement. Say that a semidirect product Γ “ A¸G is regular if the charactergroup pA is a measure space containing a Borel set that meets almost2 everyG-orbit in exactly one point. Note that if pA is countable this condition isautomatically satisfied.Let pχqrχsP pA{G be a system of representatives for the orbits of G in pA.For each rχs P pA{G, let Gχ be the isotropy group of χ, cf. A.0.2.1, and letΓχ “ A¸Gχ Ă Γ. Also let ρ be an irreducible representation of Gχ and letpi1 : Γχ Ñ A pi2 : Γχ Ñ Gχbe the canonical projection maps. Extend χ to Γχ via pi1 and extend ρ toΓχ via pi2. Finally, let θχ,ρ “ indΓΓχpχ b ρq be the induced representation ofΓ.Theorem 1.1.4 (Method of “little groups”). Let Γ be the regular semidirectproduct of a normal abelian subgroup A by a subgroup G and let V be anyirreducible continuous representation of Γ. Then with the above notation2‘Almost’ here in the sense of measure theory, i.e., the measure of the subset of orbitsnot met is zero.6(i) θχ,ρ is irreducible,(ii) θχ,ρ – θχ1,ρ1 implies rχs “ rχ1s and ρ – ρ,(iii) V is isomorphic to θχ,ρ for some χ and some ρ.This determines RpΓq as a free abelian group.Example 1.1.5. Let the circle S1 act on the additive group of complex num-bers C by multiplication of complex numbers. Then Γ “ C¸S1 is the groupof all Euclidean motions of the plane. The character group pC “ HompC,C˚qis again the additive group of complex numbers. The orbits are the circleswith center at the origin. Any ray from the origin is a Borel set meeting eachorbit only once, making Γ regular. Let tzru8r“0 be a system of representatives.If r ą 0, the isotropy group is S1zr “ t1u and if r “ 0, S10 “ S1. Applying1.1.4, we see that in addition to the obvious one dimensional representationsthere is a continuum of infinite dimensional irreducible representations, onefor each r ą 0. These exhaust the irreducible representations of Γ. 4Example 1.1.6. Think of the cyclic group Cn “ xσ | σn “ 1y of order n asthe group of roots of unity on the unit circle and let C2 “ xτ | τ 2 “ 1y acton Cn by complex conjugation. Then the dihedral group of order 2n can bedefined as the semidirect product D2n :“ Cn ¸ C2.The character group xCn “ xv | vn “ 1y is isomorphic to Cn and thuscountable making the semidirect product regular. A system of representa-tives is given by taking roots of unity lying only on the upper semicircle withendpoints included.If n is odd, then v0 “ 1 has C2-isotropy giving two one-dimensionalrepresentations 1, w of D2n with w2 “ 1. The remaining representativesv, v2, . . . , vn´12 have trivial isotropy and give two-dimensional representationsafter induction. The representations1, w, indD2nCn v, . . . , indD2nCn vn´127exhaust all irreducible representations of D2n when n is odd. Finally, whenn is even, vn{2 “ ´1 also has C2-isotropy giving two more one-dimensionalrepresentations ´1,´w of D2n and the list˘1,˘w, indD2nCn v, . . . , indD2nCn vn2´1exhausts all irreducible representations.The product structure can also be determined by techniques that will beintroduced in chapter 2. For completeness, we note here thatw ¨ indD2nCn vi – indD2nCn viandindD2nCn vi ¨ indD2nCn vj – indD2nCn pvi`jq ` indD2nCn pvi´jqwith indD2nCn p1q – 1` w. 41.2 The classification of ZG-latticesAs we will see, semidirect products T¸G are determined by lattices equippedwith a G-action. The following definitions and theorems from [CR62],[CR90], and [Lor05] make these notions precise.Let T “ pR{Zqn be a torus of rank n and let ψ P AutpT q be a smoothgroup automorphism of T . The differential dψ is an invertible linear trans-formation of the Lie algebra t “ Rn of T that commutes with the exponentialmap and must preserve the kernel Zn of exp : t Ñ T . By the naturality ofthe exponential map, cf., e.g., [BtD85, I.3], the assignment ψ ÞÑ dψ|Zn givesan isomorphismAutpT q – GLnpZq.If G is an arbitrary group, it follows that having a homomorphism φ : GÑAutpT q is the same as having a ZG-lattice, that is, a ZG-module which8happens to be a finitely generated free Z-module.Example 1.2.1. For any subgroup H Ă G with rH : Gs ă 8 we may formthe ZG-latticeZrG{Hs “àrgsPG{HZgHwhere the G-action is given by gpg1Hq “ gg1H for g, g1 P G. Thus, Gpermutes a Z-basis of the ZG-lattice ZrG{Hs. Lattices of this type arecalled permutation lattices. 4When G is finite, the group ring ZG, viewed as a module over itselfvia multiplication, is a special case of a permutation lattice, the so-calledregular ZG-lattice. Any lattice isomorphic to ZGr for some r is called a freeZG-lattice. Direct summands of free lattices are called projective.The group ring gives rise to another important ZG-lattice,IG “ ker´ZG Ñ Z : pgq “ 1, g P G¯( as the augmentation ideal.Say a lattice is indecomposable if it cannot be written as a direct sumof nonzero lattices. For example, the permutation lattices ZrG{Hs are allindecomposable; cf. [CR90, 32.14]. The set of isomorphism classes of in-decomposable ZG-lattices is usually infinite. The following result describeswhen this set is finite; cf., e.g., [CR90, 33.6]:Theorem 1.2.2. Let G be a finite group. There are only finitely manyindecomposable ZG-lattices (up to isomorphism) precisely if, for each primep dividing the order |G|, the Sylow p-subgroups of G are cyclic of order p orp2.Complete sets of indecomposable ZG-lattices are only known in rarecases. One such case is the cyclic group Cp “ xσ | σp “ 1y of prime order p.To state the main theorem, we remind the reader of the following number-theoretic facts and definitions.9Let ζ be a primitive pth root of unity and recall that the ring of algebraicintegers3 in the cyclotomic field Qpζq is Zrζs; cf. [CR62, 21.13]. Note thatQpζq is the field of fractions of Zrζs. In fact, an algebraic number field isalways the field of fractions of its ring of algebraic integers; cf. [CR62, p.104]. The ideal class group of Qpζq is defined as the multiplicative groupof fractional ideals of Zrζs in Qpζq modulo the principal ideals. The classnumber hp of Zrζs, henceforth referred to as the class number of p, is thesize of the ideal class group which is always finite; cf. [CR62, 20.6].Theorem 1.2.3 (Diederichsen, Reiner [CR62, 74.3]). Let p ą 1 be any primeand let hp be the class number of p. Then there are 2hp ` 1 non-isomorphicindecomposable ZCp-lattices and every ZCp-lattice M is isomorphic to a di-rect sum of finitely many of these indecomposables.A complete set of non-isomorphic indecomposable ZCp-lattices is exhib-ited through the following constructions.Recall that Zrζs is a Dedekind domain4 with Z-basis t1, ζ, ζ2, . . . , ζp´2u.So if A is a fractional ideal in Qpζq, then A has rank p´ 1 and we may turnit into a ZCp-lattice by definingσ ¨ a :“ ζafor all a P A. By definition, then, two ideals are isomorphic as ZCp-modulesif and only if they are isomorphic as Zrζs-modules. The latter occurs if andonly if they are in the same ideal class; cf. [CR62, 22.2].Next, consider the direct sum A‘Zy of a fractional ideal A and the freeZ-module Zy. Define a Cp-action byσ ¨ a :“ ζa, a P A, σ ¨ y :“ a0 ` y3An element α of some finite field extension of Q is an algebraic integer if its uniquemonic irreducible polynomial has integral coefficients.4More generally, the ring of algebraic integers of any algebraic number field is aDedekind domain, that is, an integral domain in which every nonzero, proper ideal factorsinto a product of primes; cf. [CR62, p. 108].10for some fixed a0 P A, a0 R pζ ´ 1qA. Let Φppxq “ 1` x` ¨ ¨ ¨ ` xp´1 be thepth cyclotomic polynomial in an indeterminate x. Thenσ2 ¨ y “ ζpa0 ` yq “ pζ ` 1qa0 ` y...σp ¨ y “ Φppζqa0 ` y “ yas required and we denote this ZCp-lattice of rank p by pA, a0q. The iso-morphism class of pA, a0q is independent of the choice of a0; cf. [CR62,§74].A complete set of non-isomorphic indecomposable ZCp-lattices is thengiven bytpA1, a1q, . . . , pAhp , ahpq, A1, . . . , Ahp ,Zuand any rank n ZCp-lattice M is isomorphic to a direct sum of these, i.e.,M – pA1, a1q ‘ ¨ ¨ ¨ ‘ pAs, asq ‘ As`1 ‘ ¨ ¨ ¨ ‘ As`t ‘ Zrwhere the isomorphism class of M is determined by the integers r, s, t obey-ing r ` sp` tpp´ 1q “ n and the ideal class of each Ai.Continuing with G an arbitrary finite group, it is often useful to replacea ZG-lattice M by a related but better behaved lattice. LetĎ : M ÑM{MGdenote the canonical map, where MG is the sublattice of G-invariants in M .Note that ĎM is a ZG-lattice and the map Ď is G-equivariant so that wehave an extension of ZG-lattices0 ÑMG ÑM Ñ ĎM Ñ 0. ( that there is an isomorphism of isotropy groupsGm – G sm ( all m P M . This follows from the fact that tensored with Q issplit exact. As a consequence, ĎMG “ t0u. We call a lattice with trivialsublattice of G-invariants effective and ĎM is called the effective quotient ofM .Let K be a commutative ring. The group algebra KrM s contains a copyof M as a subgroup of the group of multiplicative units of KrM s and thiscopy of M forms a K-basis of KrM s.Working inside KrM s, we must pass from the additive notation of Mto a multiplicative notation. We will thus write the basis element of KrM scorresponding to the lattice element m P M as xm, so x0 “ 1, xm`m1“xmxm1, and x´m “ pxmq´1.A choice of Z-basis tei : i “ 1, . . . , nu of M gives rise to a K-algebraisomorphism of KrM s with the Laurent polynomial algebra Krx˘11 , . . . , x˘1n svia xei ÞÑ xi. Under this isomorphism,M – Zn – xx1, . . . , xny. ( augmentation ideal of a lattice M is, also cf.,IM “ ker´KrM s Ñ K : pxmq “ 1,m PM¯. ( action of G on M extends uniquely to a K-algebra action viag˜ÿmPMkmxm¸“ÿmPMkmxgmand the ring of invariants KrM sG is known as the multiplicative invariantalgebra of G. In contrast, additive or algebraic invariant theory deals with12the algebra of polynomial invariants SympV ˚qG – Krx1, . . . , xnsG where V isa free K-module and V ˚ “ HomKpV,Kq its dual module with basis txi : i “1, . . . , nu and action extended to the symmetric algebra SympV ˚q. Note thatevery ZG-lattice M may be regarded as a free Z-module to form SympM˚q.Since G permutes the K-basis M of KrM s, the invariant algebra KrM sGhas a K-basis consisting of finite G-orbit sumspxmq˚ :“ÿm1PGpmqxm1( Gpmq “ thm : h P Gu is the G-orbit of m; cf. A.0.2.2. Thus,KrM sG –àmPSKpxmq˚ ( S is a system of representatives of finite G-orbits and Kpxmq˚ is thefree K-algebra generated by pxmq˚. Extend p q˚ linearly to a map KrM s ÑKrM sG and note that pfgq˚ “ fg˚ for all f P KrM sG and g P KrM s.13Chapter 2Representation Rings ofSemidirect ProductsFor the first two sections of this chapter, let G “ Cp “ xσ | σp “ 1y,let T “ pS1qn be a torus of rank n, and consider the compact Lie groupΓ “ T ¸φ G where φ : G Ñ AutpT q is an injective homomorphism; cf. 1.1.The purpose of this chapter is to compute the representation ring RpΓq forsome special cases of φ and then the general case when p is a prime of trivialclass number. The third section will examine the situation when G is thedihedral group D2p and the symmetric group Sn.2.1 From representations to invariant theoryAs explained in 1.1, given T and G the semidirect product of T by G isdetermined by the G-module φ. By 1.2, such G-modules are ZG-lattices,i.e., G Ă GLnpZq with n “ rankT . Isomorphic ZG-lattices give isomorphicsemidirect products. That is, if two ZG-lattices φ and ψ are intertwined bya matrix U P GLnpZq such thatφpgq “ U´1ψpgqU for all g P G14thenT ¸φ GÑ T ¸ψ Gpx, gq Ñ pUxU´1, gqis an isomorphism.Section 1.2 discussed the indecomposable ZG-lattices. In particular,when the ideal class group is trivial there are only three non-isomorphicindecomposable ZG-lattices. They are the trivial lattice Z, the group ringlattice ZG of rank p, and the rank p´ 1 augmentation ideal IG – Zrζs; cf., we can identify Z with ZGG and IG with the effectivequotient ĎZG in the exact sequence, cf.,0 Ñ ZGG Ñ ZGÑ ĎZGÑ 0. ( is so because there is a ring homomorphism ZGÑ Zrζs given byÿaiσi ÞÑÿaiζiwith kernel generated by Φppσq “ 1`σ`σ2`¨ ¨ ¨`σp´1. Indeed, Φppζq “ 0and if fpσq P ZG with fpζq “ 0, then fpxq “ Φppxq ¨ gpxq for some gpxq PZrxs. So fpσq “ 0 mod pΦppσqq.In matrix notation, the nontrivial lattices are given byIG : σ ÞѨ˚˚˚˚˚˚˚˝´11 ´11 ´1. . ....1 ´1˛‹‹‹‹‹‹‹‚P GLp´1pZq15andZG : σ ÞѨ˚˚˚˚˚˚˚˝111. . .1˛‹‹‹‹‹‹‹‚P GLppZq.Definition 2.1.1. Let TM denote a torus with character ZG-lattice M andlet ΓM :“ TM ¸G be the associated semidirect product.The representation rings RpΓZrq, RpΓZGq, and RpΓIGq are importantspecial cases which will be computed before moving on to the general torusT made up of copies of TZ, TZG, and TIG when p is a prime of trivial classnumber.Proposition 2.1.2 ([BtD85, II.8.3]). If H is a locally compact abelian groupand pH its character group, then there is a canoncial isomorphism of ringsRpHq – Zr pHsbetween the representation ring of H and the integral group ring on pH. Inparticular,RpTMq – ZrM s,the group algebra on the ZG-lattice M ; cf. 1.2.Example 2.1.3. By [BtD85, II.7.7], the representation ring of a product isthe tensor product (over Z) of the representation rings of the factors so thatRpΓZrq “ RpTZr ˆGq – RpTZrq bRpGq – Zrx˘11 , . . . , x˘1r srvs{pvp ´ 1qwhere v is the character vpσkq “ e2piik{p generating all p irreducible repre-sentations of G. 4To study the representation theory of compact but not necessarily con-nected Lie groups, we need an analog for what maximal tori are to compact16connected Lie groups.Definition 2.1.4 ([Seg68b, 1.1]). A closed subgroup C of a compact Liegroup H is called a Cartan subgroup if it is topologically cyclic, i.e., itcontains an element called a generator whose powers are dense in the group,and has finite Weyl group WHpCq “ NHpCq{C with NHpCq the normalizerof C in H.If H is connected, the Cartan subgroups are the maximal tori. By Kro-necker’s theorem almost every element of a torus is a generator and topo-logically cyclic Lie groups are isomorphic to the direct product of a torusand a finite cyclic group; cf. [BtD85, I.4.13,14]. Conjugacy classes of Cartansubgroups are detected by cyclic subgroups of the group of components:Proposition 2.1.5 ([BtD85, IV.4.6]). Let H be a compact Lie group and H0the connected component of the identity. There is a bijection of conjugacyclasses between Cartan subgroups of H and cyclic subgroups of H{H0.Now let M be the ZG-lattice ZGs ‘ IGt. By, there is an exactsequence0 ÑMG iÑM Ñ ĎM Ñ 0with MG – pZGsqG – Zs and ĎM – IGs`t. We can view the injection i asa map TZs Ñ TM , xm ÞÑ xipmq and, by abuse of notation, extend it to aninjectioni : ΓZs “ TZs ˆGÑ ΓM . ( 2.1.6. Let G “ Cp, where p ą 1 is any prime. If M is theZG-lattice ZGs‘IGt, then the Cartan subgroups of the associated semidirectproduct ΓM “ TM ¸G are conjugate to either TM or ΓZs as per Proposition 2.1.5 states that there are exactly two conjugacy classesof Cartan subgroups corresponding to all of G and the trivial subgroup of17G. The subgroups TM and ΓZs are topologically cyclic and not conjugate.We compute their normalizers and Weyl groups. By,pt, gq ¨ pt1, g1q ¨ pt, gq´1 “ ptgpt1qg1pt´1q, g1q.ThenNΓpΓZsq “ tpt, gq : tg1pt´1q P TZs for all g1 P Gu– tpA1, . . . , As, B1, . . . , Btqu ¸Gwhere each Ai P TZG is of the form px, ζx, . . . , ζp´1xq with x P S1 and ζp “ 1and where each Bi P TIG is of the form pθ, . . . , θq with θp “ 1. It followsthat WΓpΓZsq – Cs`tp is finite. Also,NΓpTMq “ tpt, gq : gpt1q P TM for all t1 P TMu “ ΓMso that WΓpTMq – G is finite.Theorem 2.1.7. Let G “ Cp “ xσ | σp “ 1y, where p ą 1 is any prime. IfM is the ZG-lattice ZGs‘IGt and ΓM “ TM¸G is the associated semidirectproduct, thenRpΓMq – ZrMGs b Z indΓMTM xm : rms PM{G(b Zrvs{pvp ´ 1qsubject to the relations(i) indΓMTM xm “ xm ¨ Φp for m PMG and Φp “ 1` v ` ¨ ¨ ¨ ` vp´1,(ii) indΓMTM xm ¨ indΓMTM xm1 “p´1ÿi“0indΓMTM xm`σim1,(iii) xm ¨ vj ¨ indΓMTM xm1 “ indΓMTM xm`m1.Proof. Consider the character lattice pTM – M . Since pTM is countable thesemidirect product ΓM is regular; cf. 1.1.3. Since p is prime, M has two18types of G-orbits. In multiplicative notation they are those with a singlecharacter xm, m PMG having isotropy all of G, and those with p characterstxm,xσpmq, . . . ,xσp´1pmqu with trivial isotropy.Let pG “ xv | vp “ 1y with v as per 2.1.3. Then by 1.1.4, the following isa complete list of irreducible representations of ΓM :(i) xm ¨ vj for every m PMG and 0 ď j ă p,(ii) indΓMTM xm where m runs through a system of representatives for M{G,m RMG.To understand the product structure, note that the restriction mapres : RpΓMq ÑźrCsRpCq ( rCs runs through the finite set of conjugacy classes of Cartan sub-groups of ΓM is injective because every element of a compact Lie group iscontained in some Cartan subgroup; cf. [Seg68b, 1.2]. By 2.1.6, the onlyCartan subgroups of ΓM up to conjugation are TM and ΓZs so that : RpΓMq Ñ RpTMq ˆRpΓZsqV ÞÑ presΓMTM V, resΓMΓZsV qTo calculate the image of this map we use three facts:1. By 2.1.3, RpΓZsq – RpTZsq b Zrvs{pvp ´ 1q with a restriction mapi˚ : RpTMq Ñ RpTZsqgiven by By the restriction-induction formula, cf. and,resΓMTM indΓMTMxm – xm ` xσpmq ` ¨ ¨ ¨ ` xσp´1pmq “ pxmq˚19for every rms PM{G, m RMG.3. Combining these two we getresΓMΓZs indΓMTMxm – i˚xm ¨ Φpfor every rms PM{G, m RMG and Φp “ 1` v ` ¨ ¨ ¨ ` vp´1.The irreducible representations of RpΓMq are thus mapped as follows:xm ¨ vj ÞÑ pxm, i˚xm ¨ vjq m PMGindΓMTM xm ÞÑ ppxmq˚, i˚xm ¨ Φpq rms PM{G,m RMG.The product structure between the irreducibles is thusvp “ 1,indΓMTM xm ¨ indΓMTM xm1 “p´1ÿi“0indΓMTM xm`σim1 ,xm ¨ vj ¨ indΓMTM xm1 “ indΓMTM xm`m1 .Note that m ` σim1 may be in MG but that indΓMTM xm “ xm ¨ Φp wheneverm PMG.Another way to obtain relation piiq of 2.1.7 is to express the inducedrepresentations indΓMTM xm as matrix representations and then compute theirKronecker product; cf. [CR62, §§12A,D]. Also note that dim xm ¨ vj “ 1 andthat dim indΓMTM xm “ p; cf. mentioned in appendix B, RpHq is a finitely generated ring for anycompact Lie group H. Continuing with the notation M “ ZGs ‘ IGt, ournext goal, then, is to find a finite presentation for RpΓMq, that is, to findfinitely many generators and relations for RpΓMq. It follows from the proof20of 2.1.7 that the restriction mapresΓMTM : RpΓMq Ñ RpTMqmaps onto RpTMqG – ZrM sG with kernel the ideal generated by v´1. Thus,if we can find finite generators of ZrM sG we may hope to extract finite gen-erators for RpΓMq. The next section, then, will discuss finite presentationsof ZrM sG. Before moving on, we show that for this purpose we may assumeM to be a free ZG-lattice, that is, M – ZGr for some r; cf. 1.2.Lemma 2.1.8. Let K be a commutative ring and let N be a ZG-lattice witheffective quotient sN ; cf. ThenKr sN sG – KrN sG{pxm ´ 1 : m P NGqwhere it suffices to let m run over a Z-basis of NG.Proof. The canonical map Ď : N Ñ sN extends to KrN s:KrN s Ñ Kr sN sxm ÞÑ x smIt follows from that there is a bijection of G-orbits Gpmq – Gpsmq sothat the diagramKr sN sp q˚// Kr sN sGKrN sp q˚//ĎOOKrN sGĎOOcommutes. This impliesKr sN s –ĞKrN ssince KrN sG is K-generated by finite G-orbit sums pxmq˚; cf. More-over, px smq˚ “ pxsnq˚ is equivalent to pxmq˚ “ pxmq˚xp for some p P NG. Theresult follows.21Since IG – ĎZG, cf., the lemma impliesZrM sG “ ZrZGs ‘ IGtsG – ZrZGs`tsG{pxm : m P`ZGtqG – Zt˘.When p is a prime of trivial class number, the representation ring of ΓMwith M any ZG-lattice consists of pieces we have now computed:Proposition 2.1.9. Let G “ Cp, where p ą 1 is any prime. If M is theZG-lattice Zr ‘ ZGs ‘ IGt and ΓM “ TM ¸ G is the associated semidirectproduct, thenRpΓMq – RpTZrq bR pTZGs‘IGt ¸Gq .Moreover, if p is of class number one and T is any torus of rank n on whichG acts by automorphisms with Γ “ T ¸G the associated semidirect product,thenRpΓq – RpΓMqfor some integers r, s, t obeying r ` sp` tpp´ 1q “ n.Proof. SinceTM – TZr ˆ TZGs‘IGtand the action on Zr is trivial we haveTM ¸G – TZr ˆ pTZGs‘IGt ¸Gq .The first claim now follows from the fact already used in 2.1.3 that therepresentation ring of a product is the tensor product of the representationrings of the factors. Next, let M 1 be the ZG-lattice given by the charactergroup of a general rank n torus T . Since the class number of p is trivial, by1.2.3 there is a ZG-isomorphismM “ Zr ‘ ZGs ‘ IGt –M 1for some r, s, t. It follows that T 1M – TM so that RpΓq – RpΓMq.222.2 Rings of invariantsAs mentioned towards the end of the last section, we are interested in findinga finite generating set for ZrM sG when M is a free ZG-lattice. That this ispossible is established byLemma 2.2.1. Let H be an arbitrary group and let N be any ZH-lattice.The multiplicative invariant algebra KrN sH is a finitely generated K-algebrafor any commutative ring K.Proof. It was shown in that a K-basis for KrN sH is given by finite H-orbit sums on N . In particular, KrN sH Ă KrDs, where D is the subgroup ofN consisting of all elements having finitely many H-conjugates. But N – Znfor some n, so D – Zm, for some m ď n. Thus D is finitely generated, andtherefore H acts as a finite group rH on D with KrN sH – KrDs rH . It followsfrom Noether’s finiteness theorem, cf., e.g., [Bou64, V.1.2], that KrN sH is afinitely generated K-algebra.Continuing with the free ZG-lattice M “ ZGr for some r, introduce thestandard Z-basis on ZG – Zp – xx1, . . . , xny “: xxy; cf. ThenZrM sG – Zrx˘11 , . . . ,x˘1r sG – Zrx1, . . . ,xrsGrs´1p px1q, . . . , s´1p pxrqswhere sppxiq is the pth elementary symmetric polynomial in the variables xiand G acts on xi “ pxi1, . . . , xipq by cyclic permutation. The ringZrx1, . . . ,xrsG – Sym pM˚qGis the additive invariant theory of M regarded as a Z-module; cf. 1.2. Per-mutation lattices are self-dual, i.e., M˚ –M so that we may drop the p q˚in the above isomorphism; cf. [Lor05, 1.4.3].After tensoring with a field of characteristic 0, there is an upper boundfor the number of generators of the additive invariant theory; cf., e.g., [Stu08,2.1.4]:23Theorem 2.2.2 (Noether’s degree bound). Let K be a field of characteris-tic 0 and let V be an n-dimensional K-representation of a finite group H.Then the invariant ring SympV ˚qH has an algebra basis consisting of at most`n`|H|n˘invariants whose degree is bounded above by the group order |H|.Finding a minimal set of generators—called fundamental invariants—forZrM sG as a Z-algebra and the algebraic relations, called syzygies, amongthem turns out to be rather complicated except for the following specialcase:Example 2.2.3. Consider RpΓZGq and RpΓIGq when p “ 2. In that case,ZrZGsG – Zrx˘11 , x˘12 sS2 where G “ C2 “ S2 is the symmetric group on twoletters. By the fundamental theorem of symmetric polynomials, cf., e.g.,[Stu08, 1.1.1]),Zrx1, x2sG – Zrs1, s2swhere s1 “ x1 ` x2 and s2 “ x1x2. ThusZrZGsG – Zrx1, x2sS2rs´12 s – Zrs1, s2srs´12 s – Zrs1, s˘12 s.The lattice of G-invariants, ZGG, is generated by s˘12 and s1 “ x˚1 so thatby 2.1.7 and 2.1.8,RpΓZGq – Z“indΓZGTZG x1, s˘12 , v‰{`v2 ´ 1, pv ´ 1q indΓZGTZG x1˘,RpΓIGq – Z“indΓIGTIG x1, v‰{`v2 ´ 1, pv ´ 1q indΓIGTIG x1˘.Indeed, this calculation agrees with that for RpOp2nqq, n “ 1 found in[BtD85, VI.7.7] since ΓIG – S1 ¸ C2 – Op2q. 4The reason that in this example one is readily able to write down aminimal algebra basis and syzygies is that the invariant ring SympZGqSn –Krx1, . . . , xnsSn is classically known even when K is only an integral domainsuch as Z—a fact we will make use of in 2.3.1 when discussing semidirect24products between tori and the symmetric group Sn. However, most algo-rithms for computing SympV ˚qH require that K be a field. Informationabout ZrM sH as a subring of KrM sH may be obtained by carrying out“Galois descent” as done in [Len74] but not pursued in this thesis.The symmetric group Sn—when acting by permutation representations—is an example of a reflection group1, a class of groups (really: representa-tions!) for which the invariant ring is particularly nice; cf., e.g., [Stu08,2.4.1]:Theorem 2.2.4 (Chevalley-Shephard-Todd). Let V be a vector space. Theinvariant ring SympV ˚qH of a finite matrix group H Ă GLpV q is generatedby n algebraically independent homogeneous invariants if and only if H is areflection group.This fact can be illustrated with the representation rings of compactconnected Lie groups G: in that case RpGq is isomorphic to RpT qW –Zrx˘11 , . . . , x˘1n sW with T a maximal torus of G of rank n and W the Weylgroup of T , all of which are reflection groups; cf., e.g., [BtD85, VI.2.1].Example 2.2.5. Consider the Lie group Upnq. Its rank is n and its Weylgroup is the symmetric group Sn acting by permutation of coordinates. ThenRpUpnqq – Zrx˘11 , . . . , x˘1n sW – Zrx1, . . . , xnsW rs´1n s – Zrs1, . . . , s˘1n swith si the ith elementary symmetric polynomial. Chevalley-Shephard-Toddconfirms the classic fact that tsi : i “ 1, . . . , nu forms a set of fundamentalinvariants with no nonzero syzygy. 4Unfortunately, the group G “ Cp under consideration is neither a re-flection group nor does there seem to be any reason to believe that Galois1A reflection group is a discrete group which is generated by a set of reflections of afinite-dimensional Euclidean space. Note that being a reflection group is not a propertyintrinsic to the group itself but rather to its action.25descent will be ‘nice’. However, considering KrM s instead of ZrM s when Kis a field amounts to tensoring RpΓq with K which is still valuable.There exists a plethora of algorithms for computing fundamental invari-ants and syzygies over fields, many of which use Gro¨bner bases as basicbuilding blocks; cf. [Stu08, ch. 2]. Since G is finite abelian, the mostpromising candidate for such an algorithm is [Stu08, 2.7.3] which has beenadapted for our purposes below. Before we can use it we need the follow-ing theorem. Recall that the exponent of a finite group H is the smallestpositive integer n such that xn “ 1 for all x P H.Theorem 2.2.6 ([CR62, 41.1]). Let n be the exponent of a finite group H.Then the cyclotomic field Qpζq with ζ a primitive nth root of unity is a split-ting field for H, i.e., each irreducible CH-representation has a realizationwith matrices having entries in Qpζq.Clearly, G “ Cp has exponent p.Algorithm 2.2.7 (Invariant theory of the cyclic group of prime order). LetK “ Qpζq be a splitting field for the cyclic group G “ Cp “ xσ | σp “ 1y ofprime order p and let M be a free ZG-lattice of rank r. Then the followingalgorithm produces a finite algebra basis forSym pKbMqG – Krx1, . . . ,xrsGwith xi the standard K-basis of KG.1. Choose a diagonalizing matrix MG so thatMGσM´1G “ diagp1, ζ, . . . , ζp´1q.2. Introduce new variables yi “ py1, . . . , yrq :“MGpxiq.3. For ai “ pai1, . . . , aipq P Np, i “ 1, . . . , r, consider the linear homoge-26neous congruencerÿi“1ai2 ` 2ai3 ` ¨ ¨ ¨ ` pp´ 1qaip ” 0 mod p.Compute a finite generating set H for the solution monoid. ThenKrx1, . . . ,xrsG – Kry1, . . . ,yrsMGGM´1G– Krya11 ¨ ¨ ¨yarr : pa1, . . . , arq P Hs.4. To rewrite an invariant f P Krx1, . . . ,xrsG in terms of the ya11 ¨ ¨ ¨yarrwith pa1, . . . , arq P H, note thatfpxq “ fpM´1G yq “ gpyqis an invariant polynomial in y and so can be expressed in the ya11 ¨ ¨ ¨yarr .The generating set H in step 3 is the so-called Hilbert basis. Existenceand uniqueness of the Hilbert basis for a solution monoid such as the oneabove can be established by algorithm [Stu08, 1.4.5]. The following examplesare easy enough to do by hand. Let RKpHq :“ RpHq bZ K for any locallycompact Lie group H.Example 2.2.8. Let us check that the above algorithm produces the sameanswer as example 2.2.3. A splitting field for G “ C2 is Qp´1q “ Q. Amatrix diagonalizing the generator σ “`0 11 0˘of G in the desired way isMG “`1 11 ´1˘. Without loss of generality, we may thus return to workingover Z. The new variables are then y1 “ x1 ` x2 and y2 “ x1 ´ x2. Thelinear homogeneous congruence isa2 ” 0 mod 2.27The solution monoid is generated by tp1, 0q, p0, 2qu so thatZrx1, x2sG – Zry1, y22s.By step 4 of the algorithm or otherwise, we can write the generator s2 “ x1x2of ZGG as4s2 “ y21 ´ y22so thatZrIGs – Zry1, y22sGrs´12 s{ps2 ´ 1q – Zry1s.Since y1 “ x˚1 , it then follows by 2.1.7 and 2.1.8 thatRpΓZGq – Z“indΓZGTZG x1, s˘12 , v‰{`v2 ´ 1, pv ´ 1q indΓZGTZG x1˘RpΓIGq – Z“indΓIGTIG x1, v‰{`v2 ´ 1, pv ´ 1q indΓIGTIG x1˘as before. 4Example 2.2.9. Continuing on from the previous example, we computeRpΓZG2q when G “ C2. After diagonalizing the action on each summandZG separately, i.e.,y11 “ x11 ` x12y12 “ x11 ´ x12y21 “ x21 ` x22y22 “ x21 ´ x22we obtain the linear homogeneous congruencea12 ` a22 ” 0 mod 228having solution monoid generated bytp1, 0, 0, 0q, p0, 0, 1, 0q, p0, 2, 0, 0q, p0, 0, 0, 2q, p0, 1, 0, 1qu.It follows thatZrx1,x2sG – Zry11, y21, y212, y222, y12y22s.Note that these generators are not algebraically independent sincepy12y22q2 “ y212y222.Since y11 “ x˚11, y21 “ x˚21, 4s2px1q “ y211 ´ y212, 4s2px1q “ y221 ´ y222, andy12y22 “ px11x21q˚ ´ px11x22q˚, it follows by 2.1.7 thatRpΓZG2q – Zrindx11, indx21, indx11x21 ´ indx11x22, s˘12 px1q, s˘12 px2q, vssubject to the relations v2 “ 1, v ¨ ind xm “ ind xm, andpindx11x21 ´ indx11x22q2 “`pindx11q2 ´ 4s2px1q˘ `pindx11q2 ´ 4s2px2q˘.By 2.1.8, we then also obtainRpΓZG‘IGq – RpΓZG2q{ps2px2q ´ 1qRpΓIG2q – RpΓZG2q{ps2px1q ´ 1, s2px2q ´ 1q.4Example 2.2.10. Consider RpΓZGq and RpΓIGq when p “ 3. A splittingfield for G “ C3 is K :“ Qpζq with ζ “ e2pii{3. After diagonalizing, the new29variables arey1 “ x1 ` x2 ` x3y2 “ x1 ` ζ2x2 ` ζx3y3 “ x1 ` ζx2 ` ζ2x3.The congruence relation isa2 ` 2a3 ” 0 mod 3and has solution monoid generated by tp1, 0, 0q, p0, 1, 1q, p0, 3, 0q, p0, 0, 3qu.It follows thatKrxsG – Kry1, y2y3, y32, y33s.In accordance with Chevalley-Shephard-Todd, note that the generating vari-ables are not algebraically independent sincepy2y3q3 “ y32y33. ( this is the only syzygy can be proved using Gro¨bner bases; cf. [Stu08,2.5.3]. Using step 4 of 2.2.7,s1 “ y13s2 “ y21 ´ y2y327s3 “ y31 ` y32 ` y33 ´ 3y1y2y39px21x2q˚ “ y31 ` ζy32 ` ζ2y33.Changing variables, we obtainKrZGsG – Krs1, s2, px21x2q˚, s˘13 s30and by 2.1.8,RKpΓZGq – KrindΓZGTZGx1, indΓZGTZGx1x2, indΓZGTZGx21x2, s˘13 , vssubject to the relations v3 “ 1,v ¨ indΓZGTZG xm “ indΓZGTZG xm for xm “ x1, x1x2, x21x2,and the equivalent of in these variables. Moreover by 2.1.7,RKpΓIGq – RKpΓZGq{ps3 ´ 1q.4Example 2.2.11. Let p “ 5, G “ C5, and K “ Qpζq with ζ “ e2pii{5. By[Stu08, 2.7.1], KrxsG has the following minimal algebra basis consisting of15 elementsty1, y2y5, y3y4, y2y23, y22y4, y3y25, y24y5, y2y34, y33y5, y4y35, y32y3, y52, y53, y54, y55uwith several obvious algebraic relations among them. 42.3 Semidirect products by other groupsAn analysis similar to that of the last sections can be carried out for casesother than G “ Cp. However, as discussed in section 1.2 a complete re-ducibility theorem like 1.2.3 is only available in rare instances. For instance,Reiner states in [CR90, §34] that a complete classification theorem cannot beexpected for cyclic p-groups of order pk when k ą 2. When k “ 2 there areprecisely 4p ` 1 types of indecomposable ZCp2-modules; cf. [CR90, 34.32].Another example is the dihedral group D2p of order 2p with p ą 2 a fixedprime discussed below. Some additional cases where integral representationshave been calculated are listed on [CR90, p. 753].31So letG :“ D2p “ Cp ¸ C2 “ xσ | σp “ 1y ¸ xτ | τ 2 “ 1ythe dihedral group of order 2p introduced in 1.1.6 and let ζ be a primitive pthroot of unity. The 10 indecomposable types of ZG-lattices are representedby the following list; cf. [CR90, 34.50]:(i) R :“ Zrζs, P :“ p1 ´ ζqR on which σ acts as multiplication by ζ andτ acts as complex conjugation.(ii) Z, Z1, and ZC2 on which σ acts trivially while τ acts as `1 on Z andas ´1 on Z1.(iii) The two non-split extension P Ñ X0 Ñ Z, R Ñ X1 Ñ Z1, i.e.,X0 – indGC2 Z and X1 – indGC2 Z1.(iv) The three non-split extensions R Ñ Y0 Ñ ZC2, P Ñ Y1 Ñ ZC2,R ‘ P Ñ Y2 Ñ ZC2.As before, let ΓM “ TM ¸G where pTM “M . ThenRpΓZq – RpTZq bRpGqwith RpGq given by 1.1.6. Let pCp “ xv | vp “ 1y, pC2 “ xw | w2 “ 1y, andM – Zn – xx1, . . . , xny under a choice of Z-basis; cf. Based onMackey’s method, we determine RpΓMq as abelian groups for some of theabove ZG-lattices:• The irreducible representations of ΓZ1 are (i) those of G and (ii) foreach i ą 0 and 0 ď j ă p an induced representation indΓZ1TZ1¸Cppxi ¨ vjq.• The irreducible representations of ΓZH are (i) those of G, (ii) the one-dimensional characters px1x2qi, i P Z, and (iii) for each i ă j P Z and0 ď k ă p an induced representation indΓZHTZH¸Cppxi1xj2 ¨ vkq.32• The irreducible representations of ΓR are (i) those of G, (ii) inducedrepresentations indΓRTR¸C2pχ¨wiq, i “ 0, 1, for every complex conjugationinvariant character rχs P Zrζs{G, and (iii) indΓRTR χ1 for every othernonzero character rχ1s P Zrζs{G. For instance, when p “ 5Zrζs – Zt1, ζ, ζ2, ζ3uhas conjugation invariant subspaces Zt1u, Ztζ2` ζ3u, and their directsum leading to induced representationsindΓRTR¸C2pxl1 ¨wiq, indΓRTR¸C2ppx3x4qk ¨wiq, and indΓRTR¸C2pxl1px3x4qk ¨wiq.A product structure between the listed irreducibles can be determinedas in the last sections by using Cartan subgroups of ΓM ; cf. By2.1.5, we expect T ¸D2p to have three Cartan subgroups up to conjugationso this is a manageable task which could be pursued in future work.Instead, we turn our attention to the symmetric group Sn on n letterswith its natural permutation ZSn-lattice M – Zn – xx1, . . . , xny on whichSn acts by sending xi to xspiq for all s P Sn; cf. 1.2.1.Theorem 2.3.1. Let M – Zn – xx1, . . . , xny be the natural permutationZSn-lattice and let T be a rank n torus such that pT – M . If Γ “ T ¸ Snis the associated semidirect product, then RpΓq is finitely generated as aZ-algebra by the following representations:(i) the irreducible representations of Sn,(ii) the torus characters s˘1n .(iii) for each 1 ď i ă n and for each irreducible representation ρ of Sn´iˆSi,an irreducible representation indΓT¸pSn´iˆSiqpx1 ¨ ¨ ¨ xi ¨ ρq.Proof. By Mackey’s method, the irreducible representations of Γ are givenby indΓΓmpxm ¨ ρq for each rms P M{G where Γm “ T ¸ pSnqm and ρ is anirreducible representation of the isotropy group pSnqm.33As in, the restriction map from RpΓq to the product of the rep-resentation rings of the Cartan subgroups of Γ is an injection. One of theseCartan subgroups is T withresΓT : RpΓq Ñ RpT qSnindΓΓmpxm ¨ ρq ÞÑ resΓT indΓΓmpxm ¨ ρq – dim ρ ¨ pxmq˚.Since RpT qSn – Zrs1, s2, . . . , sn´1, s˘1n s with si the ith elementary symmetricpolynomial, it follows that we may take the generators of RpΓq to be thoseirreducible representations corresponding to pxmq˚ P t1, s1, s2, . . . , sn´1, s˘1n uunder the restriction map.The preimages of pxmq˚ “ sjn, j “ ´1, 0, 1 are the representations sjnb ρwith ρ an irreducible representation of Sn. Each other si with 1 ď i ă n haspreimage!indΓT¸pSn´iˆSiqpx1 ¨ ¨ ¨ xi ¨ ρq : ρ an irreducible representation of Sn´i ˆ Si).The product structure of RpΓq may be further studied by identifying allCartan subgroups of Γ. By 2.1.5, we see that this task soon becomes difficultas n grows.Example 2.3.2. Consider n “ 3, that is, S3 – D6 and let T be as in 2.3.1.Using 2.1.5, we determine the Cartan subgroups of Γ “ T ¸ S3 to be T ,$’&’%¨˚˝xxy˛‹‚: x, y P S1,/./-ˆ C2 – pS1q2 ˆ C2,and$’&’%¨˚˝zzz˛‹‚: z P S1,/./-ˆ C3 – S1 ˆ C3.34It follows that RpT ¸ S3q is generated by (also cf. 1.1.6)RpΓq Ñ RpT qS3 ˆRppS1q2 ˆ C2q ˆRpS1 ˆ C3q– Zrs1, s2, s˘13 s ˆ Zrx˘1, y˘1, ws{pw2 ´ 1q ˆ Zrz˘1, vs{pv3 ´ 1qindpx1 ¨ wiq ÞÑ ps1, p2x` yqwi, zp1` v ` v2qqindpx1x2 ¨ wiq ÞÑ ps2, p2x2 ` y2qwi, z2p1` v ` v2qqpx1x2x3q˘1 ÞÑ ps3, x2y, z3q˘1w ÞÑ p1, w, 1qindS3C3 v ÞÑ p2, 1` w, v ` v2q.We see from this that indpχ ¨ wq “ w ¨ indpχq so that,RpΓq – Zrindpx1q, indpx1x2q, px1x2x3q˘1, w, indS3C3 vssubject to various relations. 4A similar approach may be used to find generators of RKpT¸Gq wheneverG is a reflection group and K is a splitting field for G; cf. 2.2.4 and 2.2.6.In that case, a theorem of Reichstein, cf. [Rei03] and [DR09], asserts:Theorem 2.3.3. Let G be a finite subgroup of GLnpZq and let K be a split-ting field for G. Then the ring of multiplicative invariants Krx˘11 , . . . , x˘1n sGhas a finite SAGBI basis if and only if G is a reflection group.Without going into detail, a finite SAGBI basis is in particular a finitecollection of elements of Krx˘11 , . . . , x˘1n sG that generates Krx˘11 , . . . , x˘1n sGas a K-algebra. The algorithm producing a SAGBI basis is known as thesubduction algorithm. It follows that whenever G is a finite reflection group,the subduction algorithm may be used to find finitely many generators ofRKpT ¸Gq.35Chapter 3Applications to Lie Groups andK-TheoryOnce again let G “ Cp “ xσ | σp “ 1y be the cyclic group of prime order p.In this chapter we apply the results of the previous chapter to classical Liegroups and the K-theory of the classifying space of the semidirect productsΓ “ T ¸G.3.1 Tori of compact connected Lie groupsDenote by TG and WG a maximal torus respectively the Weyl group ofa compact connected Lie group G. Consider the case when G is one ofSUpnq, Upnq, SOp2nq, SOp2n ` 1q, or Sppnq.1 By [BtD85, IV.3], there arehomeomorphismsTUpnq « pS1qn TSUpnq « pS1qn´11There is sometimes confusion in the literature about the notation of this last group.Here Sppnq denotes the compact symplectic group of isometries of the p-dimensionalquarternion algebra Hp with its standard symplectic scalar product; cf. [BtD85, I.1.11].36and the inclusions Upnq ãÑ SOp2nq ãÑ SOp2n`1q and Upnq ãÑ Sppnq inducehomeomorphismsTUpnq « TSOp2nq « TSOp2n`1q respectively TUpnq « TSppnq.Now fix a prime p and let Sp be the symmetric group on p letters. LetWrppq “ Cp2 ¸ Sp be the wreath product of C2 by Sp, and SWrppq be thesubgroup of Wrppq consisting of even permutations. Then again by [BtD85,IV.3],WUppq – WSUppq – Sp, WSpppq – WSOp2p`1q – Wrppq, and WSOp2pq – SWrppq.Up to conjugation, there is a unique subgroup Cp Ă Sp Ă WG for all G andwe can form the semidirect product TG ¸ Cp as a subgroup of TG ¸WG.Proposition 3.1.1. Let G “ Cp, where p ą 1 is any prime. If G is any ofthe Lie groups SUppq, Uppq, SOp2pq, SOp2p` 1q, or Spppq and ΓG “ TG¸Gas per above, thenRpΓGq –$&%RpΓIGq if G “ SUppqRpΓZGq else.Proof. The tori of Uppq, SOp2pq, SOp2p`1q, and Spppq are allG-equivariantlyisomorphic with G acting by cyclic permutation. It follows that TG is TZGin all of these cases. Since TSUppq is TUppq modulo the diagonal, TSUppq isG-isomorphic to TIG.These representation rings are thus known by 2.1.7. For example, 2.2.3discussed the case of TSUp2q ¸ C2 – Op2q.Next, recall that the quotient of a Lie group G by a closed normal sub-group N is a Lie group; cf. [BtD85, I.4.4]. If G is compact and connectedthen so is G{N , the latter being the continuous image of the quotient mappi : G Ñ G{N . In particular, we can consider the center ZpGq of G and37form the projectivization PG :“ G{ZpGq of G.Proposition 3.1.2. Let p ą 1 be a prime of class number one and let G beany of the Lie groups SUppq, SOp2pq, SOp2p` 1q, or Spppq. ThenRpΓPGq – RpΓGq.Proof. The group G has finite center in all cases; cf. [BtD85, V, 7.13]. Thequotient map pi is thus a finite covering space. In particular, the ranks ofPG and G are the same and their Weyl groups are isomorphic. Since p isof trivial class number there is only one nontrivial way in which Cp Ă WPGcan act on TPG and this is the same way that Cp Ă WG acts on TG.Recall thatLemma 3.1.3. The Lie groups PSUpnq and PUpnq are isomorphic.Proof. We have ZpUpnqq – S1, ZpSUpnqq “ ZpUpnqq X SUpnq – Cn, andUpnq – SUpnqZpUpnqq :“ tab : a P SUpnq, b P ZpUpnqqu.By the second isomorphism theorem of group theory, cf., e.g., [DF04, 3.3.18],PSUpnq “ SUpnq{ZpUpnqq X SUpnq – SUpnqZpUpnqq{ZpUpnqq – PUpnq.It follows that RpΓP Uppqq – RpΓP SUppqq – RpΓSUppqq. An addendumabout the case of p “ 1: since Up1q « SOp2q « S1 has trivial Weyl groupwe get RpΓUp1qq – RpΓSOp2qq – RpS1q “ Zrx˘1s. The case of Spp1q « SUp2qis covered by 3.1.1. Finally, SOp3q « PSUp2q is taken care of by Topological K-theory of the classifyingspace of semidirect productsBy the Atiyah-Segal completion theorem, B.0.4, knowing the representationring of a compact Lie group yields the complex K-theory of its classifyingspace2. Namely, all we need to do is complete the representation ring at itsaugmentation ideal. The following lemma will be helpful in doing so:Lemma 3.2.1 ([Spa81, 5.1.5]). Let R be a ring. If f1 : V1 Ñ W1 andf2 : V2 Ñ W2 are two epimorphisms of R-modules, thenkerpf1 b f2q “ αpker f1 b V2q ` βpV1 b ker f2qwhere α : kerpf1q b V2 Ñ V1 b V2 and β : V1 b kerpf2q Ñ V1 b V2 are thecanonical maps.Note that if V1 and V2 are flat R-modules then the maps α and β areinclusions. In the category of abelian groups, a Z-module is flat if and onlyif it is torsion-free.Proposition 3.2.2. Let G “ Cp, where p ą 1 is any prime. If M is theZG-lattice Zr ‘ ZGs ‘ IGt and ΓM “ TM ¸ G is the associated semidirectproduct, thenK˚pBΓMq – RpTZrqpI1 bRpTZGs‘IGt ¸GqpI2with I1 “ kerpdim : RpTZrq Ñ Zq and I2 “ kerpdim : RpTZGs‘IGt¸Gq Ñ Zq.Moreover, if p is of class number one and T is any torus of rank n on whichG acts by automorphisms with Γ “ T ¸G the associated semidirect product,thenK˚pBΓq – K˚pBΓMqfor some integers r, s, t obeying r ` sp` tpp´ 1q “ n.2cf. appendix B for a definition39Proof. By B.0.4, there is an isomorphism K˚pBΓq – RpΓqxIΓ with IΓ “kerpdim : RpΓq Ñ Zq. Let p be of trivial class number. Then by 2.1.9,there exists a ZG-lattice M “ Zr ‘ ZGs ‘ IGt with integers r, s, t obeyingr ` sp` tpp´ 1q “ n andRpΓq – RpΓMq – RpTZrq bRpTZGs‘IGt ¸Gq.Since both RpTZrq and RpTZGs‘IGt¸Gq are Z-torsion-free, it follows by 3.2.1thatRpΓMqxIΓM – RpTZrqpI1 bRpTZGs‘IGt ¸GqpI2with I1 “ kerpdim : RpTZrq Ñ Zq and I2 “ kerpdim : RpTZGs‘IGt ¸ Gq ÑZq.Recall that as a consequence of the Hilbert basis theorem, cf., e.g.,[AM69, 7.6], every finitely generated ring such as the representation ring ofa compact Lie group is Noetherian. By [AM69, 10.12], completion of finitelygenerated modules over Noetherian rings preserves exact sequences and itfollows that completion commutes with quotients. Moreover, if a Ă m Ă Aare two ideals in a Noetherian ring A, then apm – aA pm.Consequently, completion of RpTZrq and RpTZGs‘IGt¸Gq at their respec-tive augmentation ideals is not difficult since our work thus far has enabledus to express these representation rings as finitely presented algebras, atleast as long as we work over a splitting field for G.We present some examples.Example 3.2.3. Consider K˚pBpT ¸ C2qq. By example 2.2.3 we haveRpΓZGq – Z“indΓZGTZG x1, s˘12 , v‰{`v2 ´ 1, pv ´ 1q indΓZGTZG x1˘,RpΓIGq – Z“indΓIGTIG x1, v‰{`v2 ´ 1, pv ´ 1q indΓIGTIG x1˘.40Completing the former at`c1 :“ indΓZGTZGx1 ´ 2, c2 :“ s2 ´ 1, u :“ v ´ 1˘,K˚pBΓZGq – Zrc1 ` 2, c2 ` 1, u` 1s {pc1,c2,uqMpu2 ` 2u, upc1 ` 2qq– Zrc1, c2, us {pc1,c2,uqMpu2 ` 2u, upc1 ` 2qq– ZJc1, c2, uK{pu2 ` 2u, upc1 ` 2qqand similarlyK˚pBΓIGq – ZJc1, uK{pu2 ` 2u, upc1 ` 2qq.A calculation based on the representation rings of 2.2.9 revealsK˚pBΓZG2 ;Qq – QJc1, c2, d, e1, e2, uKsubject to the relations u2 ` 2u “ 0, upci ` 2q “ 0, ud “ 0 andd2 “ pc21 ` 4c1 ´ 4e1qpc22 ` 4c2 ´ 4e2qand similarly for K˚pBΓZG‘IG;Qq – QJc1, c2, d, e1, uK and K˚pBΓIG2 ;Qq –QJc1, c2, d, uK. 4Example 3.2.4. Consider K˚pBpT ¸ C3qq. Let ζ be a primitive third rootof unity and let K “ Qpζq. Then by 2.2.10,RKpΓZGq – KrindΓZGTZGx1, indΓZGTZGx1x2, indΓZGTZGx21x2, s˘13 , vssubject to relations some of which are v3 “ 1 and v ¨ indΓZGTZG xm “ indΓZGTZG xm.After a similar change of variables as in the previous example, the K-theory of BΓZG with coefficients in K isK˚pBΓZG;Kq – KJc1, c2, c3, d, uKsubject to the relations u3 ` 3u2 ` 3u “ 0, upci ` 3q “ 0, and the equivalent41of in these variables. Similarly, K˚pBΓIG;Kq – KJc1, c2, c3, uK. 43.3 Results with mod p coefficientsThe primes p which have a cyclotomic field Qpζq of class number one are2, 3, 5, 7, 11, 13, 17, 19; cf., e.g., [Was97, 11.1]. Proposition 3.2.2 can beextended to other primes p by considering mod p K-theory instead:Theorem 3.3.1. Let G “ Cp, where p ą 1 is any prime. If T is a torus ofrank n on which G acts by automorphisms, and Γ “ T ¸G is the associatedsemidirect product, thenK˚pBΓ;Cpq – K˚pBΓM ;Cpqwith M “ Zr‘ZGs‘IGt, for some integers r, s, t obeying r`sp`tpp´1q “ n.Recall that by the universal coefficient theorem, C.0.5,K˚pBH;Cpq – K˚pBHq b Cpfor any compact Lie group H. The basic idea in proving this theorem is tointroduce an object Cp-equivalent to the torus T but that does not have anyof the class number issues.The class number of the pth cyclotomic field Qpζq came into considerationin Reiner’s theorem (1.2.3) since the ring of algebraic integers Zrζs in Qpζqis not a principal ideal domain in general. However, after p-adic completionQppζq has ring of algebraic integers Zprζs: a principal ideal domain; cf., e.g.,[CR62, p. 121]. The p-adic version of Reiner’s theorem is then the statementthat every rank n ZpG-lattice is isomorphic to pZr ‘ ZGs ‘ IGtq b Zp forsome integers r, s, t such that r ` sp` tpp´ 1q “ n.It is then natural to ask which topological group has as Pontryagin dualgroup the p-adic integers Zp. The answer is given most directly by42Theorem 3.3.2 (Pontryagin Duality Theorem). Let H be a locally com-pact abelian group. Then the dual (character group) of pH is canonicallyisomorphic to H.See, e.g., [Rud62, 1.7.2] for a proof. The desired group is thus pZp. Moreexplicitly:Definition 3.3.3. Define the Pru¨fer group Cp8 to be the colimit of thegroups Cpi with respect to the homomorphisms p : Cpi Ñ Cpi`1 given bymultiplication by p. Then pZp – Cp8 ; cf., e.g., [MP12, 10.1.11]. A discretegroup isomorphic to Cnp8 is known as a p-discrete torus of rank n and denotedby T˘ .Note that T˘ is only locally compact since the Pru¨fer group is not finite.The p-discrete torus T˘ can also be identified with the subgrouptt P T : tpi“ 1 for some i P Nuof a (regular) torus T . The natural map T˘ Ñ T induces a Cp-equivalence ofclassifying spaces, that is, the induced mapH˚pBT ;Cpq Ñ H˚pBT˘ ;Cpqis an isomorphism; cf. [DW94, 6.4].One more ingredient is needed before we can prove 3.3.1:Theorem 3.3.4 (Leray-Serre-Atiyah-Hirzebruch spectral sequence). LetF ãÑ E Ñ Bbe a fibration with B a path-connected CW complex and let h˚ be a general-ized cohomology theory. Then there exists a multiplicative spectral sequenceEp,q2 “ HppB;hqpF qq ñ hp`qpEq.43Note that B is not required to be finite-dimensional. See, e.g., [Whi78,XIII.4.9* and the remarks after] for a proof. We are now ready for theProof of 3.3.1. The given semidirect product is a split extension of groups1 Ñ T Ñ Γ Ñ GÑ 1inducing a fibration of classifying spacesBT Ñ BΓ Ñ BGwith path-connected base. The Leray-Serre-Atiyah-Hirzebruch spectral se-quence of this fibration in mod p topological K-theory isEi,j2 “ HipBG;KjpBT ;Cpqq ñ Ki`jpBΓ;Cpq. (, the spectral sequence of the identity map BT Ñ BT with fiber a point˚ isEi,j2 “ HipBT ;Kjp˚;Cpqq ñ Ki`jpBT ;Cpq. ( mod p K-theory of a point isKjp˚;Cpq “$&%0 if j is odd,Cp if j is even.The associated p-discrete torus T˘ Ă T inherits a G-action from T makingH ipBT ;Cpq – H ipBT˘ ;Cpq a G-equivariant isomorphism. By the p-adicversion of Reiner’s theorem T˘ must have Pontryagin dual ZpG-isomorphicto M b Zp withM “ Zr ‘ ZGs ‘ IGtfor some integers r, s, t such that r ` sp ` tpp ´ 1q “ n. But then there is44also a G-equivariant isomorphism H ipBT˘ ;Cpq – H ipBTM ;Cpq. It followsthat there is an isomorphism of spectral sequences leading to a G-equivariant isomorphismK˚pBT ;Cpq – K˚pBTM ;Cpq.This in turn induces an isomorphism of spectral sequences leading toK˚pBΓ;Cpq – K˚pBΓM ;Cpq.45Chapter 4Adjoint Action and EquivariantK-Theory of Lie GroupsThis chapter is concerned with computing the equivariant K-theory of com-pact connected Lie groups acting on themselves by conjugation. When thefundamental group of G is torsion-free the answer is already known. Tostate it, we need the followingDefinition 4.0.5 ([Wei94, 8.8.1]). Fix a commutative ring K and let R be aK-algebra. Then the R-module Ω1R{K of Ka¨hler differentials of R over K hasthe following presentation: there is one generator dr for every r P R withdk “ 0 if k P K. For each r, s P R there are two relations:dpr ` sq “ dr ` ds dprsq “ s ¨ dr ` r ¨ ds.Also let Ω0R{K “ R and ΩiR{K “Źi Ω1R{K, the ith exterior power.Theorem 4.0.6 ([BZ00, 3.2]). Let G be a compact connected Lie grouphaving torsion-free fundamental group and acting on itself by conjugation.Then there is an algebra homomorphism between the equivariant K-theoryof G and the ring of Ka¨hler differentials of the representation ring over the46integers. In symbols,K˚GpGq – Ω˚RpGq{Z. ( shows in particular that K˚GpGq is a free RpGq-module of rankrankGÿk“0ˆrankGk˙“ 2rankG.It is an open problem what the K-theory might be in case pi1pGq hastorsion. To show that does not hold in the torsion case we inspectthe projective unitary group PUpnq – PSUpnq “ SUpnq{ZpSUpnqq, cf. 3.1.3,having fundamental group isomorphic toZpSUpnqq “ tdiagpζ, . . . , ζq : ζn “ 1u – Cn,the center of SUpnq. A maximal torus of PUpnq can be realized as the imageof a maximal torus of SUpnq under the projection mapCn Ñ SUpnqpiÑ PSUpnq – PUpnq. ( long exact homotopy sequence applied to this fibration gives the iso-morphism pi1pPUpnqq – Cn. In the non-equivariant setting:Theorem 4.0.7 ([Pet67]). Let PUpprq be the projective unitary group ofdegree pr with p an odd prime. Then the K-theory and the cohomology ofPUpprq are isomorphic as abelian groups.However, their approach in calculating the K-theory using the fibrationSUpprq Ñ PUpprq Ñ BCprand the Atiyah-Hirzebruch spectral sequence (3.3.4) does not generalize tothe equivariant setting. Instead, we shall employ, cf. [Seg68a] and [Mat73],47Theorem 4.0.8 (Equivariant Atiyah-Hirzebruch spectral sequence). Let Gbe a compact Lie group and let X be a finite G-CW complex. Then associatedto the skeletal filtrationX0 Ă X1 Ă ¨ ¨ ¨ Ă Xn Ă ¨ ¨ ¨ Ă Xthere exists a multiplicative spectral sequenceEp,q2 – HpGpX;KqGqEp,q8 – Kp`qG,p pXq{Kp`qG,p`1pXqwhere KnG,ppXq :“ ker pKnGpXq Ñ KnGpXp´1qq and where KqG is the coefficientsystem defined by G{H ÞÑ KqGpG{Hq; cf. appendix A.4.1 The projective unitary group of degree 2In this section, let G :“ PUp2q – SUp2q{C2 – SOp3q so that pi1pGq – C2.Computing K˚GpGq via the equivariant Atiyah-Hirzebruch spectral se-quence, 4.0.8, begins with a skeletal filtration of G to determine the Bredoncohomology HpGpG;KqGq; cf. appendix A. Since the category of G-vectorbundles over the homogeneous space G{H is equivalent to the category ofH-modules, cf. [Seg68a], KqGpG{Hq vanishes for q odd and is RpHq for qeven. This shows that HpGpG;KqGq “ 0 for q odd and HpGpG,KqGq “ HpGpG;Rqfor q even, where R is the coefficient system G{H ÞÑ RpHq. By A.0.3.1, theBredon cochain complex is then given byCnGpG;Rq –àrαsnRpGαqwhere rαsn are the n-dimensional G-cells and α without brackets is a repre-sentative.48LetT “#«x 00 x´1ff: x P S1+be a maximal torus of G with square brackets denoting the image of a matrixunder pi : SUp2q Ñ PUp2q; cf. Also let σ “ r 0 11 0 s be the generatorof the Weyl group W – C2 – xσ | σ2 “ 1y of G acting by permuting thecoordinates of T . Then the orbit space G{G is homeomorphic to T {W ; cf.[BtD85, IV.2.6]. It follows that under the conjugation action, a skeletalfiltration of G is equivalent to a skeletal filtration of T . To study the latter,use an explicit homeomorphism of T with S1 given byφ :«x 00 x´1ffÞÑ x2 “: ySince W acts by permutation, σ sends y to y´1. Fixed points of this actionare y “ 1 and y “ ´1. A fundamental domain for the action of W on T isthus the upper hemisphere of the circle S1 with two 0-cells and one 1-cell.Figure 4.1: The action of W on T .We can now find the isotropy groups Gα and their representation rings.There are three types. First, the identity φ´1p1q “ I in PUp2q has isotropythe whole group G with RpGq – Zrx ` x´1s; cf. 2.2.5. Second, considerφ´1p´1q “ r i 00 ´i s “ i r1 00 ´1 s and let r a bc d s P Gφ´1p´1q. Then we require«a bc dff«1 00 ´1ff“«1 00 ´1ff«a bc dffi.e.«a ´bc ´dff“«a b´c ´dff.49Solving this in PUp2q gives“a bc d‰““a 00 b‰or“0 bd 0‰. This means Gφ´1p´1q –T ¸ C2. By 3.1.2, 3.1.1, 2.2.3,RpT ¸ C2q – Zrx` x´1, vs{`v2 ´ 1, pv ´ 1qpx` x´1q˘where dimpx ` x´1q “ 2 and dim v “ 1. Third, for any value other thany “ ˘1, φ´1pyq is fixed only by T and RpT q – Zrx˘1s.It also follows from this W -CW structure that there is no cohomologyHpGpG;Rq for p ą 1 so that the spectral sequence has no nontrivial differen-tials and collapses at the E2-page. We getEp,q8 – Ep,q2 “$&%0 if q is odd;HpGpG;Rq if q is evenñ Kp`qG pGq.Then0 “ E1,´18 – K0G,1pGq{K0G,2pGq “ K0G,1pGqimpliesH0GpG;Rq – E0,08 – K0GpGq{K0G,1pGq “ K0GpGqand0 “ E0,18 – K1GpGq{K1G,1pGqmeansH1GpG;Rq – E1,08 – K1G,1pGq – K1GpGq.This can be summarized in the exact sequence0 Ñ K0GpGq Ñ C0GpG;RqδÑ C1GpG;Rq Ñ K1GpGq Ñ 0where C˚GpG;Rq is the Bredon cochain complex with differential δ. By ourcomputations thus far this yields0 Ñ K0GpGq Ñ RpT ¸ C2q ‘RpGqδÑ RpT q Ñ K1GpGq Ñ 050with δ “ i˚2 ´ i˚1 induced by the inclusions i1 : T ãÑ T ¸ψ C2, i2 : T ãÑ G; cf.A.0.3.2.We can now compute the cohomology. LetA “ÿaipx` x´1qi ` cv P RpT ¸ C2qand B “řbipx` x´1qi P RpGq and suppose that δpA,Bq “ 0, that is,ÿpai ´ biqpx` x´1qi ` c “ 0.Then ai “ bi for all i ‰ 0 and a0 ´ b0 ` c “ 0 so thatpA,Bq “˜ÿiaipx` x´1qi ` cv,ÿi‰0bipx` x´1qi ` b0¸“˜ÿiaipx` x´1qi ` cv,ÿiaipx` x´1qi ` c¸“ÿiaipx` x´1, x` x´1qi ` cpv, 1qso thatK0GpGq – ker δ – RpT ¸ C2q.Finally, the image of δ is clearly Zrx ` x´1s – RpGq. So that, as anRpGq-module,K1GpGq – RpT q{RpGq.Proposition 4.1.1. K0GpGq is not a free RpGq-module.Proof. We haveK0GpGq – RpT ¸ C2q – Zrx` x´1, vs{`v2 ´ 1, pv ´ 1qpx` x´1q˘with RpGq “ Zrx ` x´1s acting on K0GpGq in the obvious way. Clearly,RpT ¸ C2q fl RpGq. So suppose by contradiction that there is a basis E51of RpT ¸ C2q as an RpGq-module with two different elements α ‰ β P E.Then px ` x´1q ¨ α “ f and px ` x´1q ¨ β “ g for some f, g P RpGq sincemultiplication by x` x´1 kills all possible occurrences of v. But theng ¨ px` x´1q ¨ α ´ f ¨ px` x´1q ¨ β “ 0.A contradiction.Proposition 4.1.2. K1GpGq is a free RpGq-module of rank 1.Proof. We have K1GpGq “ RpT q{RpGq “ Zrx, x´1s{Zrx ` x´1s and claimthatZrx, x´1s – Zrx` x´1st1, xu.To see this, note that px` x´1qx “ x2` 1 so that x2 “ px` x´1qx´ 1. Nowargue inductively to see that all powers of x are contained in Zrx`x´1st1, xu.Likewise, x´1 “ px ` x´1q ´ x and so again by induction or by swapping xand x´1 in the expressions for xn we see that all powers of x´1 are containedin Zrx` x´1st1, xu. The claim follows which finishes the proof.Another way to establish this last fact is to observe that K1PUp2qpPUp2qq –K1SUp2qpSUp2qq since RpTPUp2qq{RpPUp2qq – RpTSUp2qq{RpSUp2qq. In sum-mary,K˚GpGq – RpT ¸ C2q ‘RpGqIn particular, we have shown that K˚GpGq is not a free RpGq-module ofrank 2 as would have been the case had G had torsion-free fundamentalgroup; cf. The projective unitary group of degree 3Inspired by the results of the last section, we attempt to compute in a similarfashion K˚GpGq for G :“ PUp3q – SUp3q{C3 with C3 generated by a primitive52third root of unity ζ. LetT “$’&’%»—–x1x2x3fiffifl : x1x2x3 “ 1, xi P S1,/./-( a maximal torus of G with square brackets denoting the image of a matrixunder pi : SUp3q Ñ PUp3q; cf. Also let W – S3 be the Weyl groupof G.Once again we find a skeletal filtration of T rather than G. To do so, usethe explicit homeomorphism of T with S1 ˆ S1 given in additive notationxi “ e2piiti , ti P R{Z, byφ : T Ñ S1 ˆ S1rdiagpt1, t2, t3qs ÞÑ pt2 ´ t1, 3t1q “: py1, y2qLet τ “”0 1 01 0 00 0 1ıand σ “”0 0 11 0 00 1 0ıgenerateW – S3 “ C3 ¸ C2 “ xσ | σ3 “ 1y ¸ xτ | τ 2 “ 1y.Then the induced action of W on S1 ˆ S1 isτpy1, y2q “ p´y1, 3y1 ` y2qσpy1, y2q “ py1 ` y2,´3y1 ´ 2y2q.A fundamental domain for this action is described by a triangle formed bythe three line segmentstp0, y2qu, tpy1, 0q : ´1{3 ď y1 ď 0u, tpy1,´3y1q : ´1{3 ď y1 ď 0uwhere the last two have been identified via σ; cf. top left square of figure4.2.53abb0-1y1y2 a bbaaabbabbbbaa bb1 (132)(123)(13)-⅓ -½(23) (23)(123)(132)(23) (12)(23)(23)Figure 4.2: Translates of the fundamental domain of T {W (top left square).The W -CW complex T thus has two 0-cells, two 1-cells, and one 2-cell. Because the fundamental domain is only 2-dimensional, there are nodifferentials on the E3-page and the spectral sequence 4.0.8 collapses. Itfollows thatK0GpGq{H2GpG;Rq – H0GpG;Rq and K1GpGq – H1GpG;Rq. ( we need to identify the isotropy groups Gα and their representationrings. This time there are four types. First, there is the 0-cell, p´1{3, 0q PS1 ˆ S1. Let”z1 z2 z3z4 z5 z6z7 z8 z9ıP Gφ´1p´1{3,0q where φ´1p´1{3, 0q “” ζ1ζ2ı. Solving»—–z1 z2 z3z4 z5 z6z7 z8 z9fiffifl»—–ζ1ζ2fiffifl “»—–ζ1ζ2fiffifl»—–z1 z2 z3z4 z5 z6z7 z8 z9fiffifl54in G gives»—–z1 z2 z3z4 z5 z6z7 z8 z9fiffifl “»—–z1 0 00 z5 00 0 z9fiffifl ,»—–0 z2 00 0 z6z7 0 0fiffifl , or»—–0 0 z3z4 0 00 z8 0fiffifl .This means that Gφ´1p´1{3,0q is isomorphic to the semidirect productT ¸ C3 Ă T ¸W.Since p “ 3 is of trivial class number, it follows by 3.1.2 and 3.1.1 that therepresentation ring of T ¸ C3 is given by 2.1.7 with M “ pT – IC3.Second, there is the vertex at the origin corresponding to the identityφ´1p0, 0q “ I in G. It has isotropy all of G. Because G is compact andconnected, we can express RpGq as RpT qW .Third, the edge labeled a in figure 4.2 has isotropy groupGφ´1paq “$’&’%»—–z1 z2 0z4 z5 00 0 z9fiffifl : z9 ¨ det˜z1 z2z4 z5¸“ 1,˜z1 z2z4 z5¸P Up2q,/./-which is compact and connected and has Weyl group xτ : τ 2 “ 1y “ C2 Ă Wso thatRpGφ´1paqq – RpT qC2 . (˜z1 z2z4 z5¸“ U˜eiθ1 00 eiθ2¸U´155with U unitary, the connectedness can be observed by defining a pathpptq “«U1ff»—–eip1´tqθ1 0 00 eip1´tqθ2 00 0 z9eitθ1eitθ2fiffifl«U´11ffwithin Gφ´1paq connecting any element to the identity.Lastly, the edge labeled b in figure 4.2 as well as the 2-cell have isotropythe torus T . The Bredon cochain complex thus becomes0 Ñ RpT ¸ C3q ‘RpGqδ1Ñ RpT qC2 ‘RpT q δ2Ñ RpT q Ñ 0withδ1pA,Bq “ p0, i˚1A´ i˚2Bq and δ2pA,Bq “ i˚3A´ σ˚B `Binduced by the inclusions i1 : T ãÑ T ¸C3, i2 : T ãÑ G, i3 : T ãÑ Gφ´1paq; cf.A.0.3.2.Proposition 4.2.1. H0GpG;Rq is not a free RpGq-module.Proof. By the proof of 2.1.7, we know that i˚1 surjects onto RpT qC3 . Byidentifying RpGq with RpT qW Ă RpT qC3 , we can interpret the kernel of δ1 tobe those elements of RpT ¸ C3q which map to RpT qW under i˚1 . The RpGq-module structure on RpT ¸ C3q—and hence also on H0GpG;Rq—is given bywriting a W -invariant f P RpGq – RpT qW as a linear combination of C3-orbit sumsřaixmi and identifying xmi with ind xmi P RpT ¸ C3q whenmi ‰ 0; cf. 2.1.7. Let f P RpGq be any linear combination of C3-orbit sumswithout constant term. Then by 2.1.7, fpv´1q “ 0 where v´1 P H0GpG;Rqsince i˚1pv ´ 1q “ 0. That is, H0GpG;Rq has RpGq-torsion and cannot befree.By, K0GpGq{H2pG;Rq – H0pG;Rq, that is, there is an extension56of RpGq-modules0 Ñ H2pG;Rq ãÑ K0GpGqpi H0pG;Rq Ñ 0.Since pi is surjective, there is a virtual bundle E :“ rLs ´ r1s P K0GpGq suchthat pipEq “ v ´ 1. Note thatpipf ¨ Eq “ fpipEq “ fpv ´ 1q “ 0so that f ¨E P H2pG;Rq – RpT q{ im δ2. Unfortunately, I do not know at thispoint how to prove that f ¨ E “ 0, i.e., that E is an RpGq-torsion elementof K0GpGq which would show that K0GpGq is not a free RpGq-module. In thetorsion-free fundamental group case one can useTheorem 4.2.2 ([Pit72]). Let H be a compact connected Lie group withtorsion-free fundamental group and T a maximal torus. Then RpT q is a freeRpHq-module of rank the order of the Weyl group W .More still, there exists an explicit algorithm by which to write down anRpHq-basis known as Steinberg basis for RpT qĂW for ĂW any reflection sub-group of W with T and W the torus respectively Weyl group of a compactconnected Lie group H with torsion-free fundamental group; cf. [Ste75].This enables one to compute Bredon cohomology groups as free RpHq-modules. It is an open problem what happens when there is torsion inthe fundamental group.What is known concerning G :“ PUppq with p a fixed prime is that ithas a 0-cell »———————–1ωω2. . .ωp´1fiffiffiffiffiffiffiffiflwith isotropy isomorphic to T ¸Cp with Cp Ă W – Sp. The representation57ring of T ¸ Cp is known by 3.1.2, at least as long as p is of trivial classnumber. By using algorithms computing Weyl invariants, it is also possibleto compute RKpGαq – RKpT qWα for Wα Ă W the Weyl group of a connectedisotropy group Gα and K a splitting field for Wα; cf. 2.2.6. For instance,using the coordinates y1, y2 on the torus of PUp3q above and since W is areflection group we could run the subduction algorithm to obtain a finiteSAGBI basis for RKpGq – RKpT qW with K a splitting field for S3; cf. 5Further DirectionsAt the end of the last chapter we discussed the possibility of comput-ing K˚PUp3qpPUp3qq explicitly. Even if that fails, it would be tempting toshow that the equivariant K-theory of PUppq for p any prime is not a freeRpPUppqq-module. It is not known, however, whether the spectral sequence4.0.8 for PUppq collapses when p ą 3: an area for future research. Exam-ples of other compact connected Lie groups with fundamental groups havingtorsion and thus lending themselves to the kind of investigations pursued inthat chapter arePSOp2p` 1q – POp2p` 1q – SOp2p` 1q, SOp2pq, PSpppqall with fundamental group C2, andPSOp4pq, PSOp4p` 2qwith fundamental group C2 ‘ C2 respectively C4. For these, an entirelydifferent approach may be necessary.Another area for further research concerns the cohomology of the classi-fying space of semidirect products T ¸Cp. By using the Leray-Serre-Atiyah-Hirzebruch spectral sequence, 3.3.4, it may be possible to make deductions.59Note, however, that this spectral sequence is usually more effective in goingfrom cohomology to K-theory and not the other way around. For instance,as mentioned in the introduction this is how the K-theory of the classifyingspace of the crystallographic groups Zn ¸ Cp was computed. Perhaps thereare also more direct ways to attack the cohomology of BpT ¸ Cpq.More work can also be done in the direction of obtaining RpT ¸ Cpq—and hence the K-theory with integer coefficients instead of Cp coefficients—when p is not of class number one. The obstacle here seems to be the classnumber quickly becoming very large. For instance, the class number of thecyclotomic field Qpζq with ζ a primitive 41st root of unity is already 121. Itis unknown whether all these different ideal classes also give rise to differentrepresentation rings. To begin with, it would be interesting to exhibit anideal class giving rise to a ZCp-lattice not isomorphic to one of the standardones.Likewise, it may be possible to use techniques such as Galois descentmentioned before 2.2.4 or an algorithm that does not require a splittingfield K to obtain the integral representation ring RpT ¸ Cpq instead of theweaker RpT ¸Cpq bK. That is, is there a way to calculate ZrM sCp with Ma ZCp-lattice without needing to pass to a splitting field of Cp?Lastly, it would be interesting to study the representation theory of T¸Gwhen G is any other finite group acting through smooth automorphisms on arank n torus T . This has been started for G “ D2p and G “ Sn in section 2.3.In particular, we have seen that the multiplicative invariant algebra ZrM sGwith M a ZG-lattice appears via Cartan subgroups. If G is a reflectiongroup, we can find a finite basis for this invariant ring tensored by a splittingfield by the subduction algorithm. 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The references are [May96], [Bre67], and [Lu¨c89].Let G be a topological group, that is, a group which is also a topologicalspace in which the group operations are continuous maps. Requiring thatG also be a smooth manifold with smooth group operations gives a Liegroup. The objects of study in equivariant algebraic topology are G-spacesand G-maps1, that is, topological spaces X equipped with a continuousaction G ˆ X Ñ X by G and maps f : X Ñ Y between them such thatfpgxq “ gfpxq. For x P X,Gx “ tg P G : gx “ xu (A.0.2.1)is called the isotropy group of x andGpxq “ tgx : g P Gu (A.0.2.2)is called the G-orbit of x.A G-CW complex is a G-space X with a decomposition X “ colimXk1Also known as G-equivariant maps.68such thatX0 “žHG{H, Xn`1 “ Xn Yϕn˜žHDn`1 ˆG{H¸(A.0.2.3)where Dn`1 ˆ G{H are called G-cells and ϕn is made up of attaching G-maps ϕn,H : Sn ˆ G{H Ñ Xn. For compact Lie groups, the situation isparticularly nice:Theorem A.0.3 ([Mat71, 4.4]). Let G be a compact Lie group. Then anycompact smooth G-manifold has a finite G-CW complex structure inducinga triangulation on the orbit space.There are several generalized equivariant cohomology theories, equivari-ant K-theory reviewed in the next appendix being one of them. Anotherprominent one which we will make use of is Bredon cohomology defined next.Let OG be the orbit category of G, that is, the category with an objectG{H for every subgroup H Ă G and morphisms the G-maps G{H Ñ G{Kbetween them. Let R be a commutative ring. A coefficient system in thecategory of R-modules is defined to be a contravariant functor from OGto the category of R-modules. Let X be a G-CW complex. We have acoefficient systemCnpXqpG{Hq “ Hn`pXnqH , pXn´1qH˘where the right term is ordinary homology. The connecting homomorphismsof the triples`pXnqH , pXn´1qH , pXn´2qHq˘specify a mapd : CnpXq Ñ Cn´1pXqof coefficient systems such that d2 “ 0. Given another coefficient system M ,define a cochain complex pC˚GpX;Mq, δq, byCnGpX;Mq “ HomOGpCnpXq,Mq and δ “ HomOGpd, idq69where HomOGpM1,Mq denotes the abelian group of morphisms, i.e., naturaltransformations of coefficient systems M 1 Ñ M . The Bredon cohomologyof X with coefficients in M , H˚GpX;Mq, is defined to be the cohomology ofthis cochain complex.Explicitly, there is an isomorphismCnGpX;Mq –àrαsnMpG{Gαq (A.0.3.1)where rαsn are the n-dimensional G-cells of X and α without brackets is arepresentative. With this description, if x P CnGpX;Mq and rβsn`1 are thepn` 1q-cells, thenδpXqβ “ÿrαsnrα : βsMpα, βqpxαq PMpG{Gβq (A.0.3.2)where rα, βs is the degree of the map induced by the attaching map of thecell β, cf. A.0.2.3 and [AGP02, 7.3.6]. If rα, βs ‰ 0, there is a G-map iα,β :G{Gα Ñ G{Gβ and Mpα, βq is defined as Mpiα,βq : MpG{Gαq ÑMpG{Gβq.70Appendix BEquivariant K-theoryIn this appendix, we review the basics of representation theory of compactLie groups and K-theory of G-spaces. The references are [AS69], [Seg68a],and [Seg68b].A representation of a compact Lie group G on a finite-dimensional com-plex vector space V is a continuous homomorphism GÑ AutpV q. The char-acter of a representation V of G is the smooth function χV : G Ñ C givenby g ÞÑ Trplgq where Trplgq is the trace of the linear map lg : V Ñ V, v ÞÑ gv.The character ring RpGq is the free abelian group generated by the irre-ducible characters of G with ring structure given by χV ¨ χW “ χVbW . By[Seg68b, 3.3], RpGq is a finitely generated ring. It can also be constructedas the Grothendieck ring of the symmetric monoidal1 abelian category offinite-dimensional representations of G (under ‘ and b).A G-vector bundle over a G-space X is a G-space E with a G-mapp : E Ñ X such that(i) p : E Ñ X is an ordinary complex vector bundle;(ii) for each g P G and x P X the map g : Ex Ñ Egx is a vector spacehomomorphism.1A symmetric monoidal category is a category with a product operation for which theproduct is “as commutative as possible”; cf. [MP12, 16.3].71If X is a compact G-space, G-equivariant K-theory K˚GpXq is definedas the Grothendieck ring of the symmetric monoidal abelian category of G-vector bundles over X under direct sum and tensor product. The naturalmap X Ñ ˚ induces a homomorphism K˚Gp˚q Ñ K˚GpXq which turns K˚GpXqinto a K˚Gp˚q-algebra. Recall that K˚Gp˚q – RpGq since G-vector bundlesover a point are representations of G.Let IG “ ker pdim : RpGq Ñ Zq be the augmentation ideal in RpGq andlet BG be the classifying space of G, that is, the quotient of a weaklycontractible space EG by a free action of G. The space EG is constructedas the infinite join of G with itself; cf., e.g., [Ros94, 5.1.15]. Denote byX ˆG EG :“ pX ˆ EGq{Gthe quotient by the diagonal action also known as the Borel construction.Theorem B.0.4 (Completion Theorem). Let G be a compact Lie group andlet X be a finite G-CW complex. Then the map pi : X ˆ EG Ñ X inducesan isomorphism of pro-algebras2pi˚ : K˚GpXqxIG Ñ K˚pX ˆG EGqwhere the term on the right is defined as limÐÝK˚GpX ˆ EGnq for a suitablefiltration of EG. In particular, if X “ ˚ one obtainsRpGqxIG – K˚pBGq,the ordinary complex K-theory of BG.2A pro-object of a category C is a cofiltered limit of objects of C. For instance, apro-(finite group) is an inverse limit of an inverse system of finite groups.72Appendix CK-theory with coefficientsWhile the reader may be versed in ordinary cohomology with coefficients,this may not be so for other generalized cohomology theories such as K-theory used in this thesis. Following [Ada74, 200ff.] we recall the maindefinitions and theorems.Given an abelian group A, construct a free resolution0 Ñ R iÑ F piÑ AÑ 0.Let S “ tΣnS0u denote the sphere suspension spectrum and take wedgesums such thatpi0˜łaPAS¸“ R and pi0˜łbPBS¸“ F.Also take a map f :ŽaPA S ÑŽbPB S inducing i. The homotopy cofiber(mapping cone) of fMA “˜łbPBS¸Yf C˜łaPAS¸73is then a Moore spectrum of type A, that is,pirpMAq “ 0 for r ă 0,pi0pMAq “ H0pMAq “ A for r “ 0,HrpMAq “ 0 for r ą 0.Finally, given a spectrum E define the corresponding spectrum with coeffi-cients in A to be the smash product E ^MA. Thus, for instance,K˚p´;Aq :“ r´, BU^MAsdefines complex K-theory with coefficients in A. K-theory with coefficientsis related to the usual K-theory byTheorem C.0.5 (Universal coefficient theorem for generalized cohomol-ogy). Let h˚ be a generalized cohomology theory. Then there exists an exactsequence0 Ñ hnpXq b AÑ hnpX;Aq Ñ TorZ1 phn`1pXq, Aq Ñ 0.It follows thatK˚p´;Aq – K˚p´q b Afor all torsion-free abelian groups A. More still, since by B.0.4 K1pBGq “ 0for any compact Lie group G we haveK˚pBG;Aq – K˚pBGq b Afor any abelian group A.74


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