Communications in Algebra, 33: 3667 3677, 2005 Copyright Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870500243312 MODULAR SPECIES AND PRIME IDEALS FOR THE RING OF MONOMIAL REPRESENTATIONS OF A FINITE GROUP # Bosco Fotsing and Burkhard Külshammer Mathematisches Institut, Friedrich-Schiller-Universität, Jena, Germany The ring of monomial representations of a finite group has been investigated by Dress (1971) and Boltje (1990), among others. It is of interest in connection with induction theorems in representation theory. Its species have recently been determined by Boltje. In this article, we will analyze the block distribution of species. As an application, we will determine the prime ideals of the ring of monomial representations. The results here constitute a slightly modified version of part of the first author s Diplomarbeit (Fotsing, 2003), written under the direction of the second author. Key Words: Character ring; Monomial representation; Representation ring; Species. Subject Classification: 19A22; 20C15. Let G be a finite group. In this article, we are concerned with D G, the ring of monomial representations of G. This ring has proved to be useful in connection with induction theorems in representation theory (cf. Boltje, 1990, to appear; Dress, 1971; Snaith, 1994). It is defined as the Grothendieck ring of the monomial category mon CG. So we start by recalling the definition of and some facts about mon CG. The objects of mon CG are pairs V L, where V is a finitely generated CG-module and L is a set of one-dimensional subspaces (called lines) ofv, such that V = L L L and gl L for g G and L L. A morphism f V L W M in mon CG is a homomorphism of CG-modules f V W such that, for L L, there exists M M with f L M. Composition in mon CG is defined in the obvious way. In contrast to mod CG, the category of finitely generated CG-modules, the category mon CG is not additive, in general. (In Boltje, 2001, a different (i.e., nonequivalent) definition of the morphisms in mon CG was given. However, this change does not affect the results below.) For objects V L and W M in mon CG, there are a direct sum V L W M = V W L M Received April 27, 2004; Accepted November 20, 2004 # Communicated by A. Turull. Address correspondence to Prof. Dr. Burkhard Külshammer, Mathematisches Institut, Friedrich Schiller-Universität, 07740 Jena, Germany; Fax: +49-3641-946162; E-mail: kuelshammer@ uni-jena.de 3667
3668 FOTSING AND KÜLSHAMMER and a tensor product V L W M = V C W L M L L M M in mon CG, with the usual properties. Moreover, for a subgroup H of G, one has restriction and induction functors Res G H mon CG mon CH Ind G H mon CH mon CG given as follows: for V L mon CG, one has Res G H V L = ResG H V L, where Res G H V denotes the CH-module obtained by restriction from CG. For W M mon CH, one has Ind G H W M = IndG H W g M g G M M where Ind G H W = CG CH W denotes the CG-module obtained by induction from CH. The effect of the functors Res G H and IndG H on morphisms is the obvious one. An object V L in mon CG with V 0 is called indecomposable if V L V 1 L 1 V 2 L 2 in mon CG implies that V 1 = 0 or V 2 = 0. Thus V L is indecomposable if and only if the action of G on L is transitive. Every group homomorphism G C defines an indecomposable object C C in mon CG, where C denotes the CG-module C, on which G acts via g = g g G C More generally, the objects Ind G H C C, where H is a subgroup of G and H C is a group homomorphism, are indecomposable in mon CG. Moreover, every indecomposable object of mon CG is isomorphic to one of this form. For subgroups H, K of G and group homomorphisms H C, K C, one has Ind G H C C Ind G K C C in mon CG if and only if K = x H and = x for some x G; here x H = xhx 1, x h = xhx 1 and x x h = h for h H. In this way one is led to consider M G = H H G Hom H C the set of monomial pairs of G. The group G acts on M G via the conjugation x H = x H x x G H M G For H M G, we denote the stabilizer of H in G by N G H = x G x H = H so that H N G H N G H, and we denote the orbit of H by H G. Also we denote the set of orbits by M G /G = H G H M G
MONOMIAL REPRESENTATIONS 3669 Then, as we saw above, the isomorphism classes V L G of indecomposable objects V L in mon CG are in bijection with M G /G. The bijection maps H G M G /G to Ind G H C C G. The Grothendieck ring D G of the category mon CG is defined in the usual way, with and V L G + W M G = V L W M G V L G W M G = V L W M G for V L W M mon CG. The isomorphism classes of indecomposable objects in mon CG form a Z-basis for the additive group of D G ; in particular, the additive group of D G is free abelian of rank M G /G. If we identify an element H G M G /G with the isomorphism class Ind G H C C G in mon CG then we can also view M G /G as a Z-basis of the additive group of D G. In this notation, the product on D G is given by H G K G = HgK H\G/K H g K g G where g H g K C is defined by g x = x g 1 xg for x H g K. For any subgroup H of G, the functors Res G H mon CG mon CH and Ind G H mon CH mon CG give rise to group homomorphisms res G H D G D H V ind G H D H D G W L G Res G H V L H M H Ind G H W M G These group homomorphisms have the usual properties. In particular, res G H is even a ring homomorphism, and the image of ind G H is an ideal in D G. Moreover, in the notation introduced above, we have ind G H K H = K G K M H res G H K G = H g K g H g K H K M G HgK H\G/K In the following, we will denote the character ring of G by R G. There is a ring homomorphism G D G R G/G defined by G H G = 0ifH<G and G H G = if H = G; here G denotes the commutator group of G, and G/G C is the group homomorphism given by gg = g for g G. For g G, there is a ring homomorphism t g R G C defined by t g = g for R G. The species of D G, i.e. the ring homomorphisms D G C, have recently been determined by Boltje (to appear). In order to construct them, it is useful to
3670 FOTSING AND KÜLSHAMMER introduce the set G = H hh H G h H Then G acts on G via the conjugation g H hh = g H g h g H g G H hh G For H hh G, we denote its orbit by H hh G and its stabilizer by Then and N G H hh = g G g H hh = H hh H H N G H hh N G H N G H N G H hh /H = C NG H /H hh In the following, we will denote the set of orbits of G on G by G /G = H hh G H hh G Each H hh G defines a species of D G as a composition of the following maps: s H hh D G resg H D H H R H/H t hh C Boltje (to appear) has shown that every species of D G arises in this way. Moreover, one has s H hh = s K kk if and only if K kk = g H hh for some g G. This means that the species of D G are in bijection with G /G. We note that the species of D G all take their values in Z, where is a primitive G -th root of unity in C. We fix a maximal ideal P of Z. Then Z /P is a finite field, and we denote its characteristic by p. We call the species s H hh and s K kk of D G P-equivalent if s H hh x s K kk x mod P for all x D G. In this case, we also call the pairs H hh K kk G P-equivalent and write H hh P K kk. Our aim is to determine the P-equivalence classes of G. In order to state our first (easy and well-known) lemma, we recall that every element g G can be written uniquely in the form g = g p g p = g p g p, where g p is a p-element and g p is a p -element in G. Then g p is called the p-factor, and g p is called the p -factor of g. Lemma 1. Let H hh G. Then H hh P H h p H.
MONOMIAL REPRESENTATIONS 3671 Proof. By the definition of s H hh and s H hp H, it suffices to show that t hh t hp H mod P for Hom H/H C. But t hh t hp H = hh h p H = h p H 1 h p H and h p H + P is a root of unity of p-power order in the field Z /P of characteristic p, so h p H + P = 1 + P, i.e. h p H 1 P, and the result follows. Lemma 1 gives an easy method to construct P-equivalent species. The following result gives another such method. Lemma 2. Let H hh G, and let K/H be a p-subgroup of N G H hh /H. Then H hh P K hk. Proof. By the definition of s H hh and s K hk, we may assume that G = K and H = 1. Then H is a normal subgroup of G, G/H is a p-group, and h Z G. Let L M G. We need to show that s G hg L G s H hh L G mod P If L = G, then s G hg L G = h = s H hh L G. So we may assume that L<G. Then s G hg L G = 0. If H L, then s H hh L G = 0 since H res G H L G = HgL H\G/L H H g L g H g L H = 0 It remains to consider the case H L<G. But then s H hh L G = ghg 1 = G L h 0 gl G/L mod P and the result is proven. We are going to show that a combination of the two methods given by Lemmas 1 and 2 determines the P-equivalence classes of G. In order to explain this, let H hh G. Then H hh P H h p H by Lemma 1. Next, let H 1 /H be a Sylow p-subgroup of N G H h p H /H. Then H h p H P H 1 h p H 1 by Lemma 2. Similarly, let H 2 /H 1 be a Sylow p-subgroup of N G H 1 h p H 1 /H 1. Then H 1 h p H 1 P H 2 h p H 2 by Lemma 2. We continue in this way until we reach a pair H n h p H n = H n+1 h p H n+1 =. Setting p G = K kk G k 0 N G K kk K mod p we have H n h p H n p G. We call the elements in p G p-regular and the pair H n h p H n a p-regularization of H hh. Note that O p H n H, that H n h p H n is uniquely determined by H hh, up to conjugation, and that H hh is
3672 FOTSING AND KÜLSHAMMER P-equivalent to each of its p-regularizations. Our next result shows that two pairs in p G are P-equivalent if and only if they are conjugate. Proposition 3. Let H hh K kk p G, and suppose that H hh P K kk. Then K kk = g H hh for some g G. Proof. We write H/H = A/H B/H with a p -group A/H and a p-group B/H. Moreover, we denote by 1 r the group homomorphisms H C containing B in their kernel, so that r = H B 0 mod p. Furthermore, we set r y = i h 1 s H hh H i G r z = i h 1 s K kk H i G Z so that y z mod P by our hypothesis. For i = 1 r, we have res G H H i G = HgH H\G/H H g H g i H g H H so H res G H H i G = gh N G H /H g i where g i H/H C is defined by g i xh = i g 1 xg for x H. Thus s H hh H i G = gh N G H /H i g 1 hg and y = gh N G H /H r i h 1 i g 1 hg. By the orthogonality relations for H/B, we have r i h 1 i g 1 hg = 0 unless g 1 hgb = hb. But h A since H hh p G, so g 1 hgb = hb is equivalent to g 1 hgh = hh, and in this case we have r i h 1 i g 1 hg = r. We conclude that y = N G H hh H r 0 mod P. This implies that 0 s H hh H i G s K kk H i G mod P for some i 1 r. In particular, we have 0 s K kk H i G and 0 K res G K H i G = KgH K\G/H K K g H g i K g H K This means that K = K g H g H for some g G. By symmetry, we have H g K for some g G. Thus H and K are conjugate in G. We may therefore assume that H = K, and it remains to show that hh and kh are conjugate in N G H /H.We assume that this is not the case. Then hb and kb are not conjugate in N G H /B, and r 0 y z i h 1 i g 1 kg gh N G H /H mod P
MONOMIAL REPRESENTATIONS 3673 by a computation similar to the one above. By the orthogonality relations for H/B, we have r i h 1 i g 1 kg = 0 for gh N G H /H, soz = 0. This is a contradiction, so the result follows. This leads us to the main result of this article. Theorem 4. Let be a primitive G th root of unity in C, let P be a maximal ideal of Z, and let p denote the characteristic of the field Z /P. Then each P-equivalence class of G contains a unique conjugacy class of p G. This way, the P-equivalence classes of species of D G are in bijection with p G /G = H hh G H hh p G As an application, we will determine the prime spectrum Spec D G of D G. We begin by investigating the prime spectrum Spec D Z G of D Z G = Z Z D G. For H hh G, the map D Z G Z a x as H hh x a Z x D G is a homomorphism of rings, which we denote by s H hh P Spec Z, again. Then, for H hh P = y D Z G s H hh y P is a prime ideal of D Z G. Moreover, s H hh induces an isomorphism of rings D Z G / H hh P Z /P y + H hh P s H hh y + P in particular, we have We prove char D Z G / H hh P = char Z /P Proposition 5. Every prime ideal of D Z G has the form H hh P for some H hh G and some P Spec Z. Proof. It is known (cf. Boltje, to appear) that the ring homomorphisms s H hh D Z G Z, where H hh runs through G, induce a monomorphism of rings D Z G Z G /G
3674 FOTSING AND KÜLSHAMMER Hence Ker s H hh Ker s H hh = 0 H hh G H hh G Now let be a prime ideal of D Z G. Then contains Ker s H hh for some H hh G. Thus P = s H hh is a prime ideal of Z. We conclude that = y D Z G s H hh y P It remains to determine the fibres of the map G Spec Z Spec D Z G H hh P H hh P We certainly have H hh P = K kk P, whenever H hh K kk G are conjugate in G. Also, we have H hh P = K kk P, whenever H hh K kk G are P-equivalent. We conclude by Corollary 6. (i) Every prime ideal of D Z G of residue characteristic zero has the form H hh 0 for some H hh G. (ii) Every prime ideal of D Z G of residue characteristic p>0 has the form H hh P for some H hh p G and some P Spec Z containing p. We are now in a position to determine the fibres of. Proposition 7. (i) Let H hh K kk G such that H hh 0 = K kk 0. Then H hh and K kk are conjugate in G. (ii) Let P Q Spec Z such that char Z /P = char Z /Q = p>0, and let H hh K kk p G such that H hh P = K kk Q. Then P = Q, and H hh K kk are conjugate in G. Proof. (i) It is easy to see that s H hh H 1 G = N G H H 0. Thus H 1 G H hh 0 = K kk 0, sos K kk H 1 G 0; in particular, we have 0 K res G K H 1 G = KgH K\G/H K K g H 1 K g H K Hence K = K g H g H for some g G. By symmetry, it follows that K and H are conjugate in G. So we may assume that H = K. Let 1 r denote the group homomorphisms H C, and let y = r i h 1 H i G D Z G It is easy to see that s H hh y = N G H hh H 0. Thus y H hh 0 = H kh 0, and s H kh y 0. The orthogonality relations for H/H imply that kh and hh are conjugate in N G H /H, and (i) follows.
MONOMIAL REPRESENTATIONS 3675 (ii) In the proof of Proposition 3, we had found a group homomorphism H C such that s H hh H G 0 mod P. Thus H G H hh P = K kk Q ; in particular, we have s K kk H G 0. As above, it follows easily that K g H for some g G. By symmetry, H and K are conjugate in G. We may assume that H = K. As in the proof of Proposition 3, we write H/H = A/H B/H with a p -group A/H and a p-group B/H. Moreover, we denote the group homomorphisms H C containing B in their kernel by 1 r.weset y = r i h 1 H i G D Z G The proof of Proposition 3 shows that s H hh y = N G H hh B 0 mod P Thus y H hh P = H kh Q, and s H kh y 0 mod Q. As in the proof of Proposition 3, we conclude that kh is conjugate to hh in N G H /H. Finally, we obtain that P = s H hh H hh P = s K kk K kk Q = Q and the result is proven. Next, we indicate how to descend from Spec D Z G to Spec D G, by using some Galois theory. In the following, we denote the Galois group of Q over Q by. There is an isomorphism of groups Z/ G Z k + G Z k such that k = k. Moreover, acts on G (considered just as a set) by k g = g k g G k + G Z Z/ G Z For H hh G, k + G Z Z/ G Z and x D G, we have k s H hh x = k t hh H res G H x = t h k H H res G H x = s H k h H x so s H hh = s H h H D G C for and H hh G. Also, acts on D Z G in such a way that a x = a x a Z x D G It is easy to verify that D Z G = y D Z G y = y for = 1 D G
3676 FOTSING AND KÜLSHAMMER For, H hh G, a Z and x D G, we have s H hh 1 a x = s H hh 1 a x = 1 a s H hh x = a s H hh x = as H h H x = s H h H a x We conclude that s H hh 1 = s H h H D Z G C for and H hh G. The action of on D Z G induces an action of on Spec D Z G. It is easy to check that H hh P = H h H P for, H hh G and P Spec Z. In the following, we will regard D G as a subring of D Z G via the monomorphism of rings D G D Z G x 1 x Let us consider the map Spec D Z G Spec D G D G Since D Z G is an integral extension of D G, is certainly surjective. Proposition 8. The fibres of the map above coincide with the -orbits on Spec D Z G. Proof. It is clear that = D G = D G = D G = for and Spec D Z G. Conversely, let Spec D Z G such that D G = D G. We assume that. Then is not contained in, sowemay choose an element y \. But now y D Z G = D G = D G so y for some. Thus y 1, a contradiction which proves the result. We are now in a position to determine the prime spectrum of D G. In the following, we set H hh P = x D G s H hh P = H hh P D G
MONOMIAL REPRESENTATIONS 3677 for H hh G and P Spec Z. We summarize our results above in the following theorem. Theorem 9. For H hh G and P Spec Z, H hh P is a prime ideal of D G. Moreover, every prime ideal of D G arises in this way. More precisely, we have: (i) Every prime ideal of D G of residue characteristic zero has the form H hh 0 for some H hh G. If H hh 0 = K kk 0 for some K kk G then there are g G and such that K kk = g H h H. (ii) Every prime ideal of D G of residue characteristic p>0 has the form H hh P for some H hh p G and some P Spec Z containing p. If H hh P = K kk Q for some K kk p G and some Q Spec Z containing p, then there are g G and such that K kk = g H h H and Q = P. ACKNOWLEDGMENT Part of the work on this article was done while the second author was visiting the University of Auckland, New Zealand, with the kind support of the Marsden fund. It is a pleasure to thank the people in the Department of Mathematics for their warm hospitality and stimulating discussions. Also, the authors would like to thank Robert Boltje for a number of useful suggestions. REFERENCES Boltje, R. (1990). A canonical Brauer induction formula. Astérisque 181 182:31 59. Boltje, R. (2001). Monomial resolutions. J. Algebra 246:811 841. Boltje, R. Representation rings of finite groups, their species, and idempotent formulae. To appear in J. Algebra. Dress, A. W. M. (1971). The ring of monomial representations I: structure theory. J. Algebra 18:137 157. Fotsing, B. (2003). Zum Ring der monomialen Darstellungen einer endlichen Gruppe. Diplomarbeit. Jena. Snaith, V. P. (1994). Explicit Brauer Induction. With Applications to Algebra and Number Theory. Cambridge: Cambridge University Press.