modular permutation representations(l) -...

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 175, January 1973

MODULAR PERMUTATION REPRESENTATIONS(l)

BY

L. L. SCOTT

ABSTRACT. A modular theory for permutation representations and their

centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras.

Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on the

fixed points of P. A detailed summary appears in [15]. A main consequence of the theory is simplification of the problem of com-

puting the ordinary character table of a given centralizer ring. Also, some pte- viously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Btauet’s theory of blocks to Green’s work on indecomposable modules.

The purpose of this article is to present proofs of the results announced in

[15]. Statements of these results have been included here, though a number of

explanatory remarks and general background references have not been repeated.

With the exception of $0, the sections of this paper have been named according

to the features of the classical modular theory with which they are most closely

related.

0. The centralizer ring. Throughout this paper G is a finite group acting

on a finite set Q (perhaps not transitively or faithfully) and p is a fixed prime.

If S is any commutative ring with identity, we define the S-centralizer ring

V,(C) = V,(G, In) to be the collection of all matrices with entries from S that

commute with the permutation matrices determined (with respect to some fixed

ordering of Q) by elements of G. In case S is the ring of rational integers, we

write only V(G) for VS(G) and refer to V(G) as the centralizer ring. The

standard basis matrices {Ai]:= are obtained from the full set {O,\:=, of orbits

of G on n x fl by setting the a, p entry of Ai equal to 1 for (a, p) E Oi and

0 otherwise. These matrices always form an S-basis for V,(G). In particular,

V,(G) is isomorphic to the tensor product XV(G).

Received by the editors August 27, 1971. AMS (MO.?) subject classifications (1970). Primary 2OC20; Secondary 16A26, 2OCO5. Key words and phrases. Indecomposable module, vertex, centralizer ring, decom-

position numbers, Brauer homomorphism, Brauer’s first fundamental theorem, Green correspondence, Brauer’s seconds fundamental theorem, defect group, defect 0 and 1, splitting field.

(I) Manuscript preparation supported by NSF-GP-9572.

101

102 L. L. SCOTT [J anuary

The notation SQ refers to the usual S-free SG module obtained from the

action of G on Q; we regard SQ, also as a V,(G) module in the obvious way. (*)

In this paper S will usually be K, R, or F where K is a p-adic number field,

R is the ring of local integers in K, and F is the residue class field R/nR. n- is

a generator of the maximal ideal of R. We use the notation X for the image of x

under some (hopefully obvious) map X --t X/rrX of an R-module X containing x.

_- (0.1) VR(G) E VF(G).

Proof. This is an immediate consequence of the isomorphism mentioned in

the first paragraph.

(0.2) Proposition 1. Let M,‘N be RG-indecomposable components of Rhz.

Then ----- --

(a) Hom,,(M, N) Z HomFG(M, N). - (b) M is indecomposable and has the same vertex as M.

-- Proof. Of course the natural map Hom,,(M, N) + Hom,,(M, N) is a mono-

morphism. Since the functor Horn is additive and VR(G) E VF(G), the map must

be an isomorphism.

Now Hom,,(M, M) is a completely primary ring, as is well known. Obviously, --

the isomorphism of_ HzmRG(M, M) with HomFG(M, M) is a ring homomorphism;

therefore HomFG (M, M) contains only one idempotent-that is, ii is indecompos-

able.

If P is a subgroup of G, then we deduce easily from the above ring isomor-

phism and D. Higman’s criterion [lo, Theorem II that M is P-projective if and

only if % is P-projective. So by definition (see Green 17, 1.21) M has vertex P - if and only if M has vertex I’.

Because of Proposition 1, many results we obtain for V,(G) contain implicit

analogues for VR(G).

1. Decomposition numbers. Let A be an R algebra (finitely generated as

an R module) and M an R-free A module such that KM is completely reducible.

Let B be an R algebra acting faithfully on M and inducing End,(M) by its

action.

We are interested in the case A = RG, M = Rfi, B = V,(G, a). However,

the results in this section are purely formal and are accordingly given a more

general treatment.

The following result is well known. The notation X 1 Y means X is

isomorphic to a direct summand of Y.

(2) All c(modules” are, by convention, finitely generated and acted upon on the

tight, with the exception that the action of a base ring such as S is written on the left.

19731 MODULAR PERMUTATION REPRESENTATIONS 103

_. (1.1) Let e, / be idempotents in B. Then the following statements are equiv-

alent:

.’ (a) e = xfy for some x, y e B.

(b) Me 1 Ml (considered as A modules).

(c) eB 1 fB (considered as B modules).

Accordingly we say that e and f are equivalent (e - f) if e = xfy and f =

.zezu for some x, y, z, w E B. When e, f are primitive then e N f whenever one of

the conditions (a), (b), (c) is satisfied.

Of course the preceding discussion is valid also for KA, KM, KB (and A, -- M, B if B induces End% 6)).

By using equivalence classes of primitive idempotents as intermediaries we

establish l-l correspondences [Mi] ++ [ujl and [xsI +-+ [Zs] between isomorphism

classes of A-indecomposable components of M and projective indecomposable B

modules, and between isomorphism classes of irreducible KA submodules of KM

and irreducible KB modules.

Also there is a well-known l-l correspondence [uj] +P [Lj] between isomor- -

phism classes of projective indecomposable B modules and irreducible B modules

(Lj is the unique irreducible quotient module of Ej).

In view of these correspondences there are three kinds of “decomposition

numbers ” we can define:

(1) Let Z,” be an R-free B module with KZ: “N Zg. Then we list the com-

position factors of Zg, writing Z,” ++ xj ;iSjLj.

(2) Set KUI % C-s dsjZ . s (3) Set KMj NN x5 ~M.x .

=I s

(1.2) Theorem 1. Assume K, F are splitting fields for KB, i respectively.

Then zsj = dSj = dtj for all s, j. Also M x @ xj(dimF Lj)Mj.

Note that the last assertion is a modular-theoretic version of the familiar

KM “N z;s (dim, Zs)Xs. w

Proof. The equality dSj = dSj is well known [2, IX, 81. Let u E B be a prim-

itive idempotent with Mu NN Mj. Now the following two assertions are easily

verified:

(i) For each nonnegative integer d, dZS 1 uKB if and only if there exists a

set of d orthogonal primitive idempotents in uKBu each equivalent to an idempo-

tent f with fKB “N Zs.

(ii) For each nonnegative integer d, dXS 1 KMu if and only if there exists a

set of d orthogonal primitive idempotents in uKBn each equivalent to an idempo-

tent f with KMf E Xs.

dS j = dzj.

By definition fKB X Zs if and only if KMf “N Xs. Thus


To finish the proof we make two further observations:

(iii) For each nonnegative integer I, ZMj\M if and only if there exists a set

of 1 orthogonal primitive idempotents in B each equivalent to U.

(iv) The multiplicity of u, as a component of B is dimF Lj (see the argu-

ment on p. 419 of [41).

The final assertion of the theorem is an obvious consequence.

The following proposition clarifies the hypothesis of Theorem 1.

(1.3) End,,(Zs), EndE (Lj) are anti-isomorphic to End,, (Xs),

End, (Mi)/Rad (End, (Mj)) respectively,

In particular K is a splitting field for KB if and only if each XS is absolutely

irreducible, and F is a splitting field for B if and only if each Mj is absolutely

indecomposable.

Proof. Let e E B be a primitive idempotent. Then of course eBe X End,(Me).

But also the left multiplications by members of eBe form Ends(eB). Hence

End, (Me) is anti-isomorphic to End, (eB).

Since eB is projective, End, (eB/e Rad (B)) is a homomorphic image of

End, (eB). Since the latter is completely primary,

EndB(eB)/Rad (EndB(eB)) xEndB(eB/eRad (B)).

This proves the assertion regarding Ends (Lj); the statement regarding

End,, PSI is established by the first paragraph of the same argument.

2. Defect groups. Let Ai be a standard basis matrix. We define “the” defect

group Di of Ai to be a Sylow p-subgroup of GaP where (a, p) E 0;. (Defect groups

are well defined only up to conjugation by elements of G.)

Set AiAj = xk aijkAk (in V(G)). The quantities aijk are the “intersection

numbers” defined by D. Higman [ll]. We have

(2.1) aijk = lOi nOT( where (a, p) E 0,.

Here Oi(a) = !p\ (a, /3) e Oil and 0: = Oj*= I(& a)) (a, p) E Oil. The proof

of (2.1) is a direct matrix calculation from the definition of the Ai’s.

(2.2) rf aijk f o(p) t/J en Dk LG Di and D, & Dj. (The notation pf~c99

means “5 a G-conjugate of.“)

Proof. G ap acts on O$o) n OT(/3) and hence so does (a suitable) Dk. If

lOi n O;(P)\ & O(p) h t en D, must fix a letter y E Oi(o) C-I O;(p). Thus DK

fixes (a, y) E Oi and (y, ,!3) E Oj. Th e result now follows from Sylow’s theorem.

For any p-subgroup P of G we define IF(P) = I,(P; G, fi) to be the set of

F-linear combinations of Ai’s satisfying Di sG P.

Lemma 1. FOT P, Q, p-subgroups of G,

(2.3)


Proof. Immediate from (2.2).

Now, given a primitive idempotent e E V,, there is an index i such that e E

r&D;) and Di SC P whenever e E IF(P). We set D(e) =G Di and call D(e) “the”

defect group of e. (See Green’s proof [7, 3.3b] for a discussion of how Lemma 1

guarantees the legitimacy of our definition of defect group.)

D(e) is also the defect group of any idempotent equivalent to e. This is easily

verified directly, or may be seen as a consequence of the next result.

Proposition 2. D(e) is the vertex of File.

To prove Proposition 2 we need an alternate description of the ideals IF(P).

Define N, .(x), for H 5 G and x E V,(H, a), to be c xg where g ranges over a

set of.right coset representatives of H in G (here xg E VF(Hg, a) is defined in

the obvious way).(3) Clearly N, .(x) E V,(G, a).

(2.4) @‘; G, fi2) = N, .(V,iP, a)).

Proof. For o, p E Cl I& easp denote the Is11 x (al matrix with 1 in the a, p

position and 0 everywhere else. Obviously, e aB E V,(H, n) whenever H 5 Gap.

It is easy to calculate that, for (a, p) E Oi and D; 5 G ap9 Ai = NG D ,(x) where

x = [G ap : Di]- *e Qp. Thus if Di 5 P we have Ai = N,,,(N, .i(x))‘t’

N (-,(V,(P, Cl)). c onsequently I,(P; G, Q) E N, ,(V,(P, n,i. ,

Next let cz E V,(P, Q2) b e a standard basis matrix. As above we calculate a =

NP ,d(“ad h w ere (a, p) belongs to the orbit of P on fi x fi corresponding to a

and d = Pba. Now

N G, ,(a) = NGp dkap) = N,, Gap(NGap, ,(e,p)) = NG, Gap([Gap : dleap).

Thus N, p (u) = 0 if d is not a Sylow subgroup of Gap, and N, p(a) is a multiple

of the stakdard basis matrix in V,(G, fi) corresponding to (a, p’)G if d is a Sylow

subgroup of Gap. In either case N, e ,(a) E I,(P; G, 0) and so N,,,(V,(P, a)) _C

r,(P; G, i-i).

(2.5) Proof of P p ro osition 2. This follows from (2.4) and the definition of D(e)

because of D. Higman’s criterion [lo, Theorem 11. See also Green 19, p. 1431.

(2.6) Remarks. (a) Note that if e,, E V,(G, fl2) is a primitive idempotent with

e = 0

e, then D(e) is also a vertex of Rae0 by Proposition 1.

(b) An alternate proof of Lemma 1 can be given by (2.4) and Green’s formula

[9, 4.111.

If F is a splitting field for V, and e E V, is a primitive idempotent, then

there is a unique irreducible modular character(*) X of V, satisfying h(e) = 1.

-- (3) We follow Feit’s notation [5]. Th

TH,G(x) in Green [9].

e analogous notation in Green [7] is TGzH(z), and

(4) Throughout this paper the term “character” refers to the trace function obtained

from a module.

106 L. L. SCOTT II anuary

This leads to a second characterization of defect groups.

Proposition 3. Assume F is a splitting field for V,. ,Let e, e’ E V, be primi-

tive idempotents, and let X, X” be the associated modular characters. Then

(1) &A;) = 0 /or all x 6 V, unless Di zc D(e).

(2) X(Ai) f 0 for some i with Di =G D(e).

(3) e is equivalent to e’ i[ and only i/ D(e) =G D(e”), and h(Ai) = X”(Ai) /or

all i with Di =G D(e).

Before giving the proof we establish a general fact which will be used again

in $6.

(2‘7) Suppose h is the character afforded by an absolutely irreducible repre-

sentation of a finite dimensional F-algebra A, and e E A is a primitive idempotent

with X(e) f 0. Then the restriction of X to eAe is an algebra homomorphism,

Proof. Let 2 be a matrix representation affording h. Then s(e) is still a

primitive idempotent in S(A), as is well known. Thus by suitably choosing f we

may assume that 63(e) h as a 1 in the upper left-hand corner and O’s everywhere

else. Hence for x E eAe, X(X) is the upper left-hand corner entry of S(X), and all

other entries are 0. The result is now obvious.

(2,8) Proof of Proposition 3. (1) If Di & D(e) then e 4 I,(D+ G, a) by defi-

tion of D(e). Thus fF(Di; G, Q2) n eV, e is contained in the kernel of hlevpe.

So X(exAi) = X(exAie) = 0 for all x E V,. Clearly the same equation holds if e

is replaced by an equivalent idempotent. But an idempotent not equivalent to e

is in the kernel of any representation affording h. Since X(xAi) = h(lxA i) and 1

is a sum of primitive idempotents, the proof of (1) is complete.

(2) By definition e is an F-linear combination of standard basis matrices A,

with Di sG D(e). Since X(e) f 0, and X(AiJ = 0 whenever Di CG D(e) by (l), we

must have h(Ai) f 0 for some i with Di zG D(e).

(3) Suppose D(e) =c D(e’) and h(Ai) = X”(Ai) for all i with Di =c D(e). Since

A and h’ vanish on matrices Ai with Di CG D(e), and e is an F-linear combination

of matrices Ai with Di Fc D(e), we have h”(e) = X(e) = 1. Thus e” must be equiva-

lent to e. The converse is almost trivial.

The following theorem depends on Theorem 3 in $4.

(2.9) Theorem 2. A necessary condition that D be the vertex of an indecom-

posable component of FQ is that there exists a E Cl and y E NG(D) such that D

is a Sylow p-subgroup of GaaY.

Proof (assuming Theorem 3 (and Proposition 5)). By Theorem 3 and the

remark following its proof we can assume G = NG(P). Since any indecomposable

component of Ffl is a component of Fr for some orbit r of G on fi, we can

assume G is transitive. By Proposition 2 and the definition of defect group, P is

a Sylow subgroup of Gap for some a, p E Q.

19731 MODULARPERMUTATIONREPRESENTATIONS 107

_.

:

3 The Brauer homomorphism. Suppose P < H < NG(P) where P is a p-group, - - and let Q2, denote the set of fixed points of P in fi. Then there is a natural homo-

morphism f = f,,, n,p ,Hj mapping V,(G, a) into V,(H, a,): First define f in the

special case G = H-if Oi 5 fi,, x a,, define /(Ai) to be the standard basis matrix

for. (H, fi2,) corresponding to Oi, and otherwise set /(Ai) = 0; then efftend the

definition of f to V,(H, n) b y rnearity. f is now defined in the general case as 1’

the composite of the natural inclusion V,(G, a) -+ V,(H, fi) and lcH R p H)

We give the verification that f,, R p H) is a homomorphism: Not: &a; Oi _C

Q, x a2, if and only if P 5 Di. Set i =’ x’p, D, I,(Di; H, a) and let B denote the

set of all F-linear combinations of standard bisis matrices Ai in V,(H, a)

satisfying Oi 2 fl, x a,. Then B is an algebra isomorphic to V,(H, fip) in an

obvious way, and J is an ideal in V,(H, Q2) by Lemma 1. Trivially, V,(H, a) =

J + B and ] n B = 0. The map f is clearly the composite of the natural projection

V,(H, fi) --+ B and the obvious isomorphism B -+ V,(H, fip). Thus f is a homo-

morphism.

If A E VF(G) then the restriction VF(G) -+ V$H) carries A into J if and

only if A = E:= 1 ciAi where Oi n Q2, x fl, = @-that is, P & Di-whenever ci f

0. Now the above proof gives

(3.1) The kernel of f,, n p Hj is CPdGD. I,(P; G, a). 1 , , I

The relationship between f and the classical Brauer homomorphism is de-

scribed below. (See also [15, footnote 81.)

Proposition 4. Assume C,(P) (_ H and let s: Z(FG) -+ Z(FH) be the standard

Brauer homomorphism. Then the following diagram is commutative:

Z(FG) - V.JG Cl)

Z(FH) - V&T a,)

,’

If C C G we write _C for the R-sum of all permutation matrices corresponding

to members of C. Thus, if C is a conjugacy class of G, C is the image of the

class sum C E Z(RG) under the map Z(RG) -+ VR(G). To prove Proposition 4 we

need to calculate c in terms of standard basis matrices. This has essentially

been done by Tamaschke (for the transitive case, in the formalism of S-rings [16]).

We give an alternate approach more suitable to the present formalism by using

the ‘*orbital character” notation.

For i= l,..., T we define 13;(g) = I{ a E Ci2( (a, ag) E OilI for each g E G (as

in Scott [14]). Of course ei is identically 0 unless Oi 2 a’ x a’ for some a E Q.

(3.2) (Scott [13, 2.21) e;( g is the trace of CAT acting on RQ. )

The proof is a direct calculation from the definition of A’.

108 L. L. SCOTT [I anuary

(3.3) _c = Ci(lOi( -*cxec di(x))Ai whenever C is a union of G-conjugacy

classes.

Proof. C ertainly we can express C = ci ciAi for some coefficients ci E R. -

A particular coefficient ci is computed by multiplying _C by A’ and taking the trace

(the trace of AjAT on Rfi is easily calculated to be 6ij[0,1; see Wielandt [lg, V]).

(3.4) (Scott 113, 2-7.91) S uppose C is a union of G-conjugacy classes. Fix a,

p E Oi. Then (O,]-* cXcc 0$x) = ((x E C( ax = PI].

Proof. Count the number of triples (y, 6, x) with (y, 6) E Oi, x E C, and yx =

6. Counting in terms of x gives a total CXec e,(x). Counting in terms of (y, 6)

gives ]O,l](x E Cl ax = @]I.

(3.5) Proof of Proposition 4. Fix a conjugacy class C of G and a standard

basis matrix a E V,(H, n,). Let a, p E fi,, be such that (a, P)H corresponds to

a.

?; Then the a-coeffic-(c) is ({x E ClaX = PI] by (3.3) and (3.4). Now

s(C) is by definition C n C,(P). The a-coefficient of C n C,(P) is

(Ix E C n C,(P)(a” = p]I, again by (3.3) and (3.4). Since P fixes a, p and

centralizes no member of C - C n C,(P), we deduce

({x E Clax = p]I z 1(x e C n C,(P)\ ax = PII (p).

Thus the two a-coefficients are equal. Since C and a were arbitrary, the commu-

tativity of the diagram is established.

The following two propositions describe elementary properties of ltG, R,P “).

(3.6) Proposition 5. Let e E VF(G) be a primitive idempotent. Then f(e) f 0

i/ and only if D(e) & P.

Proof. If f(e) = 0 then e E c,cco. I,(D,; G, Q2) by (3.1). Hence e E

I,(Di; G, (;l)-consequently D(e) LG D;-?or some i with Di & P by Green 17,

3.3a]. Since P -& Di we have P PC D(e).

If f(e) f 0 then f (IF(D(e); G, Q)) f 0, since e E ItJO( G, a), and so

P SC Dj by (3.1).

(3.7) Proposition 6. Let e E V,(G) b e a primitive idempotent with f(e) f 0.

If N is an indecomposable component of Fop/(e), then some D(e) contains a

vertex of N. If C,(P) 5 H then N lies in a block b of H such that File is in

b”.(s)

Proof. It is easy to check that, for any i, /(Ai) is a sum of standard basis

matrices with defect group contained in a G-conjugate of Di. ,Thus f(e) E

c Q cG~(e) b(Q; H, %)-

(5) See Brauer [3, $21 for a definition of b G .


Set f(e) = xj j h e w ere the ejss are primitive orthogonal idempotents. Then

N “N Ffipej for some i by the Krull-Schmidt theorem, and the vertex of N is

D(ej) by Proposition 2. Since f(e)ej = ej we have e. E 2 osGD(e) $JQ; H, In,).

So ei E I,(Q; H, fi,) for some Q SC D(e) by Green f7, 3.3al. Hence D(ej) LH Q

sG D(e).

Now suppose C,(P) 5 H so that the standard Brauer homomorphism s: Z(FG)

-+ Z(FH) is defined. Let c E Z(FG) b e a primitive idempotent whose image c

under the map Z(FG) -+ V,(G) satisfies ec = e (thus (Fae)c = Ffie and so Ffie

belongs to the block associated with cr. Then f(e) = f(es) = f(e)/@) = f(e)s(c)

- by Proposition 4. Hence ej = ejf(e) = ejf(e)s(c) = ejsM; consequently

(Ffipej)s(c) = Ffipej. By [3, paragraph foll owing 4.161, the proof is complete.

The following fact is also worth noting.

(3.8) Any indecomposable component of Flip is isomorphic to a component

of Fn,f(e) for some primitive idempotent e E VF(G).

Proof. Since f (1) = 1, a decomposition 1 = 2, ek into primitive orthogonal

idempotents in VF(G) leads to a decomposition 1 = 2 , (e+Of w into ,orthogonal

idempotents in V,(H, fip). Th e result now follows from the Krull-Schmidt theorem.

At this stage of our development we sacrifice a little simplicity for the sake

of obtaining more detailed information.

The l-1 condition. A p-group Q zG P satisfies the 1-l condition with respect

to (G, fi2, P, H) if P SC Di whenever Di & Q.

The onto condition. A p-group Q zG P satisfies the onto condition with respect

to (G, fi, P, H) if oG n a, x 0, = o whenever o is an orbit of H on fi2, x fi2,

whose associated defect group d satisfies d & Q.

If P contains no proper subgroup conjugate to any Di, then P satisfies the

l-1 condition with respect to (G, fi, P, H).

If H = NG(P), an d if P = Q (or more generally, if P is weakly closed in Q

with respect to G), then Q satisfies the onto condition with respect to (G, a, P, H).

(3.9) We insert here a proof that Q satisfies the onto condition with respect

to (G, fi, P, H) when H = NG(P) and P is weakly closed in Q with respect to G.

Let o be an orbit of H on fi2, x Q2, whose associated defect group d is contained

in a conjugate of Q. Then a straightforward argument shows P is weakly closed

in d. In particular NG(d) 5 H, so d is a full Sylow subgroup of Gap for (a, p) E

0. We now easily deduce that any G-conjugate of P contained in G 4

is G ,Bcbn-

jugate to P. By the Jordan-Frattini argument (see [18, proof of 3.51) H is transi-

tive on the fixed points of P in o G. That is, oG n 52, x fi2, = o.

If e is any idempotent in VJG), we let ‘? e denote the character of G afforded

by KfleO where e0 is an idempotent in VR(G) satisfying To = e. In addition we

define ye = 0 for e = 0.


Proposition 7. Suppose Q satisfies the 1-l condition with respect to

(G, a, P, H). Let e, e’ be idempotents in V,(G), and assume that e is primitive

with D(e) =G Q. Then

(Ye7 Ye& L yqey Y,(,&’

Proposition 8. Suppose Q satisfies the onto condition with respect to

(G, Cl, P, H). Let e, e’ be idempotents in VF(G), and assume that e is primitive

with D(e) =G Q. Then

(Ye3 Ye& L (y,(e)T Y,(&)H.

The following standard fact is needed for the proofs of Propositions 7 and 8.

(3.10) Suppose e, e’ E VF(G), and e* = e, (e’)* = es. Then (Ye, Ye,) =

dim, eVF(G)e’.

Proof. The assertion is trivial in case e or e’ is 0; consequently, we may

assume e, e’ are idempotents. Let U, u’ be idempotents in VR(G) with U = e,

ii’ = e ‘. Let ;? be a splitting field for VK(&). Then obviously (Ye, Ye,) =

djmz(Hom;‘G (%I,, EQu’)), and HomzG (%I,, %I,‘) ~uV;u’. Since Vz Z

K mR VR and uV,u’ is an R-direct summand of V,, we have

dim% uV,U’ = rankR uVRu’ = dimF uvR”(= dim, eVFe’. K

(3.11) Proof of Proposition 7. By (3.1) and our hypothesis, the kernel of f intersects the ideal !,(Q; G, a) trivially. Since e E r,(Q; G, 0) we have eVFe’

2 !,(Q; G, a). Thus eVFe’ z f (eVFe’) _C f (e)VF(H, Q2,)f (e’). Application of

(3.10) now finishes the proof.

(3.12) Proof of Proposition 8. Set ] = CdCGB I,(d; H, a,). Then ] is an -

ideal in V,(H, fi2,), and I _C f(VF(G)) by hypothesis. By Proposition 6 (or direct

calculation) we have f(e) E 1. Thus

f (e)VF(H, Cl,)/(e’) 2 f(e)Jf(e') cf(e)f(V,(G))f(e')c_f(e)VF(H, f12,)f(e').

Therefore f (e)VF(H, fip)f (e’) = f (e)f (V,(G))f (e’) = f (eVF(G)e’). Again applica-

tion of (3.10) completes the proof.

The next section is devoted to further consequences of the onto condition.

4. Brauer’s first fundamental theorem. We assume throughout this section

that Q satisfies the onto condition with respect to (G, Q, P, Hj. P 5 H 5 NG(P)

and f = fCG, R,P,H) as in 93.

(4.1) Theorem 3. (a) Let e, e’ be idempotents in V,(G, a) with e primitive

and D(e) =G Q. Then f(e) is primitive, and ksle\Ffle” if and only if Fnpf (e)lFClpf(eE).

(b) In case H = NG(P) and F E V,(H, fip) is a primitive idempotent with

D(z) = P, then 2 is equivalent to f(e) f or some primitive e E VF(G) with D(e) =G P


Proof. (a) We shall use the fact established in (3.12) that f (eVF(Gk’) = f(e)VF(H, fi,)f(e’). Of course we also have f (e’VF(G)e) = f (e’)VF(H, a,)f (e).

Since eVF(G)e is completely primary, so is f(eVF(G)e) = f (e)VF(H, ap)f (e).

Hence f(e) is a primitive idempotent. (f(e) f 0 by Proposition 5.)

If Ffif (e)\FQf (e’) we have f(e) = zf (e”)w where z E f(e)VF(H, fip)f(e’) and

w E f(e”)VF(H, Qp)f (e). Ch oose x E eVF(G)e’ and y E e’VF(G)e with f(x) = 2

and f(y) = w. Then xe’y E eVF(G) e, and no power of xe’y is 0 since [(xe’y) =

zf (e’)e = f (e). Th us xe’y is a unit in eVJG)e. Consequently FfieiFQe’.

Conversely if FQe (Ffie’, then e = ue’v for some u, u E VF(G). Hence /(e) =

f b)f (e’)f (v) and so Faf (e)lFfiff(e’).

(b) By the remark following Proposition 6 we have 2 = zf (e)w for some z E

pVF(H, ap), w E V,(H, ap) 2 and a primitive idempotent e E VF(G). Since 2

belongs to the ideal IF(P; H, fip) by hypothesis, we have z, w E I,(P; H, fip).

By Sylow’s theorem P is a Sylow p-subgroup of Gap if and only if P is a

Sylow p-subgroup of Hap; when this occurs H is transitive on the nonempty set

(a, PJG n a, x 52,. Now we easily deduce that /(fF(P; G, a)) = IF(P; H, Q).

Hence we can choose x, y E I,(P; G, a) such that z = f(x) and w = f(y).

Thus eyxe E eVF(G)e f~ ZF(P; G, n). Since /(xey) = .z/(e)w = 2 we see that

no power of eyxe is 0. Consequently eyxe is a unit in the completely primary

ring eVF(G)e. Therefore eVF(G)e = eVF(G)e f? IF(P; G, Q), and we conclude that

e E I,(P; G, a).

By part (a), f(e) is a primitive idempotent. Clearly f(e) is equivalent to 2.

Since e.E Z,(P; G, Q2) we have D(e) LG P. But f(e) f 0 so D(e) kG P by

Proposition 5. Thus D(e) =G P and the proof is complete.

The previous theorem and Proposition 5 show that f establishes a l-1 corre-

spondence between equivalence classes of primitive idempotents in V,(G, a) and

VF(N,(P), fip) with defect group P. This correspondence will now be identified

with the help of a lemma in the next section.

Proposition 9. Suppose H = NG(P) and e E VF(G) is a primitive idempotent

with D(e) =G P. Then Fn,f(e) is the G reen correspondent(6) of FQe. In par’ticular

Ye(l) = [G : HlY,,,)(l) v ([G :PI)+I

(p p ).

The following preliminary result is an easy application of Proposition 1; the

details are left to the reader.

(4.2) Suppose the RG modules X, Y are direct summands of RCl. Then X is

isomorphic to a direct summand of Y if and only if !? is isomorphic to a direct

summand of y.

--

(6) See [8, Theorem z].

112 L. L. SCOTT [J anUary

Now we use (4.2) to prepare for the proof of Proposition 9.

(4.3) Let X be an NG(P)-indecomposable component of Rfi NG (P)’

and Y a

G-indecomposable component of RCl with vertex P. Then X is the Green corre-

spondent of Y if and only if I? is the Green correspondent of y.

- Proof. Of course Y is indecomposable with vertex P by Proposition 1.

If X is the Green correspondent of Y, then X( YNGcp) and X has vertex P.

Thus xIYNG(p) - and X is indecomposable with vertex P by Proposition 1, and SO

x is the Green correspondent of y. --

Conversely if x is the Green correspondent of Y, then XIYNGtP) and x has

vertex P. Therefore X has vertex P by Proposition 1, and X(YNccp) by (4.2)

applied to NG(P). Hence X is the Green correspondent of Y.

(4.4) Proof of Proposition 9 (assuming Lemma 2). Lemma 2 gives

FQfM\mk)H.

Trivially P 5 D(f(el) , and @f(e)) sG P by P roposition 6. Thus D(f(e)) =G P,

and so FQp/( ) e is the Green correspondent of Fne.

Now choose primitive idempotents U, II in VR(G), V,(H, a,) respectively,

with iT = e and V = f(e). Thus R&L z File and RClpv y Fflp/(e). By Proposi-

tion 1, R&L has vertex P, and by (4.3) Ra pi is the Green correspondent of R&L.

Thus (Rs2pv)G is the direct sum of R!Ju and indecomposable modules with vertex

conjugate to a proper subgroup of P. The stated congruence on character degrees

now follows from Green [7, Theorem 31.

(4.5) Theorem 4. Assume F is u splitting field for V,(G, Q) and VC@, 0,).

Suppose e E VF(G) is a primitive idempotent with D(e) =G Q and let X, h be the

modular chracters associated with e, f(e) respectively. Then h = xo f.

- Proof. Let S and 2 denote matrix representations affording X, x” respec-

tively. xhese representations have the same degree by Theorems 1 and 3. Thus

2 and 2 0 f are repre%entations of V,(G) of the same-degree. Since x(/(e)) = 1,

6 is a constituent of y 0 f. Thus 2 is equivalent to 2 0 f.

5. Braaer’s second fundamental theorem. Again, P 5 H 5 NG(PJ where P is

a p-subgroup of G, and f denotes fcG R p “, 19 is the permutation character of 7 I 1

G, and if B is a p-block of G, we set 8, = x xEB(e, X)X (where the X’s are

absolutely irreducible characters of G).

Lemma 2. Let e E VF(G) be an idempotent. Then (FQe)H x F!Jpf(e) CB T

where T is a summand of /(Cl - (2,).

(5.1) Before the proof, we clarify the condition on T. Suppose P 4 G. Let

M be an indecomposable component of FQ. Then MIFflp if and only if P is con-

tained in the vertex of M.

. .


Proof. Obviously P is contained in the defect group of every standard basis

matrix of (G, a,) and is contained in the defect group of no standard basis matrix

of (G, fl - ap). So the result follows from Proposition 2, the Krull-Schmidt

theorem, and the way defect groups for idempotents were defined.

(5.2) Proof of Lemma 2. Obviously we may assume that G = H and e is primi-

tive. By (5.1) and Proposition 5 we have /(e) = 0 if and only if FQe\F(o - Qp).

Thus it remains to show FQpf(e) z. FQe in case f(e) f 0.

Observe that f(e) is a primitive idempotent since f is onto (/(e)V,(H, Qp)f(e)

= f(eVF(H, fi)e) is completely primary).

To complete the proof we identify V,(H, Qp) with the algebra f3 described

in the first paragraph of 93. The map f is now just the projection VH(H) - B.

In particular /(l(e)) = f(e) and so f(ef(e)e> = f(e) f 0. Consequently ef(e)e is a

unit in eVF(H, n)e. Therefore e is equivalent to f(e) and the proof is complete.

(5.3) Theorem 5. Suppose x E G and a power xn = z is a p-element. Let e E

V,(G) be an idempotent, and take P = (z), x E H. Then ye(x) = ylfej(x). In

particular Ye(z) is a nonnegative rational integer, and [ye(;)I 5 ye!z) 5 0(z).

ff e is primitive, then Ye(z) > 0 if and only if z is conjugate in G to un

element of D(e).

Proof. First we show Y,(X) = Y,,e, kc). Again we may assume that G = H

and e is primitive. By Lemma 2 we need only show Y,(X) = Y,,,,(X) = 0 when

FQe(F(Q - Q,,). Let e0 E VR!H) be a primitive idempotent with e,, = e. Then P

is not contained in the vertex of Rae0 by Proposition 1 and (5.1). Therefore, the

p-part of x is not conjugate to an element of the vertex of RfieO, and so Ye(x) =

0 by Green [7, Theorem 31. Of course ‘J’,,,,(X) = 0 since f(e) = 0 by Proposition

5.

Since P acts trivially on Qp, Y,(e,(z) = Y,(e) (1) > 1 Y,vIce,(X)I.

Y,(Z) f 0 precisely when Y ,(e) f O-that is, f(e) f 0. By Proposition 5 this

occurs if and only if P is conjugate in G to a subgroup of D(e). This completes

the proof.

As an immediate consequence of the above theorem (3.8) and Proposition 6,

we have -u

(5.4) Corollary A. Supp ose x E G and a power xn is a p-element. Let 8 be

the permutation character for the action of C,(x”) on the set of fixed points of

X* in a. Then

e,(x) = c ii,(x).

bG= B

In particular, O,(xn) is a nonegutive tationul integer, and )6,(x)\ 5 6,(x”)

< e(P).


6. Defect 0 and 1. We state all the main results of this section before at-

tempting any proofs.

Assume K, F are splitting fields for V,, V,, and let x,, vIli denote the

characters of G afforded by Xs, KMj respectively (see $1 for notation). Let the

symbol cp x,” denote the sum of all distinct p-conjugates of X, and write SN t

if x, is P-conjugate to x1.

Set a = v,((C\), and define vg = vp(xs(l)). Put e = max (vp!lOil)I:,l.

By a theorem of Wielandt (see Keller [12]), we have us 5 e for all s. We are

interested here in the cases u = e and u s =e-1. s

Lemma 3. Suppose the number of p-conjugates of x, is divisible by py. Then

(a> Y 5 e - us.

(b) If y = e - vs then x, has exactly py p-conjugates, and CD x0= YJj for

some j. If x, E Y, then k = j. The vertex of Mj has order pa-‘.

Theorem 6. Suppose vg = e.

Then x, = ‘Pi for some j. If x, s Y, then k = j. The vertex of Mj has

order paWe.

Also, x, is p-rational.

Theorem 7. Suppose us = e - 1. Then we have either A or B below:

(A) x, has exactly p p-conjugates. c, xF= yj for some j, and if x, _C Y,,

then k = j. The vertex of Mj has order paPe.

(B) The number of p-conjugates of x, divides p - 1. lf x, 5 Yj then one of

the following occurs:

(i) Yj = C, ~0 Th ‘a-e+ 1

e vertex of Mj has order p .

(ii) Yj = C, x,“+ 2, xf where t + s, ZJ~ = e - 1, and the number of p-con-

jugates of xt divides p - 1. The vertex of Mj has order pace. If Yj = Yk, then

k = j.

(iii) Yj = Cp Xz + T where T f 0 is u character such that, whenever xt _C T,

the number of p-conjugates of x, is divisible by p, ut 5 e - 2, and each p-conju-

gate of xt appears in T with the same multiplicity as x1. The vertex of Mj has

order pame. If Yj = Y,, then k = j.

(iv) p = 2 and Yj = 2x,. The vertex of Mj has order pame. If Yj = Y,, then

k = j.

Corollary B. Suppose the block B has defect 1. Let (x) be u defect group

of B, and let b be the block of NG( (x)) satisfying bG = B. Let g be-the per-

mutation chardcter of NG(( x)) on the fixed points of x in Q, and set 8 b =

c, 1,:,7 where Sk is absolutely irreducible and appears in gb with multiplicity 1,.


Then we may find distinct indices jk such that 8, = 2, lkYi, + @ where

(1) @ is the sum (with multiplicities) of characters Yj whose affording module

Mj is projective and lies in B.

(2) Yjk (x) > 0 f or each k, and Yjk is either a nonexceptional character of B

or the sum of all exceptional characters of B.

Now we prepare to give the proofs. Several arithmetic preliminaries are required.

(6.1) There exists a p-adic number field K with the following properties:

(i) K is obtained by adjoining an mth root of unity, where p $m, to the p-

adic completion of the rational field.

(ii) K contains a primitive lth root of unity, where 1 is the PI-part of the

group order ICI.

(iii) Each character x, is afforded by u K(x,) representation.

(iv) Each indecomposable component of Rfi is absolutely indecomposable

(where R denotes the ring of local integers in K).

Proof. By Fong’s result(‘) ( see [G, 16.51) a field K, satisfying the first three

conditions is obtained by adjoining an lth root of unity (or (31)th root of unity if

p = 2 and 371) to the p-adic completion of the rational field. Let F,, denote the

residue class field of K,, and select a finite field GF(q) which is a splitting

field for VFO!G, n). N ow let K be the field otained by adjoining a primitive

(q - 1)th root of unity to K 0. Then the field K satisfies condition (iv) by (1.3).

Obviously K satisfies the other three conditions.

(6.2) Notation. Note that it suffices to prove Lemma 3, Theorems 6 and 7 and

Corollary B for any particular p-adic field satisfying condition (iv) above. Conse-

quently we shall assume throughout the rest of this section that K is a field satisfy-

ing all the conditions of (6.1).

We let g, denote the Galois group of K(x,)/K, and let 9,” denote the Sylow

p-subgroup of 9,. Set pws = \gz\.

(6.3) (i) Y’,” ranges over all the distinct p-conjugates of x, as u ranges over

%

(3 Z.mcg2 y” = 0 (p”“) for any local integer y of K(x,).

Proof. By (6.lii) G C K, where 5 is the field of lth roots of unity. Then

K(x,) E K( ) h o w ere w is a primitive path root of unity. K(w)/K is a fully ramified

abelian extension of degree (p - l)p’-’ (see [17, proof of 7-d-11). The p-conju-

gates of X$ are by definition the characters x,“,

of 5(x,)/Q. Since &0)/~ = (p - I)p”- ‘,

o ranging over the Galois group

it follows from our description of

(7) Quoting this result to prove (6.1) is, in a historical sense, putting the cart before

the horse. The reader familiar with algebras over local fields (see Albert [l, IX, Theorem

231) can easily prove (6.1) directly.


K(w)/K that K(x,)/K is a fully ramified extenskn of dzgree [~(x,):~], and 9,

is naturally isomorphic to the Galois group of Q(x,)/Q. In particular (i) has now

been proved.

Let ws be a primitive p w,tl

th root of unity. K(ws) 5 K(o) since

P ws = I~~ll[K(x,): KlI[K(& Kl = (p - I)p”-’

We claim K(x,) 2 K(u).(*)

of &J

Let 6 denote the Galois group of K(w)/K, and let 6x, 6, be the subgroups

which Galois correspond to K(x,), K(ws) respectively. Since 9,” is a Sylow

subgroup of 9, and g, z E/6x, the power of p in 16 xI is pa-l-ws. Since

G/GU is isomorphic to the Galois group of K(ws)/K and [K(os): Kl = (p - l)pw”,

we have IGo\ = P~-I-~~. Thus GU 2 gx and so K(ws) 2 K(x,).

Now let 95 denote the group of all automorphisms o of K(ws)/K such that

Cd,“= Cd”, for some integer n = 1 (p). We claim 9: is isomorphic to 9,” by restric-

tion to K(x,).

Let $$f denote the Galois group of K(os)/K. From the definition of 9: we

have [sz : 951 = p - 1

striction map gf -+ s,

and so si is a Sylow p-subgroup of 9:. The natural re-

But \$j;\ = pw” =

is an epimorphism and consequently sends 95 onto 9:.

I$$,“\ and so gi . . IS rsomorphic to 9,” by restriction as claimed

Thus to prove (ii) it is enough to show cnEg~ y” 5 0 (p”“) for any local in-

teger y of K(os).

Now the powers of ws contain an integral basis for K(wS)/K [17, proof 7-5-

31. Thus we need only verify the congruence in case y = w”, for some integer k.

Then

The final sum is pw” if apk = 1 and (oik)pWs - l/oPsk - 1 = 0 otherwise. This s

proves (ii).

(6.4) If s - t then dsj = dtj for all j.

Proof. By (6.1 iv) each character Yj is afforded by an RG representation.

Hence (Yj, x,) = (Yj, x, “) for each o E g, and each j. The result now follows

from (6.3 i).

(6.5) If Yj = Y, and Mj, M, both have a vertex of order pace, then j = k.

Proof. Let P be a vertex of Mj. Then P satisfies both the 1-l condition

and the onto condition with respect to (G, fi, P, NG(P)). Let ej, ek E V,(G, a)

(*)-This argument fails if p = 2. But in this case the sum in (ii) is just

&d : Q(x,)l-'tr &&(y).


be primitive idempotents with zi m FQei, Mk E K?ek. Then Propositions 7 and

8 give

W lkj)' y/(ek) ) = we ,, Yeg) =tq, Yk) J

= (Yj, Yj) (f 0)

= (Y Be;)’ y,(ej)).

Since the inner product is not zero, we have D(e,) & P by Proposition 5.

By hypothesis ID( = \P[. H ence D(ek) =G P. By symmetry we now have a

second equality

(Y /cy )’ y,(ek ,) = Qk )’ y,(ek )).

It follows that Y ffej) = ‘fCek). Since Fnp/(ej) and Ffip/(ek) are projective

indecomposable F(NG(P)/P) modules (Theorem 3), we deduce that Ffip/(ej) *

Fflpf(ek) (from Proposition 1 and [4, 7-14 or 8.4.111). Thus ej is equivalent to

ek by Theorem 3 (or Proposition 9) and so j = k.

(6.6) Proof of Lemma 3. Select an index j such that x, _C Yj, and choose a

primitive idempotent u E VR with Mj “-R&.

Let ‘;i denote the ring of local integers in K(x,) = “K. By (6.1 iii) there exists

a primitive idempotent /, E UV~U such that ~ %2/S affords x,. By a suitable

choice we can insure that f, belongs to an R order of uV?u containing EL!+.

Let <, denote the absolutely irreducible character of!;; associated with f’

(that is, [,(/,) = 1). Then 5,(/,x) = [,(u/,nx) = L,(/,uxu) E R for all x E Vs, and

[(f,x) = 0 for any character r of Vz which does not contain l,.

We let g, act on Vz in the obvious way. For u E $$, let 6: denote the

character associated with /O Thus [,“(x”) = Ls(x)= for all x E Vz.

Since the characters xz, u E s,, are distinct, the idempotents /zare orthog-

onal. In particular C a+$ f,” = / is an idempotent in uV?u.

Now set f= x tiAi. We compute the coefficients ti as follows: Let T denote

the character of Vz afforded by ?kI. Then T = ct x,( l)L,, and r(Aj A”) = c?ijlOil

(as in the proof of (3.3)). Thus (0; (ti = 7(/A:) = x,(l)~~cg~ <,(fAT)o for each i.

Hence

>Vs+W - s - u,(lO,l) by (6.3ii)

2 us + ws - e.

118 L. L. SCOTT [J %l”WY

Since / is an idempotent, vp(ti) 5 0 for some i. This proves part (a) of

Lemma 3.

Now suppose ws = e - us.

Then vp(ti) 2 0 for all i and so f E uVzu. Hence f = II. In particular Yj =

c OE92 x,“. By (6.4) g,“= ss-that is, x, has exactly pw” p-conjugates.

If vp(ti) = 0 then 0 2 zis + w - s v,(lOi 1) 2 0 and so v,(lOi 1) = us + wus = e.

So by definition ID(U)\ = paWe. D(U) is the vertex of Mj by Propositions 1 and 2.

Since the choice of i was arbitrary except for x, _C Yj, we have Yj = Y, =

cp x,” whenever x, 2 Yy,. Also the vertex of M, has order pame; thus x, E Y,

implies k = i because of (6.5). Th is completes the proof of part (b) and the lemma.

(6.7) Proof of Theorem 6. Immediate from Lemma 3.

The next result is a technical preliminary to the proof of Theorem 7. Its as-

sertions are almost trivial in case VK is commutative.

(6.8) Suppose x, + xt 2 Yj where either s = t or s + t. Let u E VR be a -x,

primitive idempotent with Rh w M. I’

and let K be a p-adic field containing

K(x,), K(x,). D enote the ring of local integers in 2 by z.

Then there exist primitive idempotents f,, 4 E V; such that ?a/, affords

x,, %l/l affords xt, and

(i) f, E UV KcX )u and /, E uVKfx )u;

(ii) f ,” is ortho,~onal to f 1 forzacb u E 9, and r E 9,;

(iii) cS(fSw) E R and L,(f,w) E R for all w E V;y, where [,, ct are the abso-

lutely irreducible characters of I$ satisfying [,(f,) = <,(f,) = 1. Furthermore,

~s(,sw) = z,(/,w, _hi-

- uwu) where X is the absolutely irreducible character of Vz

(with uaZues in R) satisfying h(Z) = 1.

Proof. From (the proof of) Theorem 1 we know that uVKfX )u has an abso-

lutely irreducible representation module Ys of degree dsj sucg that K(x,)fif

affords x, for any primitive idempotent f E uV,(, )u with Y,f f 0. If < is the

character of VK(,,,) associated with f then Ls(fw)Sis easily seen to be tie trace

of the action of fuwu on Ys (W E V KC x,,). Moreover, if fl, fZ, *. * are a full set

of dSj orthogonal primitive idempotents in uVKCX )u which do not annihilate Yg,

then we may view the quantities L,(flw), <s(/2~f,. . * as diagonal coefficients

for uwu in a matrix representation associated with Ys.

Let RS denote the ring of local integers in K(x,). Now the algebra uVR u

has a unique irreducible representation, s

which is linear and may be realized as

the restriction to uV~ u of the irreducible character h of ‘JR associated with U

(see (2.7)). Thus we Lay choose a matrix representation as&iated with Y so 5 that any x E UVR,U is represented integrally with all diagonal coefficients having

residue class h(x). Let fly f2, -*- E uVKfX ) u be primitive idempotents which s

give rise to such a matrix representation.

19731 MODULARPERMUTATIONREPRESENTATIONS 119

In case s = t we take f, = f 1, f, = fz and it is clear from the preceding dis-

cussion that (i) and (iii) are satisfied. (ii) is a consequence of the mutual orthog-

onality of the idempotents (fl + f,)“, (J E 9,. In case s + t we take f, = f 1 and repeat the whole procedure (with xt in-

stead of x,) to obtain ft. Here (ii) is automatic, while (i) and (iii) are satisfied

as before.

(6.9) Proof of Theorem 7. If the number of p-conjugates of x, is divisible by

p, then we have (A) by Lemma 3. Hence we may assume the number of p-conju-

gates of x, is prime to p-that is, divides p - 1.

Suppose x, _C Yj and let u E VR be a primitive idempotent with RQu e Mj.

We distinguish two main cases.

Case 1. Yj = Xp x,“.

Then 7.1) f 0 (p”). H ence by Green [7, Theorem 31 (or Theorem 3) the ver-

tex of Mj does not have order pa-‘. We calculate u = Xi ti Ai where ( Oi Iti =

x,(l) TTE gs [s(uAT)“. Hence up 1 (t.) > 0 if 1Di1 > pnme+‘. So by definition (D(ii)( a--e+ 1

IP . Thus D(U) (= vertex of Mj) has order exactly pace + I.

Case 2. Yj f CD xz.

Here we must show that we get one of the types B (ii), (iii), or (iv). ,Note

that x’p x,” 2 Yj by hypothesis and (6.4). Let cp x,” + xt 2 Yj where s = t or

s + t.

Set z = x,(l), x1 = x,(l)-

Let K be a p-adic field containing K(x,), K(x,), and choose f,, f, as in

(6.8). Then we calculate

(pe/xJfs = c (p”/loZl)5s(,sA~~A i E uv-uu

i R

and

(pe/xl)ft = c (pe/(Oi()it(ftA;Mi e uv:u. i

Thus 1---1 -I_

(p’/~~)f~ = (pe/xt)ft = 2 (pe/jOilh(df~)Ai E uV-U. i R

c

We show (pe/xs)fs f 0: If not, then (p”-‘/xs) xCEss fz E uVRu and so

f” E uVRu. Therefore =‘E%- =

C ,,g,f:=u. B ut t h is implies Yj = X0x:, a contradiction.

In particular h(uATu) f 0 for some i with (Dil = pa-“. By Proposition 3 the

vertex of Mj has order pnee. Note also that this gives K = i whenever Yj = Yk

because of -(6.5).

Suppose now that the number of p-conjugates of xt divides p - 1. We will

show that this leads to type (ii) or (iv).


Here we may assume that is obtained from K by the adjunction of a primi-

tive pth root of unity (this may be seen directly or from the proof of (6.3)). Set s

= 9(2/K) and let rS = \g,l, rr = Is,\. Write [ for CoEg, f,O and /; for

zi cEgt f T. Then

(P - 1 /rS) CPe/x,fs - (P - 1 /rt) be/x,)/, = C ((pe/x,)f, - (pe/xl)ft)m E 0 tp). uet3

Hence [ + c[ E uV,u where c = - r,x,/rtxt. Now vp(xI) = vt 5 e - I by

Theorem 6, and so c E R. - - --

Set v = [ + cl. Then h(r) = (s(fsu) = (,(f,) = 1; also h(c) = [,(b~) = rl(cft) =

C. Therefore C = 1. In particular, ut = _~,(xr) = e - 1, and c = 1 + (p”/x,)k for

some k E R. Now c + /; = v - k(pe/xt)fl E uVRu. By (6.8) c + 6 is an idempotent,

and so [ + [ = u. Consequently Yj = x0 x,” + x’p xy. If s + t we have type (ii).

If s = t then p = 2, since C = 1 and c = - 1; so we have type (iv).

Finally, w,e are reduced to the case where the number of p-conjugates of xt

is divisible by p whenever cp x,“+ xr _C Yj. By Lemma 3, ut 5 e - 2. By (6.4)

all p-conjugates of xr appear in Yj with the same multiplicity. Thus we have

type (iii), and the proof is complete.

(6.10) Proof of Corollary B. Suppose Mj is a nonprojective indecomposable

component of RQ which lies in B. Thus the vertex of Mj is (x) by Green [7,

Corollary to Lemma 4.la]. In particular Yj(x) > 0 by Theorem 5.

Let x, & Yj. Then us = a - 12 e - 1. If vs = e then x = Yj is p-rational

by Theorem 6. If us = e - 1, then a - e = 0 and so we have :ype B(ii), since

Mj is not projective. Hence Yj = cp x,“.

Note that the latter case does not occur when (x) acts trivially on a, be-

cause e is obviously less than a. So by Theorem 6 the 1,‘s are the multiplicities

of the various indecomposable components of Rflx which lie in b (R. suitably

large). Application of Theorem 3 now finishes the proof.

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DEPARTMENTOF MATHEMATICS,YALE UNIVERSITY,NEWHAVEN CONNECTICUT 06520

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