K-Theory 7: 575-578, 1993. 575 © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

A Simple Counterexample to the Hambleton- Taylor-Williams Conjecture

DAVID WEBB Department of Mathematics and Computer Science, Dartmouth College, Hanover, NH 03755, U.S.A. e-mail: david.l.webb(~dartmouth.edu

and

D O N G Y U A N YAO Department of Mathematics, Washington University, St. Louis, MO 63130, U.S.A. e-mail: yao(a~jezebel.wustl.edu

(Received: 26 August 1992)

Abstract, It is shown that the conjectured formula of Hambleton, Taylor, and Williams for the G-theory of the integral group ring of a finite group does not hold for the symmetric groups Ss.

Key words. G-theory, group rings, symmetric group.

O. Introduction

Let G be a finite group. Consider the G-theory G,(ZG) of the integral group ring of G, i.e., the K-theory of the category of finitely-generated ZG-modules. In [L], H. Lenstra gave a beautiful formula for Go(ZG) when G is finite Abelian. Lenstra's formula was shown in [W1] to hold for the higher G-theory, and the formula was generalized to nilpotent groups in [HTW] and [W2]. Based upon these cal- culations, I. Hambleton, L. Taylor and B. Williams conjectured in [HTW] a general formula expressing the G-theory of the integral group ring of any finite group G as a direct sum of the G-theory of the factors of a maximal order with suitably chosen rational integers inverted in the factors; the integers which must be inverted are those which appear in the Jacobinski conductor formula. ([R], (41.3)). Their conjecture was shown to hold in some special cases in [W3], [LW], and [K].

The purpose of this short note is to record the fact that the conjecture fails for the symmetric group $5. This shows that the conjecture must be revised; more is involved than the conductor formula. Briefly, one uses well-known facts about representations of symmetric groups to calculate easily the rank of GI(ZSs) predicted by the Hambleton-Taylor-Williams conjecture; but Keating [Ke] com- puted Gt(ZG) for any finite group G, and the rank obtained by Keating differs from

576 DAVID WEBB AND DONGYUAN YAO

that predicted by the conjecture. For the basic facts we use concerning representa- tions of symmetric groups, as well as our notation and terminology, see James I-J].

1. The Hambleton-Taylor-Wiiliams Conjecture

Let G be any finite group. Then its rational group algebra decomposes as

QG ~ H Endo~(Vp) ~ H M.,(D~P).

where p ranges over the irreducible rational representations of G, Vp is the representation space of p, and Dp is the associated division algebra Endoa(Vp). For such a representation p: G--, Aut(Vp), let wp = Jim pl/gp, where gp is the degree of any of the irreducible constituents of the complexified representation C ® o Vp. Let Ap be a maximal Z[1/w p ]-order in Dp. Then the conjecture of Hambleton, Taylor and Williams is

G,(ZG) = (~ G,(Ap). (i.i)

where p ranges over the irreducible rational representations of G. Now consider the case of G = S,. It is well known I-J, Theorem (4.12)] that the

irreducible rational representations of S, are in bijective correspondence with the partitions of n: as 2 ranges over the partitions of n, the Specht modules S ~ furnish a complete set of irreducible rational representations. Moreover, the Specht modules are absolutely irreducible. The dimensions of the Specht modules are given by the hook length formula [J, Theorem (20.1)]:

dim(S z) = n!/Fl(hook lengths).

The meaning of this is the following: given a partition 2 of n, consider the associated Young diagram. The hook length of an entry in the diagram is the number of entries in the hook consisting of all entries below and to the right of the given entry (in the same row or column as the given entry) in the diagram. For example, in the Young diagram

O X •

O *

associated to the partition (3, 2) of 5, the hook length of the entry labeled X is 3, its hook consisting of the X and the asterisks. The Specht module S (3'2~ has dimension 5!/3"4.2 = 5. From the hook length formula, it is easy to determine the degrees of the irreducible rational representations of Ss; they are tabulated below:

Parti t ion;t: (5) (4,1) (3,2) (3,12 ) (22,1) (2,13 ) (15 )

dim(S~): 1 4 5 6 5 4 1

THE HAMBLETON-TAYLOR-WILLIAMS CONJECTURE 577

Thus the rational group algebra decomposes as

QSs = Q × M4(Q) x Ms(Q) x M6(Q) x Ms(Q) x M4(Q) x Q;

the first factor (associated to the partition (5)) corresponds to the one-dimensional trivial representation, and the last factor (associated to (15)) corresponds to the one-dimensional parity representation. Since each representation except the trivial representation StS) and the parity representation S tl~) is faithful, each of the integers jim p~[ except for these two is just 5!, while lim P(5)[ = I and jim p~15)[ = 2. Since each irreducible representation is absolutely irreducible, each integer g4 defined above coincides with dim(S~). From this, one easily tabulates the integers wz appearing in the conjecture; for example, w(g.t) = 5!/dim(S ¢4,t)) = 5!/4 = 2.3-5. The result is:

)t: (5) (4,1) (3, 2) (3, 12 ) (22 , 1) (2,1 a) (15 )

w~: 1 2"3"5 23"3 22"5 23'3 2"3"5 2

Since GI (Z[1/wp~]) is the group of units of Z[1/wp~], its rank is the number of prime divisors of wpz. Thus, the conjecture (1.1) predicts that the rank of G1 (ZS5) should be 13.

2. Keating's Computation

In [Ke], Keating computed GI(ZG) for any finite group G; his formula for the rank of GI(ZG) is the following. If the decomposition of the rational group algebra is QG - Z1 x --. x Zk, let ~3~ be the maximal Z-order in the center of Y.,. Let ri be the rank of the group of units of ~ , and let v, be the number of primes of ~3, which divide I GI. Finally, let e denote the number of isomorphism classes of simple ZG modules annihilated by t GI. Then Keating's rank formula is:

rank(Gl(ZG)) = r l + ... + rk + Vl + "'" + Vk -- e. (2.1)

In the case G = $5, each ~i is just Z, so each ri is 0 and each v~ = 3. To determine e, it suffices to determine the number of simple FSs-modtfles, where F is the finite field of order 2, 3, or 5. For this, recall that for any field F, there is a natural bilinear form on the permutation module M ~" over F arising from the Young subgroup Sz of $5; the Specht module S x is a subspace of M ~', and one defines D x = SX/(S ~ c~ Sx±).

For a prime p, a partition 2 = (21,22, 23, ...) of n is p-s ingu lar if there is some i for which 0 ¢ ;ti + 1 = 2i + 2 . . . . . 2i + p; ,~ is p- regu lar otherwise. By [J, Theorem ( 11.5)], the simple FSs-modules are precisely the spaces D z as ;~ ranges over all p-regular partitions of 5, where p is the characteristic of F. Since the 2-regular partitions of 5 are (5), (4, 1), and (3,2), there are precisely three simple FzSs-modules. Similarly, there are five simple F3Ss-modules and six simple FsSs-modules. Thus e = 14.

Substituting into (2.1), it follows that the rank of G1 ( Z S s ) is 7. Since this disagrees with the prediction of (1.1), the conjecture cannot hold in general.

It may be reasonable to conjecture that (1.1) holds for solvable groups.

578

References

[HTW]

[J]

EK] I-Ke] ILl

[LW]

FR] [Wl]

[W2] [W2]

DAVID WEBB AND DONGYUAN YAO

Hambleton, I., Taylor, L., and Williams, B., On G~(RG) for G a finite nilpotent group, J. Algebra 116 (1988), 466-470. James, G. D., The Representation Theory of the Symmetric Groups, Lecture Notes in Math. 682, Springer, Berlin, Heidelberg, 1978. Ktck, B., Higher K'-groups of integral group rings, K-Theory 4 (1991), 177-187. Keating, M. E., G1 of an integral group ring, J. London Math. Soc. 14 (1976), 148-t52. Lenstra, H. L., Grothendieck groups of abelian group rings, J. Pure AppL Algebra 20 (1981), 173-193. Laubenbacher, R. C. and Webb, D. L., Go of integral group rings for groups with cyclic Sylow subgroups, Contemp. Math. 126, AMS, Providence RI (1992), pp. 97-I l l . Reiner, L, Maximal Orders, Academic Press, New York (1975). Webb, 13. L., The Lenstra map on classifying spaces and G-theory of group rings, Invent. Math. 84 (1986), 73-89. Webb, D. L., Higher G-theory of nilpotent group rings, J. Algebra 16 (1988), 457-465. Webb, D. L., G-theory of group rings for groups of square-free order, K-Theory I (1987), 417-422.