Most primitive groups have messy invariants

ADVANCES

32, 118-l

IN MATHEMATICS

Most

Primitive

27 (1979)

Groups w.

Department

c,

of Mathematics,

Have Messy Invariants HUFFMAN*

Loyolu

University,

Chicago,

Ill.

606 26

AiTD

N. J. A. Bell Laboratories,

Murray

SLOANE

Hill,

New

Jersey

07974

Suppose G is a finite group of complex n x n matrices, and let AC be the ring of invariants of G: i.e., those polynomials fixed by G. Many authors, from Klein to the present day, have described RG by writing it as a direct sum rfzl ~@[8, ,..., 8,]. For example, if G is a unitary group generated by reflections, 6 = 1. In this note we show that in general this approach is hopeless by proving that, for any E > 0, the smallest possible d is greater than 1 G ln-1--c for almost all primitive groups. Since for any group we can choose S Q 1 G In--l, this means that most primitive groups are about as bad as they can be. The upper bound on 6 follows from Dade’s theorem that the 19~can be chosen to have degrees dividing I G I,

Let G be a finite group of complex n x n matrices, If f(xr ,..., x,) is a polynomial in x1 ,..., x, with complex coefficients, and A = (uij) is an element of G, then Af(.r, ,..., x~) =f(C a,,.~~ ,.,., C apzjxj) is the polynomial obtained by letting A act as a linear transformation on the variables x1 ,.,., x, . The ring RG of inoariunts of G consistsof all polynomialsf with Af = f for all A E G. The central problem is to find a description of RG that is conciseand easy to use. Several types of basesfor RG have been considered([4, Ch. XVII], [32]), but we are concerned here with finding a direct sumdecomposition

whereOr ,..., 19,are algebraicalIy independenthomogeneousinvariants andq1 = 1, Q ,..,, r], are certain other homogeneousinvariants. It is known from the theory of Cohen-Macaulay rings ([IS, Prop. 131)that RG alwayshasa direct sum decomposition (1). Then RC is a finitely generatedfree @[O,,..,, 8,]-module of rank 8. We are interested in the smallestvalue of S that can be attained by any choice of 6, ,..,, 0,: call this value 6(G). We shall give examplesto show that 6(G) may * Work

done

while

at Union

College,

SchenectadT,

118 OOOI-8708/79/050118-10$05.00/O Copyright All rivhta

8 1979 by AcademicPress, of rrnroduction in arm form

Inc. reserved.

N.Y.

12308.

MOST

PRIMITIVE

GROUPS

HA%VE

MESSY

119

INVARIANTS

be large, and then show that for most primitive groups S(G) is about as large as it can be. In [33, Section 3.4.121 T. A. Springer has proved a somewhat similar result for the ring of invariants of binary forms of given degree. The 8, and rlj in (1) form a polynomial basis for RG. In the past century polynomial bases have been given for a number of groups ([l], [lo], [13], [16], [21], [23]-[25], [27], [30]-[38]). E. Noether had shown in 1916 ([29]; see also [32], [42]) that any group has a polynomial basis containing no more than (l”ii”‘> invariants, i.e. about 1 G I”/R! for large / G /. We had always thought that this was a very weak bound (cf. also [I 3]), b u t in fact it is not far off, at least for primitive groups. WC make use of the Afolien series of G, i.e. the generating function

in which the coefficient of Xj is the number of Iinearly independent homogeneous invariants of degree j. Molien ([ZS]; see also [4]> [27], [32]) showed that the series can be calculated from the identity

If R" has the decomposition

(1) then the Molien

series may be written

as

where

THEOREM

1,

Proof. If we equate (2) and (3), multiply obtain l/l G 1 = S/d, ... c&, . EXAMPLE 1. is a free ring and 1 G 1 (Shephard The next two

EXkMPLE

2.

by (1 - ,I)“, and let h --f I we

If G is a finite unitary group generated by reflections then RC may be expressed in the form (1) with 8(G) = 1 and d,d, .*a d, --. and Todd 1307; see also [5]). examples are six-dimensional groups from Lindsey’s list [23]. Let

G be a six-dimensional

representation

of a nonsplitting

120

HUFFMAN

AND

SLOANE

central extension of Z, by the alternating group A$~, of order 3 * 360. This group is of interest because it is generated by matrices with two eigenvalues -1 and the rest 1. Such groups are a natural generalization of the class of groups given in Example 1, ([18], [41]). We shal1 see that B(G) > 54, which suggests that the invariants of these groups do not have such a simple description as those of Example 1. The Molien series may be calculated from (2), and when written in simplest terms (with numerator and denominator relatively prime) is equal to (5) where f,(A) = 1 - A3 + 3h” + x9 + 4P + 3A15+ 5P + 3x21 + 4p4 + A37 + 3A30- A33 + A36 and #7 = #,(A) is the r-th cyclotomic polynomial. The numerator of (5) (and of (6)-(10) below) is a symmetric polynomial: this is an easy consequence of the fact that all the group elements have determinant one. The TayIor series expansion of the MoIien series is 1 + 2x3 + 7x6 + 16hg + 38h’2 + 71915 + ]46#8 + . . To obtain a lower bound on S(G) we make use of Theorem 1. Let the Molien series be written in the form @(3)(x) = (1 - ~DI)( 1 _“,‘ti)

. ., (1 _ ADo)

where f3(h) has nonnegative coefficients and the product DID, ... D, is a minimum. From Theorem 1, S(G) > S,(G) =

““y;.;

D6 .

There are somesubtletiesin the determination of the minimum value of 4 ... De, and it seemsworthwhile to sketch a method for calculating it. By equating (5) and (6) we seethat the #i’s in the denominator of (5) can be partitioned into six sets, say kJfll 9---r5&l) . .. Mt,, I’**>A,,)

MOST

PRIMlTIW

GROUPS

HAVE

MESSY

iNVARIANTS

I21

such that the product of the j-th set divides 1 ~ hDj. (Thus the cjd’s in each set are distinct.) Let F, 7. l.c.m.{tj, Then the product of thej-th

Consequently,

,..., tjr,}.

set of +;‘s divides 1 - AF9,Fj divides Dj , and

if we multiply the numerator and denominator of @r)(h) by

we obtain

(7) which is a form of the Molien series intermediate between @(r)(A) and @(s)(h). Then @a)(A) is obtained by multiplying the numerator and denominator of CD(~)(X) by

In the present example it is not difficult to see that only three partitions of the +i’s need he considered, namely:

F6 = 30.

122

HUFFMAN

AND

SLOANE

Suppose partition (I) occurs. Then PyX)

= @(l)(A) 1 -F+3P+“‘+P = (1 ~ h3)3(1 ~ P)(l

~ P)(l

- P) ’

and pIa = 1. Since the numerator of P)(h) contains a negative coefficient, pz3(X) # 1. If we take P&) = 1 + P, we obtain 1+2Xe+4Xg+5X12+‘..+h3g @‘3’(h)

=

(1

-

h3)2(]

-

x6)2(1

-

X12)(1

-

(8)

A15)

with nonnegative coefficients in the numerator, and D, *a. R, = 32 76s * 12 7 15 = 58320. Any other choice of ps3(X) will give a larger value of D, ‘*a D, . Suppose instead that partition (II) occurs. Now Plzo) = A0930 and 1~ @(2)(A)

=

(1 -h3)4(1

2A3 + ... + -A'")(1

P _ pl)

'

Again p&X) # 1, and so D, ... D, 2 58320. Similarly for partition (III). conclude that the minimum value of D, I*. D, is 58320, and therefore

We

58320 S(G) 2 S,(G) = m = 54. EXAMPLE 3. In this example we class of groups, a six-dimensional ([9, Section 381). Using th e method partitions to be considered) we find minimum D, ..f D, is

@(3)(h) = =

consider a member of another important representation of S&,(13) og order 2184 described in Example 2 (there are now 29 that the Molien series when written with

1 + P + 5P + 6h1* + 12hl” + 18P + 25h2” $- ... + X7* (1 - A”)(1 - X8)(1 - P)S(l ~ X14)(1 - P) 1 +

A4 + 2P

-j- Al" +

9P2

+

W"

-f- 22P

+ 27X'S

(9)

+ 54P* f .I.

Consequently S(G) 2 S,(G) = 768. EXAMPLE 4. Let G be the 24-dimensional representation of the Conway group ‘0, of order 83 15553613086720000 ([q), The Molien series is

MOST

PRIMITIVE

GROUPS

HAVE

MESSY

I23

INVARIANTS

where p@)

_

1 $-

p3

+

jp

+ 40]07gpO

+

2h28

+ p

+ p

+ 3A:JJ t . .

-I- ..- .I. AD'"

and q(h)

:~- (I ~ (1 -

h”)( I ~ A=)“(

P)( 1 ~

1 ~

P)(

,b”G)“( 1 ~

I pJ)(

FO)(l 1 -

“pp)(

A”2)( I 1 -

p?)(l

(I ~ A”“)( I ~ A”“)( I - A’“)( I - X7x)( 1 - P),

P)3( -

I p2)(

PO)? 1 -

AS”)

of degree 940.

JVe have not attempted to minimize D, ... D,, in this example. However, if (10) is the minimum-and our experience suggests it should be close to the minimum then 6(G) ‘: 8,(G) = D, ... I),,/1 G 1 :- 205679393714995200. I
Kemnrk. Once such an R-sequence has been found it is straightforward to sholv that there are further invariants Q ,., ., vfi such that (I) holds (see for example [351)ProoJ. Let A, consist of the homogeneous linear polynomials in K. Choose any non-zero fr E R, , and let f 11 = f7 , f,? ,.,., flo, be the distinct images of fi under G. Set

124

HUFFMAN

AND

SLOANE

Suppose 0, ,..., Bi (1 < i < n) have been found. Choose fi+l E R, such that fi+r is not in the prime ideal (fie, , fir, ,...,fic,)

VI)

for all 1 < c, ,< a, ,.*., 1 < c$ < n, . (Since there are only finitely many of theseideals,and each intersectsR, in a vector spaceof dimensioni < n, such an fi+r can aIways be found.) Let fi+l,l =fi+r , fi+rs2 ,...,f+l,cai+, be the distinct imagesoff,+r under G, and set

The construction implies thatfi+r, j is not in the ideal (11) for all 1 < cr < a, ,. _, 1 < C~< ai . Repeat until 0r ,..., 8, have been obtained. Clearly the 0, are homogeneousinvariants of degrees dividing ] G 1. That 0r ,..., 8, is an Rsequence follows from N. Bourbaki, Groupes et Algebras de Lie, Chap. 5, Exercise 5, Hermann, Paris, 1968, p. 137. COROLLARY

S(G) < 1G In--l. Immediate from Theorems 1 and 2.

Proof. THEOREM

lent, n

x

3. Let p1be fixed, and let % be any infinite family n complex groups that satisfies

of finite,

inequiwa-

(Al)

every elementoj%?isirreducible,

(A2)

there are finitely many abstract groups HI ,..., H, such that ;f GE % g Hk f OP some h, where Z(G) is the center of G.

then G/Z(G)

Then for any f&d

and

E > 0,

and so 6(G) > 1 G ln-1-s for almost all G E ‘X. hoof. For G f % let Z(G) have order 7p~.Then by [l 1, Theorem 1.41, Z(G) = (~$1: 0 < j < m}, where w == 8,%/m. Suppose G/Z(G) z Ht. Then / G ) = m ) Hk I < mh, where h = maxi] HI I,..., / H, 1). The degree of any homogeneousinvariant of G must be divisible by m, soin (4) each dj > M. Then

MOST PRIMITIVE

GROUPS HAVE MESSY IKVARIANTS

12.5

As 1G ] + c$, we must have ] Z(G)] = m -+ cci (since there are onIy finitely many H,‘s), which proves (8) and the theorem. Let G be an irreducible finite group of 1~x n matrices acting on a complex n-dimensional vector space V. Then G is imp&z&e if I’ is a direct sum of vector spaces:

where m > 2, each Vi j; 0, and for all A E G and all i, A(FJ = Vj for some j = j(A,i). 0th erwise G is primitive. G is quasiprimitiwe if for any normal subgroup N of G, P is a direct sum (14) where the Vi are invariant under N and afford equivalent representationsof N. By Clifford’s theorem ([S], [9], [14]) a primitive group is quasiprimitive. In the classification of linear groups it is customary to restrict attention to primitive or quasiprimitive groups ([l], [2], [12], [16], [l 81, [22], [40]) since imprimitive groups are induced from a representation of smaller degreeof a proper subgroup (cf. [9, Th. 14.1 (4)]). THEOREM 4. The conclusionof Theorem 3 applies to the family n X n primitive groups OT quasiprimitive groups.

of &equivalent

Proof. Suppose G is quasiprimitive, and let H be any abelian normal subgroup of G. By hypothesis V is the direct sum of one-dimensionalsubspaces that are invariant under I-r, and the elements of H can be taken to be scalar matrices. Hence H C Z(G). Th erefore by Jordan’s theorem ([l], [9]), ] G/Z(G)] is bounded, and so (A2) holds, proving the theorem. Remark. It seemslikely that a similar result will hold for the class of al1 n x n groups, or all rz x 71irreducible groups, The conclusion will not be as strong, however, since for any PZ2 2 there is an infinite family of irreducible imprimitive unitary groups generated by reflections with S(G) = 1 (the groups Gh P, 4 in [301). ACKNOWLEIIGMENTS We thank J. H. Conway for providing the data used to calculate the Molien series of his group, E. C. Dade for permission to include his theorem, and R. L. Graham, C. L. Mallows and R. P. Stanley for helpful discussions. Some of the computations were carried out on the ALTRAN and MACSYMA systems ([3], [26]).

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PYOC.C’anrbridge Philos. Sm. 74 (1973), 435-440. 25. H. MASCHKE, The invariants of a group of 2 . 168 linear quaternary substitutions, in “International Mathematical Congress 1893,” pp. 175-186, Macmillan, New York, 1896. 26. MATHLAB Group, M. I. T., “MACSYMA Reference Manual,” Project MAC, M. I. T., Cambridge, Mass., Version 8, 1975. 27. G. A. MILLER, H. F. BLICHFELDT, m L. E. DICKSON, “Theory and Applications of Finite Groups,” Dover, hTcw York, 1961. 28. T. MOLIEN, Uber die Invarianten der linearen Substitutionsgruppe, Sitzwzgsber. Kiinig. Pmuss. A&d. Wiss. (1897), 1152-3156. 29. E. lYoETrmt, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1916), 89-92. 30. G. C. SHEPHARD AND J. A. TODD, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304.

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31, N. J. A. SLOANE, Weight enumerators of codes, in “Combinatorics” (M. Hall, Jr., and J. H. van Lint, Eds.), pp. 11 S-142, Reidel, Dordrecht, 1975. 32. N. J, A. SLOANE, Error-correcting codes and invariant theory: New applications of a nineteenth-century technique, Amer. Math. Monthly 84 (1977), 82-107. Lecture Notes in Mathematics No. 585, 33. T. A. SPRINGER, “Invariant Theory,” Springer-Verlag, Berlin/New York, 1977. 34. R. P. STANLEY, Relative invariants of finite groups generated by pseudorcflections, J. Algebra 49 (1977), 134-148. 35. R. P. STANLEY, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. SW., New Series, 1 (1979), 475-511. 36. C. W. STROM, On complete systems under certain finite groups, Bull. Amer. Mush. SIX. 37 (1931), 57cF574. 37. C. W. STROM, A complete system for the simple group G,,‘, Bull. Amer. Math. SW. 43 (1937), 438-440. 38. C. W. STROM, Complete systems of invariants of the cycbc groups of equal order and degree, Proc. Iozca Acad. Sci. 55 (1948), 287-290. 39. J. A. Toon, The invariants of a finite coIlineation group in five dimensions, Proc. Cambridge Philos. Sac. 46 (1950), 73-90. 40. D. B. WALES, Finite linear groups of degree seven-I, Canad. 1. Math. 21 (1969). 1042-1056; II, PaciJic J. Muth. 34 (1970), 207-235. 41. D. B. WALES, Linear groups of degree n containing an invoIution with two eigenvalues - 1, II, J. Algebra 53 (1978), 58-67. 42. H. WEYL, “The Classical Groups,” Princeton Univ. Press, Princeton, N. J., 1946. 43. 0. ZARISKI AND P. SAMUEL, “Commutative Algebra,” Vol. 1 and 2, Van Nostrand, Princeton, N. J., 1958 and 1960.