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NOETHER'S BOUND IN THE INVARIANT THEORY OF FINITE GROUPS
NOETHER'S BOUND IN THE INVARIANT THEORY OF FINITE GROUPS
NOETHER'S BOUND IN THE INVARIANT THEORY OF FINITE GROUPS
NOETHER'S BOUND IN THE INVARIANT THEORY OF FINITE GROUPS
NOETHER'S BOUND IN THE INVARIANT THEORY OF FINITE GROUPS
AND AND AND AND AND
VECTOR INVARIANTS OF ITERATED WREATH PRODUCTS OF SYMMETRIC GROUPS
VECTOR INVARIANTS OF ITERATED WREATH PRODUCTS OF SYMMETRIC GROUPS
VECTOR INVARIANTS OF ITERATED WREATH PRODUCTS OF SYMMETRIC GROUPS
VECTOR INVARIANTS OF ITERATED WREATH PRODUCTS OF SYMMETRIC GROUPS
VECTOR INVARIANTS OF ITERATED WREATH PRODUCTS OF SYMMETRIC GROUPS
LARRY SMITH
LARRY SMITH
LARRY SMITH
LARRY SMITH
LARRY SMITH
# AND
SCHOOL OF MATHEMATICS
UNIVERSITY OF MINNESOTA
ORDWAY LECTURES
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : Let # : G # GL(n, IF) be a faithful representation of the finite group G over the
field IF. In 1916 E. Noether proved that for IF of characteristic zero the ring of invariants
IF[V ] G is generated as an algebra by the invariant polynomials of degree at most ïGï. This
result has been generalized to the case where the characteristic of IF is greater than ïGï, or
when the characteristic of IF is prime to the order of G and the group G is solvable. In this
note we show how to refine Noether's proof to yield a more general nonmodular result. In
particular we prove that Noether's bound holds for the alternating groups in the nonmodular
case.
MATHEMATICS SUBJECT CLASSIFICATION : 13A50 Invariant Theory
Typeset by LS T E X
Let G be a finite group and # : G # GL(n, IF) a faithful representation of G over the field
IF. Via # the group G acts on the vector space V = IF n , and hence also on the graded algebra
IF[V ] of homogeneous polynomial functions on V . The subalgebra of functions, IF[V ] G , fixed
by the action of G is called the ring of invariants of G. As a general reference on invariant
theory see [3].
In [2] E. Noether proved that, whenever the characteristic of IF is zero, then IF[V ] G is a finitely
generated as an algebra over IF. In addition she gave an algorithm to construct a system of
generators using polarizations of elementary symmetric polynomials (see also [7] VIII.B.15 or
[3] §3.3). From this she deduced that IF[V ] G is generated by invariant polynomials of degree
at most ïGï. This upper bound for the degrees of the generators in a minimal generating set
is referred to as Noether's bound. This result has been extended to fields of characteristic
p #= 0 which satisfy p ³ ïGï (called the strong nonmodular case) in [6], and to solvable
groups G, whose order ïGï is not divisible by p (called the nonmodular case) in [4]. It is
known that if Noether's bound fails in the nonmodular case then it fails for a finite nonabelian
simple group (see [5]). For this reason the alternating groups have become a test case for the
conjecture that Noether's bound holds for all finite groups in the nonmodular case.
In this note we show how to refine one of Noether's argument in [2] using ideas comming
from permutation representations. This leads to results in the nonmodular case, that, for
example, apply to the alternating groups A n . For a different discussion of Noether's bound
for the alternating groups see [5].
This research was done while preparing a lecture on Noether's bound for a Crash Course on
Invariant Theory given by the author as Ordway Visitor at the School of Mathematics of the
University of Minnesota. I would like to thank the more than twenty participants in this
lecture series for their attentive and critical attitude, and probing questions.
I am very grateful to Kay Magaard for a number of tutorials on permutation representations.
In particular, the idea to use that the regular representation of a finite group preserves the
partition comming from a subgroup is due to him.
§1. Review of Noether's Proof
In this section we review the essential steps in one of E. Noether's proofs [2] of the finite
generation of rings of invariants of finite groups over the complex numbers (see also [3]
Chapter 3). So, let # : G # GL(n, IF) be a represnetation of a finite group over the field IF.
Let V = IF n and V * its vector space dual, which we regard as the space of linear forms on V .
Denote by IF(G) the group algebra of G over IF, and consider the action map
a : IF(G) Ä IF V *
# V *
defined by
a(g Ä z) = gz g ÎG , z ÎV * .
If we let G act on IF(G) Ä IF V * by
h(g Ä z) = h g Ä z h, g ÎG and z ÎV * ,
then the map a is Gequivariant. Let us write W for the vector space dual of IF(G) Ä IF V * .
Then
W = (IF(G) Ä IF V * ) * = (F(G)) * Ä IF V ** = (IF(G)) * Ä IF V = Hom IF (IF(G), V ) = map(G, V ) ,
LARRY SMITH
where map(G, V ) is the vector space of maps of G into V . Dual to the map a is the Noether
map
g : V ## map(G, V )
which is a Gequivariant map, and hence, induces a Gequivariant map
g * : IF[W] # IF[V ] ,
which we also refer to as the Noether map, though no confusion should arise. This is the map
induced by a if we regard IF[V ] as S(V * ), and similarly IF[W ] as S(W * ), where S( ) is the
symmetric algebra functor.
The action of G on map(G, V ) = W is by permutation of the elements of the underlying set W of
G, and hence extends to an action of the full permutation group S d , where d = ïGï. Choose a
basis z 1 , . . . , z n for V * , then n g Ä z i #i = 1 , . . . , n, and g ÎG o may be identified with a basis
for W * . Define the operator
E : IF[V ] # IF[W]
by the formulae
E( f ) = X A
k A E(z A )
E(z A ) = X gÎG
( g Ä z 1 ) a 1 · · · ( g Ä z n ) a n
where A = (a 1 , . . . , a n ) ÎIN 0 × · · · × IN 0 is a multiindex of nonnegative integers, and
f = X k A z A
expresses f as a sum of monomials z A = z a 1 · · · z a n with coefficients k a ÎIF. The following
lemma is a direct consequence of the definitions (for a definition of the transfer homorphism
Tr G and its properties see e.g., [3] Chapter 2).
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1 : With the preceeding notations, the operator E satisfies:
(i) E( f ) ÎIF[W] S d for all f ÎIF[V ], and
(ii) g * (E( f )) = Tr G ( f ) for all f ÎIF[V ].
As a consequence we obtain:
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2 : Let IF be a field of characteristic p and G a group of order d = ïGï.
Suppose that p does not divide d. Then, with the preceeding notations, we have that the
composition
u : IF[W] S d # IF[W] G (g * ) G
####### IF[V ] G
is onto.
PROOF PROOF PROOF PROOF PROOF : Since ïGï = d ÎIF × the averaging operator
p := 1
ïGï X gÎG
g = 1
ïGï
Tr G : IF[V ] # IF[V ] G
is defined, and is a splitting for the inclusion IF[V ] G
# IF[V ] (see e.g., [3] Section 2.4).
Therefore the transfer
Tr G : IF[V ] # IF[V ] G
is onto, and the result follows from Lemma 1.1.
2
NOETHER'S BOUND AND WREATH PRODUCTS
For a graded connected commutative Noetherean algebra A over IF let us introduce the
notation b(A) for the maximal degree of a a generator of A in a minimal generating set. This
is nothing but the degree of the Poincar’ e series, which in this case is a polynomial, of the
module of indecomposables
QA := •
A Ä A IF = •
A / ( •
A) 2 ,
where A denotes the augmentation ideal of A. (See for example [3] Chapter 4 and Section
5.1) For a representation # : G # GL(n, IF) then we write b(#) for b(IF[V ] G ), and set
b IF (G) = max n b(#)## : G # GL(n, IF) o .
Proposition 1.2 reduces finding an upper bound for b(#) to finding one for b(IF[W ] S d ). To do
this, we note that W regarded as a S d representation has the following description:
W = map(W , V ) = map(W , F n ) = Å n
map(W , IF) ,
(remember W denotes the underlying set of G) and map(W , IF) is just the defining represen
tation s of S d over the field IF. If we denote this representation by X then
IF[W ] S d = IF[Å
n
X ] S d ,
which is the ring of vector invariants of vectors of dimension n for the defining representation
of the symmetric group S d . For these invariants we have the FIRST MAIN THEOREM OF
INVARIANT THEORY FOR S d [3] Theorem 3.3.1 (the proof is in Section 3.4; for a proof in
characteristic zero see [7] VIII.B.15), which we state here in a form directly applicable to our
discussion of Noether's proof.
THEOREM 1.3
THEOREM 1.3
THEOREM 1.3
THEOREM 1.3
THEOREM 1.3 : Let d ÎIN be a positive integer and IF a field of characteristic p. Let
s : S d # GL(d, IF) be the defining representation of G over the field IF. If p > d then for any
positive integer n, b(Å n
s) = d.
Since
u : IF[W] S d = IF[Å
n
X ] S d # IF[V ] G
is onto we therefore conclude: if p > d then b(#) £ b(Å n
s) = d = ïGï.
§2. Refining Noether's Proof
In this section we show how to refine Noether's proof to yield an improved condition on
the characteristic of the ground field for the validity of b IF (G) £ ïGï (Noether's bound). We
continue to employ the notations of Section 1, in particular, # : G # GL(n, IF) is a fixed
representation of a finite group G of order d = ïGï over the field IF of characteristic p. For
the moment we make no assumptions about the relation between p and d. As in Section 1
we let W = map(W , V ), where W is the underlying G set of G, and we have the composite
u : IF[W ] S d # IF[W] G (g * ) G
####### IF[V ] G ,
where G has been embedded in S d via the regular representation reg(G) : G # S d . Notice
that if S is any subgroup of S d that contains the image of G under reg(G) in S d , then the
factorization of u
u : IF[W ] S d
# IF[W ] S
# IF[W ] G (g * ) G
# IF[V ] G ,
3
LARRY SMITH
shows that the map
IF[W] S
## IF[V ] G
is onto for ïGï ÎIF × . Thus in Noether's argument we could replace S d by S, if only we could
find an S for which we could show b(IF[W] S ) £ d. We proceed to show how to do this.
Call a subgroup K £ S d regular if K acts transitively on W and the isotropy groups K x are all
trivial, x ÎW . Regular subgroups of S d cannot sit arbitrarily in S d . To explain what is meant
by this, choose a chain of maximal subgroups
1 = G k < G k-1 < · · · < G 1 < G 0 = G
in G, i.e., G i is a maximal subgroup of G i-1 for i =1 , . . . , k - 1. Let d i =ïG i ï and e i =ïG i : G i+1 ï
for i = 0 , . . . , k - 1. Note that ïGï = d =
k-1
Q
i=1
e i . Let
W = W 1 # , . . . , #W e 0
be the decomposition of W into the left cosets of G 1 in G. This partition of W must be preserved
by the action of G on W via the regular representation, and therefore reg(G)(G) must sit in
the largest subgroup of S d that preserves this partition. This subgroup is
S d 1
× · · · × S d 1
###### e 0 ##### # S e 0
= S d 1 # S e 0
.
If we denote by G the underlying set of G 1 then the representation W as S d 1 #S e 0
representation
may be described as follows: Note
W = map(W , V ) = map(W 1 # · · · # W e 0
, V ) = Å e 0
map(G , V )
and that map(G , V ) is just the direct sum of n copies of the defining representation of S d 1
over
IF. Therefore
IF[W] S d 1 ×···×S d 1 = Ä n
IF[Y ] S d 1 ,
where Y is the linear representation defined over IF from the representation of S d 0
as the
permutations of G . Let us write A = IF[Y ] S d 1 so that
IF[W ] S d 1 #S e 0 ~ = IF[W ] S d 1 ×···×S d 1 S e 0 = Ä n
IF[Y ] S d 1 S e 0
= Ä n
A S e 0
.
Notice that
Ä n
A S e 0
are just the invariants of S e 0
acting by permutation of the tensor factors. We can employ the
linearization process of [4] and [5] to obtain information on b( Ä
n
A S e 0
) provided p # ïS e 0
ï,
which is equivalent to p > e 0 . Here is how this works.
Let A be generated by homogeneous forms of degree less than or equal to b = b(A) say. Let
Z be the vector space dual of
b
Å
i=1
A i , where A i denotes the homogeneous component of A in
degree i. Note that the natural map Z * ## A extends to an epimorphism IF[Z] ## A. Hence
there is an induced map of algebras with S e 0
action
Ä e 0
IF[Z] ## Ä e 0
A
4
NOETHER'S BOUND AND WREATH PRODUCTS
and, if p > e 0 , i.e., ïS e 0
ï is not divisible by p, then the induced map of the invariant subalgebras
Ä e 0
IF[Z] S e 0
## Ä e 0
A S e 0
is also surjective. Finally, if U denotes the IFlinear representation corresponding to the
defining representation of S e 0
, then
Ä e 0
IF[Z] S e 0
~ = Ä n
IF[U] S e 0
,
which are the vector invariants of the defining representation of S e 0
on vectors of dimension
n. Therefore, since we have already assumed that p > e 0 , we can apply Theorem 1.3 to
conclude that b( Ä
e 0
IF[Z] S e 0
) = be 0 , where we have reaccounted for the fact that b = b(A) is
the maximal degree of an element in Z. Hence b((Ä e 0
A) S e 0 ) £ be 0 . We therefore obtain
b(IF[W] S d 1 #S e 0 ) £ b(A)e 0
provided only that p > e 0 .
If, in addition, p > d 1 , we could apply Theorem 1.3 to S d 1
also, and could conclude that
b(IF[W] S d 1 ×···×S d 1 ) £ d 1 . Combining all this would then give
b(#) £ b(IF[W] S d 1 #S e 0 ) £ d 1 e 0 = d .
If k > 1 the partition
W = W 1 # · · · #W e 0
can be refined further by using the maximal subgroup G 2 < G 1 to partition the sets W i , (Recall
that the W i are the left cosets of G 1 in G 0 = G) into the left cosets of G 2 in G 1 . Denote the
resulting partition of W i by
W i = W i,1 # · · · #W i,e 1
.
note that the action of G on W must preserve the multipartition
W =
e 0
G i=1
e 1
G j=1
W i, j ,
so reg(G)(G) must belong to the subgroup of S d consisting of all permutations preserving this
multipartition. This subgroup is:
(S d 2 # S e 1
) # S e 0
.
Continuing in this way, we find that the image in S d of G under the regular representation
preserves the multipartition
W =
e k-1
G
i k-1 =1
G · · ·
e 0
G
i 1 =1
W i k-1 ···i 1
obtained from the full chain of maximal subgroups, and therefore is a subgroup of
· · · (S e k-1 # S e k-2
) # · · ·) # S e 1
) # S e 0 .
Inductively applying the preceeding arguments, we obtain the degree bound
b IF[W ] ···(S e k-1 #S e k-2 )#···)#S e 1 )#S e 0 £
k-1
Y i=0
e i = d ,
provided that e 0 !, e 1 ! , . . . , e k-1 ! are invertible in IF. Therefore we have shown:
5
LARRY SMITH
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1 : Let IF be a field of characteristic p, G a finite group, and
1 = G k < G k-1 < · · · G 1 < G 0 = G
a chain of maximal subgroups in G, i.e., G i is a maximal subgroup of G i-1 for i = 1 , . . . , k - 1.
Let e i = ïG i : G i+1 ï for i = 0 , . . . , k - 1. If
p > max e k-1 , . . . , e 0 ,
then b IF (G) £ ïGï.
REMARK REMARK REMARK REMARK REMARK : Note that the condition
p > max
e k-1 , . . . , e 0 ,
in the preceeding theorem implies that p # ïGï.
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2 : Let IF be a field of characteristic p, n ÎIN a positive integer, and A n the
alternating group on n letters. If p > n then b IF (A n ) £ ïA n ï, i.e., Noether's bound holds for
the alternating groups A n , n ÎIN, in the nonmodular case.
PROOF PROOF PROOF PROOF PROOF : Apply Theorem 2.1 to the chain of maximal subgroups
1 = A 2 < A 3 < · · · < A n-1 < A n
and note that p > n is equivalent to p # ïA n ï = n!
2 .
§3. The Fine Structure of Orbits and Orbit Chern Classes
The method employed in Section 2 leads to a refinement of the orbit Chern classes, a tool
introduced in [6] for constructing invariants. (We make use of several standard constructions
for permutation representations, and refer to [8] for basic facts about permutation represen
tations.) To explain this, suppose that G is a finite group and W is a finite transitive G set. A
system of imprimitivity for W is a partition of W ,
W = W 1 # , . . . , #W e ,
that is preserved by the action of G. This means, for each g ÎG and 1 £ i £ e, that g(W i ) = W g(i) ,
where 1 £ g(i) £ e, and the correspondence i # g(i) is a permutation of the set
1 2 , . . . , e
.
The subsets W i , i = 1 , . . . , e, are called the blocks of the system of imprimitivity. Since G
acts transitively on W it is easy to see that the blocks all have the same number of elements,
say c. In fact, if K = G x 1
is the isotropy group of a fixed point x 1 ÎW 1 , then W may be identified
with the Gspace G / K , and the blocks with the cosets of K in H , where
H = n h ÎG#h(W 1 ) = W 1 o .
So a system of imprimitivity for W corresponds to a chain of subgroups K < H < G.
Next, let # : G # GL(n, IF) be a representation of a finite group G and B # V * an orbit of G
in the space of linear forms V * on V = IF n . Suppose the permutation representation of G on
B is imprimitive and
B = B 1 # · · · # B e
is a system of imprimitivity for B. For each integer j, 1 £ j £ c introduce the polynomial
u B j
(X ) = Y
bÎB j
(X + b j ) ÎIF[V ][X ]
6
NOETHER'S BOUND AND WREATH PRODUCTS
and write
u B j
(X ) = X
r+s=c
c r (B j )X s ,
where c is the block size. Call the forms c i (B j ) the block Chern classes. If I = (i 1 , . . . , i c ) is
a multiindex of nonnegative integers let r I denote the Ith polarized elementary symmetric
polynomial in e vector variables, each of dimension c. Then
c I (B 1 , . . . , B e ) := r I (c i 1 (B 1 ) , . . . , c i e (B e )) ÎIF[V ] G ,
and is called the Ith Chern class of the system of imprimitivity
B = B 1 # · · · # B e ,
or a fine orbit Chern class of B. The proof of Theorem 2.1 shows:
THEOREM 3.1
THEOREM 3.1
THEOREM 3.1
THEOREM 3.1
THEOREM 3.1 : Let IF be a field of characteristic p, G a finite group, and
1
= G k < G k-1 < · · · G 1 < G 0 = G
a chain of maximal subgroups in G, i.e., G i is a maximal subgroup of G i-1 for i = 1 , . . . , k - 1.
Let e i = ïG i : G i+1 ï for i = 0 , . . . , k - 1. If
p > max
e k-1 , . . . , e 0 ,
then, for any representation # : G # GL(n, IF), the ring of invariants IF[V ] G is generated as
an algebra by fine orbit Chern classes.
Let us illustrate this with an example. The example is chosen because it is a generic example
(see [3] Section 3.2 Example 1 and the discussion preceeding Example 2) of a ring of invariants
that is not genertaed by orbit Chern classes.
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1 (L. E. Dickson) : Consider the subgroup of GL(2, IF 3 ) generated by the matrices
A = 0 1
-1 0 , B = -1 1
1 1 ÎGL(2, IF 3 ) .
Set
C = AB = 1 1
1 -1 .
Since
A 2 = B 2 = C 2 = -I ,
where I is the identity matrix, the subgroup of GL(2, IF 3 ) generated by A and B is isomorphic
to the quaternion group Q 8 of order 8.
Inspection of the 8 matrices in Q 8 shows that every nonzero vector in IF 2
3 occurs exactly once
as a first column, so that Q 8 acts transitively on IF 2
3 \ 0 . There are only two orbits of Q 8
acting on V * , and they are {0}, and V * \ {0}. The only Chern classes are therefore the two
Dickson polynomials
d 2,1 = x y 9 - x 9 y
x y 3 - x 3 y
, d 2,0 = (x y 3 - x 3 y) 2 ,
where {x, y} is the dual of the canonical basis of IF 2
3 . These polynomials cannot generate
IF[x, y] Q 8 , which therefore is not generated by ordinary orbit Chern classes.
7
LARRY SMITH
Denote by W the orbit V * \ 0 of G. This orbit is imprimitive, and the system of imprimitivity
corresponding to the subgroup Z / 4 in Q 8 generated by A consists of the two blocks
W 1 =
±x, ± y
W 2 = ±(x + y), ±(x - y) .
Each block has two nonzero block Chern classes:
c 2 (W 1 ) = -(x 2 + y 2 )
c 4 (W 1 ) = x 2 y 2
c 2 (W 2 ) = (x 2 + y 2 )
c 4 (W 2 ) = (x 2 - y 2 ) 2 .
These give the following fine Chern classes for the orbit W :
F = c 4 (W 1 ) + c 4 (W 2 ) = (x 2 + y 2 ) 2
Q = c 4 (W 1 )c 2 (W 2 ) + c 2 (W 1 )c 4 (W 2 ) = -(x 2 + y 2 )(x 4 - y 4 ).
There is another system of imprimitivity for W corresponding to the subgroup Z / 4 generated
by B, whose blocks are
L 1 = ±x, ±(x - y)
L 2 = ± y, ±(x + y) .
From the block Chern classes of this system of imprimitivity we obtain the fine orbit Chern
class
Y = c 4 (L 1 ) + c 4 (L 2 ) = x 2 (x - y) 2 + y 2 (x + y) 2 .
The fine orbit Chern classes F , Y ÎIF 3 [x, y] Q 8 form a system of parameters and (see the
discussion in [3] of Example 1 in Section 3.2)
IF 3 [x, y] Q 8 = IF 3 [F , Y] Å IF 3 [F , Y] · Q .
Therefore the fine orbit Chern classes F , Y , and Q generate IF[x, y] Q 8 as an algebra.
8
NOETHER'S BOUND AND WREATH PRODUCTS
References
[1] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, with computational
assistance from J.G. Thackray, Atlas of Finite Groups, Claerndon Press, Oxford, 1985.
[2] E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77
(1916), 89--92.
[3] L. Smith, Polynomial Invariants of Finite Groups, (second printing) A.K. Peters Ltd.,
Wellesley, MA 1995, 1997.
[4] L. Smith, Noether's Bound in the Invariant Theory of Finite Groups, Arch. der Math.
66 (1996), 89 -- 92.
[5] L. Smith, Putting the Squeeze on the Noether Gap: The Case of the Alternating Groups
A n , Preprint, The University of Minnesota and AGInvariantentheorie, 1998.
[6] L. Smith and R. E. Stong, On the Invariant Theory of Finite Groups: Orbit Polynomials
and Splitting Principles, J. of Algebra 110 (1987), 134--157.
[7] H. Weyl, The Classical Groups (second edition), Princeton Univ. Press, Princeton, 1946.
[8] H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
Larry Smith
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
email:SMITH@MATH.UMN.EDU
and
# Mathematisches Institut der Universit ˜
at
D 37073 G˜ ottingen
Federal Republic of Germany
email: LARRY@SUNRISE.UNIMATH.GWDG.DE
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