Unspecified Journal
Volume 00, Number 0, Pages 000-000
S ????-????(XX)0000-0
ON THE HILBERT IDEAL
MARA D. NEUSEL
Abstract.We prove the Hilbert number conjecture.
1.Introduction
Let F be a field, G a finite group. Consider a faithful representation
ae : G ,! GL(n, F).
It induces a G-action on the vector space V = Fn by matrix multiplication and t*
*hus
on the dual space V *. We extend this action additively and multiplicatively to*
* the
symmetric algebra on V *, denoted by F[V ]. Once we choose a basis x1, . .,.xn *
*for
V *we can identify
F[V ] ~=F[x1, . .,.xn]
as the ring of polynomial functions in n indeterminates over the field F. We are
interested in the subalgebra
F[V ]G ,! F[V ]
of G-invariant polynomials. By a classical result due to E. Noether the ring of*
* in-
variants F[V ]G is finitely generated as an F-algebra. Furthermore, since our a*
*ction is
F-linear we can assume that the generators of F[V ]G are homogeneous polynomial*
*s.
We define fi(F[V ]G ) to be the maximal degree of an algebra generator in a min*
*imal
algebra generating set. This number is usually called the Noether number. If the
group order, |G|, is invertible in F, the nonmodular case, then fi(F[V ]G ) |*
*G|,
independent of the representation, see Theorem 2.3.3 in [8]. In the modular cas*
*e,
i.e., |G| = 0 2 F, the value of fi(F[V ]G ) usually depends on the degree of th*
*e rep-
resentation or the order of the field and the group order, see [3] for an overv*
*iew on
degree bounds, see [4] and [5] for recent results.
In this paper we consider not the ring of invariant polynomials but the Hilbe*
*rt
ideal ______
h(V, G) = (F[V ]G) F[V ]
which is by definition generated by all homogeneous invariant polynomials of po*
*si-
tive degree. The Hilbert number, denoted by fi(h(()V, G)) is the maximal degree*
* of
an ideal generator in a minimal ideal generating set. In [1] it has been conjec*
*tured
that the Hilbert ideal is, in contrast to the ring of invariants, always genera*
*ted by
invariants of degree at most group order. Of course, in the nonmodular case this
____________
Received by the editors December 3, 2007.
2000 Mathematics Subject Classification. Primary 13A50 Commutative Rings.
Key words and phrases. invariant theory of finite groups, degree bounds, Hil*
*bert ideal, Hilbert
number.
Oc0000 (copyright holder)
1
2 MARA D. NEUSEL
follows from Noether's bound. Furthermore, this conjecure was proven for permu-
tation representations in [2] and for indecomposable representations of Z=p in *
*[9].
In this paper we prove this conjecture for all representations. While typing t*
*his
manuscript I learned that the same conjecture has been proven simulteneously by
Symonds, [10].
2. The Proof
Consider the induced FG-module FG F V and the canonical inclusion
V ,! FG F V .
This induces a G-equivariant projection
jG : F[FG F V ] -! F[V ]
of F-algebras. Upon restriction to the respective rings of invariants we obtain*
* the
classical Noether map
jGG: F[FG F V ]G -! F[V ]G .
This map remains surjective in the classical nonmodular case. However in the
modular case it often fails to be surjective, see [6] and [7] for a thorough st*
*udy of
the Noether map.
In the nonmodular case the ring of invariants and hence the Hilbert ideal is
generated by polynomials of degree at most group order. Thus we restrict our
attention to the modular case and obtain a commutative diagram as follows.
jG
F[FG F V ] i F[V ]
[ [
F[V ]G
[
jGG
F[FG F V ]G i Im(jGG)
Note that G acts by permutations on FG F V . Thus by Fleischmann's result,
[2], we have
fi(h(FG F V, G)) |G|.
Therefore
_______
(?) fi(Im (jGG)) |G|,
_______
where Im (jGG) F[V ] denotes the ideal generated by the invariants of positive
degree in the image of the Noether map.
Furthermore, the kernel of jG is generated by linear forms. Moreover, Ker(jG )
F[FG V ] is a prime ideal of height n|G|-|G|. Thus we obtain the following chai*
*ns
of ideals
jG
F[FG F V ] i F[V ]
[_______ ___[___
Ker(jG ) j-1G(Im (jGG)) Im (jGG)
[
h(FG V, G)
ON THE HILBERT IDEAL 3
Let I h(V, G) F[V ] be the ideal generated by all G-invariant polynomials
of positive degree at most group order. We want to show that I = h(V, G). By
Inequality (?) we have
_______
Im (jGG) I h(V, G) F[V ].
Thus we can extend the above diagram to the following
jG
F[FG F V ] i F[V ]
[ [
j-1G(h(V, G)) h(V, G)
[ [
j-1G(I) I
[_______ ___[___
Ker(jG ) j-1G(Im (jGG)) Im (jGG)
[
h(FG V, G)
We collect some properties of the ideals involved in the following lemma.
Lemma 2.1. The ideals I h(V, G) F[V ] as well as the ideals j-1G(I)
j-1G(h(V, G)) F[FG V ] are closed under the action of the group G.
Proof.The first two ideals are by definition generated by invariant polynomials.
Thus they are closed under the G-action. Since the map jG is G-equivariant, the
latter two are also closed under the G-action.
Next we present a series a reduction arguments. The first one shows that it is
enough to prove that the inverse images of I and h(V, G) in F[FG V ] are equa*
*l.
Lemma 2.2. If j-1G(h(V, G)) = j-1G(I) then h(V, G) = I.
Proof.By construction we have I h(V, G). To prove the reverse inclusion let
f 2 h(V, G). Then there exists an element F 2 j-1G(h(V, G)) = j-1G(I) with
f = jG (F ) 2 jG (j-1G(h(V, G))) = jG (j-1G(I)) = I
as desired.
The third lemma shows that it is enough to consider elements that map onto
G-invariant polynomials under the Noether map.
Lemma 2.3. If F 2 j-1G(I) for all F 2 F[FG V ] with jG (F ) = f 2 F[V ]G , th*
*en
j-1G(I) = j-1G(h(V, G)).
Proof.The inclusion is true by construction. To prove the reverse inclusion l*
*et
H 2 j-1G(h(V, G)) be an arbitrary element. Then
X
jG (H) = ffifi2 h(V, G),
where fi 2 F[V ]G have positive degree and ffi 2 F[V ]. By assumption for each i
there exists an Fi2 j-1G(I) such that jG (Fi) = fi. Thus
X
H = AiFi+ K 2 j-1G(I),
for some jG (Ai) = ffi and K 2 Ker(jG ).
4 MARA D. NEUSEL
We write
(x1g1 . . .x1g|G|)
x2g1 . . .x2g|G|
FG V = spanF .. .
. ..
xng1 . . .xng|G|
for some enumeration of the group elements G = {g1, . .,.g|G|}. Withoutloss of
generality we assume that g1 = 1 2 G is the identity element. Then the group G
acts on the first index, and the Noether map is given by
jG (xigj) = gjxi2 F[V ].
We choose the reverse lexicographic orderPx1g1< . .<.x1g|G|< x2g1< . .<.xng|G|.
We note that for any polynomial f = aixi11. .x.inn2 F[V ], where ai2 F, the*
*re
exists an inverse image
X
F = j-1G(f) = aixi111.x.i.nn12 F[FG V ]
whose terms are monomials in the elements of the first column of (xigj)igj.
Lemma 2.4. Let f 2 h(V, G), and assume that the inverse image F of f consisting
of terms in the first column is an element in j-1G(I). Then all inverse images *
*of f
lie in j-1G(I).
Proof.Since two elements F and F 0of j-1G(f) differ by an element in the kernel*
* of
jG which in turn is contained in j-1G(I), the result follows.
We want to prove our statement degree-wise by induction on the term order.
The next result constitutes the induction start for every degree.
Proposition 2.5. Let F 2 j-1G(h(V, G)) have degree d. Let F be minimal with re-
spect to the reverse lexicographic order. Assume that the ideals j-1G(I) and j-*
*1G(h(V, G))
coincide in degree less than d. Then F 2 j-1G(I).
Proof.The unique minimal monomial in F[FG V ] of degree d is xd11. If G is a
p-group we assume without loss of generality that G fixes x1, then x1 2 I and t*
*hus
x112 j-1G(I).
Now, let G be an arbitrary group with p-Sylow subgroup P fixing x1. Order the
elements of G such that g2 2 P is a nontrivial element. Then jG (x1g2) = g2x1 =*
* x1.
We have to consider two cases:
CASE: xd1162 j-1G(h(V, G))
In this case we have
F = xd-111(x11- x1g2) 2 Ker(jG ) j-1G(h(V, G)).
Its leading term is xd-111x1g2and is thus the smallest leading term of an eleme*
*nt
of degree d in j-1G(h(V, G)). Furthermore, j-1G(I) contains the kernel of jG a*
*nd
hence it contains xd-111(x11- x1g2). Finally, note that F is the unique polynom*
*ial
in j-1G(h(V, G)) with this leading term.
CASE: xd112 j-1G(h(V, G))
Then jG (xd11) = xd12 h(V, G). Thus if d |G| it follows that xd12 I by defi*
*nition
of I, and hence x112 j-1G(I). Otherwise we have
X
xd1= ffifi
i
ON THE HILBERT IDEAL 5
for some G-invariant polynomials fi. If the degree of the ffi's is positive, we*
* have
fi2 I by assumption, and thus
X
xd11= AiFi+ K 2 j-1G(I)
i
for jG (Fi) = fi, jG (Ai) = ffi, and an element K in the kernel. Finally assume*
* we
have X
xd1= f0 + ffifi.
i
Then the relative transfer gives
X
TrGP(xd1) = |G : P |f0 + TrGP(ffi)fi.
i
Since d > |G| we can apply Lemma 2.2 in [2] and obtain
TrGP(xd1) 2 I.
Hence X
f0 = __1___|G(:TPr|GP(xd1) - TrGP(ffi)fi) 2 I.
i
So, finally we get
X
xd112 j-1G(xd1) = j-1G(f0 + ffifi) j-1G(I)
i
as desired.
We need one more preparation:
Lemma 2.6. Let M, M0 2 F[FG V ] be monomials of the same degree. Assume
that M consists of elements in the first column, and assume that M > M0 in the
reverse lexicographic order. Then for all g 2 G we have that gM > gM0.
Proof.M is a monomial in the first column, say
M = xi11,1.x.i.nn,1.
Thus
gM = xi11,g.x.i.nn,g
is a monomial in the gth column.
We turn to the monomial M0. Since M0 is smaller it must look like
M0 = xinn,1.x.i.j+1j+1,1xkjj,1N
where N is a monomial avoiding the columns j through n and kj < ij. Thus gM0
looks like
gM0 = xinn,g.x.i.j+1j+1,gxkjj,ggN
where gN still avoids columns j through n. Thus gM0 < gM.
The following result proves the Hilbert ideal conjecture for p-groups in char*
*ac-
teristic p.
Theorem 2.7. Let P be a p-group, and assume that ae(P ) GL(n, F) is in lower
triangular form, i.e., gxi2 spanF{x1, . .,.xn}. Then
h(V, P ) = I.
6 MARA D. NEUSEL
Proof.By Lemma 2.2 it is enough to show that j-1G(h(()V, G)) = j-1G(I). The
inclusion " " is valid by construction. To show the reverse inclusion we proc*
*eed
by double induction on degree and term order.
Let F 2 j-1P(h(V, P )) be an element of degree d. If d |G| there is nothing*
* to
show. Thus we assume that the degree of F is strictly larger than the group ord*
*er.
By Lemma 2.4 we can assume that F lives in the first column and by Lemma
2.3 we can assume that jP (F ) is invariant.
Since the induction start is proven in Proposition 2.5 we assume that for all
elements of smaller degree or same degree and lower in term order we have shown
that it is in j-1G(I). We write F as a sum of monomials
F = M0 + M1 + . .+.Mk
and without loss of generality we assume that M0 > M1 > . .>.Mk.
Let r = max{1, . .,.n} such that the variable xr1 appears in F .
CASE: M0 = xdr1
We consider the top orbit Chern class of xr
ctop(xr) 2 F[V ]P
which is a polynomial of degree, say, t with leading term xdr. Note that t cann*
*ot
exceed the group order. We find an inverse image C 2 j-1P(ctop(xr)) in the first
column
C 2 F[x11, x21, . .,.xr1].
We note that this polynomial is by construction in j-1P(I). Then
F - Cd-t
lies in j-1P(h(V, P )) and has lower leading term than F . Thus by induction
F - Cd-t 2 j-1P(I)
and thus F 2 j-1P(I) as desired.
CASE: Assume that M0 = xir1N0, where N0 has positive degree and xr1 does
not divide N0, M1, . .,.Mk.
Since jP (F ) is invariant we have that gF - F 2 Ker(jP ) j-1P(I) for all g*
* 2 P .
Since xr1appears solely in M0, we have that xr|jP (M0) = m0 but it does not div*
*ide
any of the other jP (Mi) for i = 1, . .,.k. Since jP (gF - F ) = gjP (F ) - jP *
*(F ) = 0
the terms with the same xr-degree must cancel. Thus
jP (gM0 - M0) = g(xr)ign0 - xirn0 = xir(gn0 - n0) + R,
where the remainder R has lower xr-degree and jG (N0) = n0. Thus gn0 - n0 = 0
and n0 is an invariant of strictly smaller degree than m0. Thus N0 2 j-1P(I),
therefore M0 2 j-1P(I). Hence
F - M0 2 j-1P(h(V, P ))
has strictly smaller term-order, and is by induction in j-1P(I). Thus F 2 j-1P(*
*I).
CASE: xr1 divides M0, . .,.Mj but none of the other terms of F
Write
M0 = xi1r1N0, . .,.Mj = xijr1Nj
If i1 > max{i2, . .,.ij}, then the preceding argument goes through without chan*
*ge.
Thus we assume that i1 = i2 = . .=.il for some 1 < l j. Since M0 > M1 >
ON THE HILBERT IDEAL 7
. .>.Ml there exists a largest index s, 1 s r - 1, such that the monomials
differ. Hence we can write
M0 = xirr1xir-1r-1,1.x.i.s+1,1s+1,1xis,1s1N0, . .,.Ml= xirr1xir-1r-1,1.x.i.*
*s+1,1s+1,1xjs,1s1Nl.
Similar to the above we obtain that
xis,1s1N0 + . .+.xjs,1s1Nl
maps under jP to an invariant. By construction it has lower degree than F and is
thus in j-1P(I). Here we can apply the argument of the preceding case to find t*
*hat
N0 2 j-1P(I).
Finally, we are prepared to prove the Hilbert number conjecture for all groups
and representations.
Theorem 2.8. Let ae : G ,! GL(n, F) be a faithful representation of a finite gr*
*oup
over an arbitrary field F. Then
h(V, G) = I
and hence the Hilbert number is bounded above by the group order
fi(h(V, G)) = fi(I) |G|.
Proof.By Lemma 2.2 it is enough to show that j-1G(h(()V, G)) = j-1G(I). The
inclusion " " is valid by construction. To show the reverse inclusion we proc*
*eed
as above by double induction on degree and term order.
Let F 2 j-1G(h(V, G)).If degree if F is at most group order then F 2 j-1G(I) *
*by
construction. Thus we assume that deg(F ) = d > |G|.
The polynomial of minimal leading term of degree d in j-1G(h(V, G)) is contai*
*ned
in j-1G(I) by Proposition 2.5.
Thus assume that the leading term of F is not minimal, and all polynomials of
degree less than d or of degree d with smaller leading term than F are in j-1G(*
*I).
We denote the leading term of F by LT(F ). We note that by Lemmata 2.3 and
2.4 we may assume that F is a polynomial in the first column mapping under jG
onto an invariant polynomial.
We note also that the leading term of gF is strictly larger than F for all g *
*6= 1,
because F lives in the first column, while gF lives in the gth column.
Since F maps to an invariant polynomial under jG we have that gF - F 2
Ker(jG ) j-1G(I). Thus
LT (gF - F ) = LT(gF ) 2 LT(Ker(jG )) LT(j-1G(I)),
where LT(I) denotes the ideal of leading terms of the ideal I.
Hence there exists an element H 2 j-1G(I) such that LT(H) = LT(gF ). If H = gF
we are done by Lemma 2.3.
Otherwise, consider gF - H 2 j-1G(h(V, G)) with LT(gF - H) < LT(gF ).
If LT(gF - H) < LT(F ), then gF - H 2 j-1G(I) by induction on the term order,
and thus gF 2 j-1G(I). Hence F 2 j-1G(I). Otherwise, we consider the polynomial
(gF - F ) - H which is an element in j-1G(I).
If LT(gF -F -H) > LT(F ), we pick an element K1 2 j-1G(I) such that LT(K1) =
LT(gF - F - H). Then
LT(gF - F - H - K1) < LT(gF - F - H)
8 MARA D. NEUSEL
and we can proceed inductively and find K1, . .,.Kl2 j-1G(I) such that
LT(gF - F - H - K1 - . .-.Kl) LT(F ).
Thus we may assume without loss of generality that LT(gF - F - H) LT(F ). If
there exists an element H such that LT(gF - F - H) = LT(F ), we have
LT(gF - F - H - F ) < LT(gF - F - H) = LT(F ).
By construction gF - F - H - F 2 j-1G(h(V, G)) and thus by induction on the
term order in j-1G(I). Since H as well as gF - F are in j-1G(I) we conclude that
F 2 j-1G(I).
Finally we have to take care of the case LT(gF - F - H) < LT (F ) for all
H 2 j-1G(I) and all g 2 G.
Recall that F is a polynomial in the first column and that jG (F ) = f is inv*
*ariant.
We write it as a sum of monomials
F = M0 + M1 + . .+.Mk
and assume without loss of generality that M0 > M1 > . .>.Mk in the term-order.
Then
gF = gM0 + gM1 + . .+.gMk
has leading term gM0 by Lemma 2.6. Since the leading term of gF - F - H is
strictly smaller than the leading term of F we have
X
H = gM0 - M0 + Ki2 j-1G(I),
Ki