Unspecified Journal
Volume 00, Number 0, Pages 000-000
S ????-????(XX)0000-0
DEGREE BOUNDS AND THE REGULAR REPRESENTATION
DEDICATED TO THE MEMORY OF ANDERS J. FRANKILD [1971-2007].
MARA D. NEUSEL
Abstract.Let ae : G ,! GL(V ) be a permutation representation of a finite
group G. We describe universal algebra generating set for the associated*
* ring
of polynomial invariants over any field as long as the invariants are Co*
*hen-
Macaulay. This has many consequences for permutation invariants as well *
*as
for invariants of arbitrary representations. In particular, it implies *
*Killius'
Conjecture for permutation invariants and Schmid's Inequality for the ge*
*neral
nonmodular case.
1.Introduction
Let ae : G ,! GL(n, F) be a faithful representation of a finite group G over *
*a field
F. The representation ae induces an action of G on the vector space V = Fn of
dimension n and hence on the ring of polynomial functions F[V ] = F[x1, . .,.xn*
*],
where we chose a basis x1, . .,.xn of the dual space V *. Our interest is focus*
*ed on
the subring of invariants
F[V ]G = {f 2 F[V ]G |gf = f 8g 2 G},
which is a graded connected Noetherian commutative algebra, see [11] for more
information on rings of polynomial invariants of finite groups.
We denote by fi(F[V ]G ) the maximal degree of an F-algebra generator of F[V *
*]G
in a minimal homogeneous generating set. By Noether's Bound we have that
fi(F[V ]G ) |G|
in the nonmodular case (where |G| 2 Fx ), see Theorem 2.3.3 in [11]. Many re-
finements of this statement have been proven over the past years as well as many
results valid in the modular case (where |G| = 0 2 F), see [8] for a survey on *
*these
matters. This paper has been motivated by the question whether there is a "wors*
*t"
representation.
In [12] Schmid showed that
fi(F[V ]G ) fi(F[FG]G )
whenever the ground field F has characteristic zero. It is known that this ineq*
*uality
does not remain valid in the modular case, where char(F) = p||G|, see Example 1*
* of
Section 3.2 in [11] for a counterexample. However, it was a long standing conje*
*cture
____________
Received by the editors February 12, 2007.
2000 Mathematics Subject Classification. Primary 13A50, Secondary.
Key words and phrases. Invariant Theory of Finite Permutation Groups, Permut*
*ation Repre-
sentation, Cohen-Macaulay, Regular Representation.
I would like to thank the referee for many helpful comments.
Oc0000 (copyright holder)
1
2 MARA D. NEUSEL
that Schmid's inequality holds in the general nonmodular case. Indeed, it was
generalized to the strong nonmodular case (i.e., char(F) = 0 or char(F) > |G|) *
*by
Smith, see [15], and for abelian groups G in the general nonmodular case by Sez*
*er,
see [13]. Furthermore, Knop proved this result for nonmodular representations
where the characteristic of the ground field is zero or p > 3=8|G| + 1, see Cor*
*ollary
7.5 in [5].
In this paper we prove Schmid's inequality for the general nonmodular case.
Indeed, the proof follows easily from Corollary 2.12. Section 2 of this paper *
*is
devoted to the proof of that result as well as of Theorem 2.11 which gives us
precise information on the set of algebra generators. In Section 3 we deduce se*
*veral
results on degree bounds for polynomial invariants. In particular we prove Kill*
*ius'
Conjecture in Theorem 3.1 for permutation representations and we prove Schmid's
Inequality in Theorem 3.2 for the general nonmodular case.
2. Rings of Invariants of Permutation Representations
Let ae : G ,! GL(n, K) be a permutation representation of a finite group G. We
note that ae factorizes through the prime field F = Fp, resp. F = Q:
ae : G,! GL(n, K)
& "
GL(n, F).
We obtain for the respective rings of invariants
K[V ]G = K F F[V ]G .
Thus a minimal set of F-algebra generators of F[V ]G minimally generates K[V ]G
as an K-algebra as well. Therefore we can assume in what follows that the fields
involved are prime fields.
Furthermore, any permutation representation ae : G ,! GL(n, Q) factorizes thr*
*ough
GL(n, Z)
ae : G,! GL(n, Q)
& "
GL(n, Z).
Taking coefficients mod p we obtain a faithful representation G ,! GL(n, Fp) wh*
*ich
we will call ae also. Our goal is to explicitly describe a common algebra gener*
*ating set
for both rings, Q[V ]G and Fp[V ]G , in the case that both rings are Cohen-Maca*
*ulay.
(Of course, Q[V ]G is always Cohen-Macaulay, see Theorem 5.5.2 in [11].)
We note that any element in a ring of permutation invariants is a sum of orbit
sums. Therefore, we can consider any invariant f 2 Fp[V ]G as an element in
Z[V ]G Q[V ]G . On the other hand, after clearing denominators we can consider
an invariant f 2 Q[V ]G as a (possibly zero) element f 2 Fp[V ]G .
Let s1, . .,.sm 2 Z[V ] Q[V ]. We note that s1, . .,.sm are algebraically i*
*nde-
pendent in Z[V ] if and only if they are algebraically independent in Q[V ]. We*
* have
the following lemma.
Lemma 2.1. Let s1, . .,.sm 2 Z[V ] Q[V ]. If s1, . .,.sm are algebraically in*
*de-
pendent in Fp[V ], then they are algebraically independent in Q[V ].
SCHMID'S RESULT 3
Proof.Let s1, . .,.sm 2 Fp[V ] be algebraically independent. Assume there is a
polynomial P (X1, . .,.Xm ) 2 Q[X1, . .,.Xm ] such that
P (s1, . .,.sm ) = 0.
By clearing denominators we can assume that P (X1, . .,.Xm ) 2 Z[X1, . .,.Xm ].
Without loss of generality we assume that the greatest common divisor of the
coefficients of P (X1, . .,.Xm ) is 1 (or P (X1, . .,.Xm ) = 0). Since the sj's*
* are alge-
braically independent over Fp we have that P (X1, . .,.Xm ) 0 mod p. This
implies that the coefficients of P (X1, . .,.Xm ) are divisible by p > 1. Hence
P (X1, . .,.Xm ) = 0 2 Z[X1, . .,.Xm ].
We note that a ring of polynomial invariants F[V ]G of a permutation represen-
tation is a finite module over the ring of invariants of the full symmetric gro*
*up in
n letters, n, in its defining representation, i.e., the extension
F[V ] n ,! F[V ]G
is finite, see Exercise (1) in Section 12.6 of [9]. Denote by e1, . .,.en the e*
*lementary
symmetric functions in x1, . .,.xn. Then
F[V ] n = F[e1, . .,.en]
is a Noether normalization for F[V ]G . We note that this is true for any groun*
*d field,
i.e., the elementary symmetric functions form a universal system of parameters *
*for
any ring of permutation invariants over any field F. Furthermore we have
Xk
F[V ]G = F[e1, . .,.en]fi
i=1
for some homogeneous invariants f1, . .,.fk. We note that this sum is direct if*
* and
only if F[V ]G is Cohen-Macaulay, see Theorem 5.1.3 in [11].
Next we show that we can choose also a universal set of secondary invariants
across characteristics as long as the rings of invariants involved remain Cohen-
Macaulay.
Proposition 2.2. Assume that Fp[V ]G is Cohen-Macaulay. Let s1, . .,.sn, f1, . *
*.,.fk
be homogeneous invariant polynomials of positive degree. If
Mk
(o) Fp[V ]G = Fp[s1, . .,.sn]fi,
i=1
then
Mk
(O) Q[V ]G = Q[s1, . .,.sn]fi.
i=1
L k
Proof.Assume that Fp[V ]G = i=1Fp[s1, . .,.sn]fi. Consider the extension
Xk
Q[s1, . .,.sn]fi Q[V ]G .
i=1
We note that Q[s1, . .,.sn] remains a polynomial algebra by Lemma 2.1. Since
Poincar`e series of rings of permutation invariants are independent of the char*
*acter-
istic of the ground field (see Proposition 3.2.7 in [11]), it is enough to show*
* that
this sum is direct.
4 MARA D. NEUSEL
We proceed by induction on k. Let k = 1 and assume that
(1) ff1f1 = 0
for some ff1 = ff1(s1, . .,.sn) 2 Q[s1, . .,.sn]. By clearing denominators we *
*can
assume that
ff1(X1, . .,.Xn) 2 Z[X1, . .,.Xn].
Without loss of generality we assume that p 6 |ff1(X1, . .,.Xn). From Equation *
*(1)
we have that ff1f1 0 mod p. By assumption it follows that ff1(s1, . .,.sn)
0 mod p. Since s1, . .,.sn are algebraically independent in Fp[V ]G it follows *
*that
ff1(X1, . .,.Xn) 2 Fp[X1, . .,.Xn] is the zero polynomial. Since p does not div*
*ide
ff1(X1, . .,.Xn) we have that ff1(X1, . .,.Xn) = 0 and hence ff1(s1, . .,.sn) =*
* 0 2
Z[s1, . .,.sn] Q[s1, . .,.sn] as claimed.
Next let k > 1 and assume there is a relation
Xk
(2) ffifi= 0
i=1
for some ffi = ffi(s1, . .,.sn) 2 Q[s1, . .,.sn], i = 1, . .,.k. By clearing de*
*nomina-
tors we assume that the polynomialsPffi(X1, . .,.Xn) have integer coefficients.*
* By
Equation (2) it follows that ki=1ffifi 0 mod p. Since the sum (o) is direc*
*t it
follows that
ffi(s1, . .,.sn) 0 mod p.
Since s1, . .,.sn are algebraically independent in Fp[V ]G , it follows that th*
*e poly-
nomials ffi(X1, . .,.Xn) 2 Fp[X1, . .,.Xn] are zero. Thus
ffi(X1, . .,.Xn) = pdiff0i(X1, . .,.Xn) 2 Z[X1, . .,.Xn]
for some di 2 N with p 6 |ff0i(X1, . .,.Xn). Without loss of generality set dk*
* =
min{d1, . .,.dk}. Dividing the Equation (2) by pdk leads to
Xk
ff00ifi= 0
i=1
where p does not divide ff00k= ff0k(s1, . .,.sn). Hence ffk(s1, . .,.sn) = 0 an*
*d we are
done by induction.
Remark 2.3. The preceding result shows that a set of invariants s1, . .,.sn for*
*ming a
homogeneous system of parameters for Fp[V ]G , forms also a system of parameters
for Q[V ]G . Furthermore once we have chosen a universal system of parameters
we obtain a universal set of homogeneous secondary generators f1, . .,.fk across
characteristics as long as the ring of invariants remains Cohen-Macaulay.
Remark 2.4. The converse of the preceding result remains true if and only if
s1, . .,.sn remain algebraically independent as elements in F[V ].
Remark 2.5. If the system of parameters s1, . .,.sn 2 Fp[V ]G is not homogeneou*
*s,
but none of the si's has a constant term, then the preceding result remains true
with some slight modification: By assumption the system of parameters s1, . .,.*
*sn
is contained in the irrelevant ideal m Fp[V ]G . Thus by localizing we obtain
Mk
Fp[V ]Gm= Fp[s1, . .,.sn]nfi
i=1
SCHMID'S RESULT 5
where n = m \ Fp[s1, . .,.sn], see Proposition 2.2.11 in [2]. The preceding pr*
*oof
then shows that
Mk
Q[V ]Gm0= Q[s1, . .,.sn]n0fi
i=1
for m0 Q[V ]G the irrelevant ideal and n0= m0\ Q[s1, . .,.sn] its contraction.
Thus we have found a common set of primary and secondary invariants. Of
course it follows that
Fp[V ]G = F[s1, . .,.sn, f1, . .,.fk]
and
Q[V ]G = Q[s1, . .,.sn, f1, . .,.fk].
However, in both cases the set of algebra generators given should not be minima*
*l.
Our next goal is to show that in both rings the same secondary invariants can be
deleted from the list of algebra generators.
Proposition 2.6. We continue with the same notation as above. Assume that the
invariants s1, . .,.sn, f1, . .,.fk are homogeneous. Then
fk 2 Fp[s1, . .,.sn, f1, . .,.fk-1]
if and only if
fk 2 Q[s1, . .,.sn, f1, . .,.fk-1].
Proof.Assume that
fk 2 Q[s1, . .,.sn, f1, . .,.fk-1].
We write
X
(?) fk = pI(s1, . .,.sn)fI
I
where fI = fi11. .f.ik-1k-1and pI are polynomials in the elements of the system*
* of
parameters. For degree reasons we have for each I appearing in the sum (?)
k-1X
(*) fI = ~Ifk + qIi(s1, . .,.sn)fi
i=1
for some coefficients ~I 2 Q. By linear independence we obtain
X
~I = 1.
I
Thus for all I0 such that ~I06= 0 we have
k-1X
fk = ~-1I0fI0+ ~-1I0qI0,ifi
i=1
and hence
k-1M
Q[V ]G = Q[s1, . .,.sn]fi Q[s1, . .,.sn]fI0
i=1
6 MARA D. NEUSEL
whenever ~I0 6= 0. We claim that there exists at least one I0 so that we can do
the same over the prime field Fp. For that we multiply Equation (*) by a minimal
~ 2 Z such that
k-1X
~fI0 = (~~I0)fk + ~qI0,ifi
i=1
is an equation with integer coefficients. We reduce modulo p. If ~ 0 mod p t*
*hen
~~I0 0 mod p
as well as
~qI0,i 0 mod p
because the sum (o) is direct. This contradicts that ~ was chosen to be minimal.
Thus ~ is not divisible by p. We find
k-1X
~fI0 = (~~I0)fk + ~qI0,ifi 6= 0 mod p.
P i=1
Since ~I0= 1 and no denominator of Equation (*) is divisible by p (recall ~ 6*
*= 0
mod p), there exists an index I0 such that ~~I06= 0 mod p. Therefore
k-1X
fk (~~I0)-1~fI0- (~~I0)-1~qI0,ifi mod p,
i=1
and
k-1M
Fp[V ]G = Fp[s1, . .,.sn]fi Fp[s1, . .,.sn]fI0
i=1
as desired.
We prove the converse and assume that
X
fk pI(s1, . .,.sn)fI mod p,
I
for some polynomials pI(s1, . .,.sn) 2 Fp[s1, . .,.sn] and I = (i1, . .,.ik-1).*
* For
every multi-index I appearing in this sum we have
k-1X
fI ~Ifk + qI,i(s1, . .,.sn)fi mod p
P i=1
with ~I 1 mod p. Thus for all I0 such that ~I06= 0 mod p we can write
I
k-1X
fk = ~-1I0fI0- ~-1I0 qI0,i(s1, . .,.sn)fi mod p.
i=1
Lifting to Z[V ]G gives the equation
k-1X
fI0 = ~I0fk + qI0,i(s1, . .,.sn)fi+ pH
i=1
for some invariant H 2 Z[V ]G . Thus as a rational invariant we can write H as
k-1X
H = ~fk + qH,i(s1, . .,.sn)fi
i=1
SCHMID'S RESULT 7
for suitable ~ 2 Q and qH,i(s1, . .,.sn) 2 Q[s1, . .,.sn]. Thus we have
k-1X k-1X
fI0 = (~I0+ p~)fk + qI0,ifi+ pqH,ifi.
i=1 i=1
If ~I0+ p~ 6= 0 for some index I0 we are done. Otherwise we have ~ = -~I0_pand
thus
k-1X k-1X
fI0 = qI0,ifi+ pqH,ifi
i=1 i=1
for all I0. We multiply this equation with the minimal ff 2 N such that
k-1X k-1X
fffI0 = ffqI0,ifi+ ffpqH,ifi
i=1 i=1
has integer coefficients. Modulo p this relation has to become trivial (for oth*
*erwise
fI0 2 k-1i=1Fp[s1, . .,.sn]fi) and thus ff 0 mod p. Let ff = psff0such that*
* p 6 |ff0.
Then ps+1 is the maximal p-power dividing a denominator in qH,i. Thus there is
an i0 such that ffpqH,i0is not zero modulo p. This is a contradiction, because *
*the
sum (o) is direct.
Remark 2.7. The preceding result actually shows that a secondary invariant fk t*
*hat
is redundant as an algebra generator can be replaced by some fI0 = fi11. .f.ik-*
*1k-1.
Thus inductively we find
Fp[V ]G = Fp[s1, . .,.sn, f1, . .,.fl]
if and only if
Q[V ]G = Q[s1, . .,.sn, f1, . .,.fl].
Remark 2.8. Assume that the invariants s1, . .,.sn, f1, . .,.fk are inhomogeneo*
*us,
but have no constant terms. Then they are contained in the maximal ideal gener-
ated by all homogeneous invariants of positive degree. By localizing at that ma*
*ximal
ideal, the preceding proof extends to that case, cf. Remark 2.5.
We are now prepared to prove that a minimal algebra generating set of Fp[V ]G
generates also Q[V ]G .
Proposition 2.9. Let Fp[V ]G = Fp[h1, . .,.hm ]. Then
Q[V ]G = Q[h1, . .,.hm ].
Proof.By assumption we have
Mk
Fp[V ]G = Fp[s1, . .,.sn]fi= Fp[h1, . .,.hm ].
i=1
Without loss of generality we assume that none of the hi's is redundant. Since *
*this
ring of invariants is Cohen-Macaulay there are Fp-linear combinations of the hi*
*'s
Lj(h1, . .,.hm ) j = 1, . .,.n
such that L1, . .,.Ln 2 Fp[V ]G forms a system of parameters. If the Li's are n*
*ot
homogeneous we need to localize at the irrelevant ideal in order to be able to
8 MARA D. NEUSEL
proceed. To simplify notation we assume they are homogeneous and write
mM Ml
Fp[V ]G = Fp[L1, . .,.Ln, hn+1, . .,.hm ] = Fp[L1, . .,.Ln]hi Fp[L1, . *
*.,.Ln]Fi
i=n+1 i=1
for suitable invariants Fi. By Proposition 2.2 we have
mM Ml
Q[V ]G = Q[L1, . .,.Ln]hi Q[L1, . .,.Ln]Fi.
i=n+1 i=1
Thus by Lemma 2.6 we have
Q[V ]G = Q[L1, . .,.Ln, hn+1, . .,.hm ]
as desired.
Remark 2.10. We could try to reverse the preceding proof, and construct linear
combinations of the algebra generators L1, . .,.Ln 2 Q[V ]G . The proof then go*
*es
through if and only if L1, . .,.Ln remain algebraically independent in Fp[V ]G .
Let
{x1, . .,.xn} = B1 t . .t.Br
be a decomposition into disjoint G-orbits. We denote by eijthe jth elementary
symmetric function in the elements of Bi. Then the eij's form a homogeneous
system of parameters for F[V ]G over any field F. Thus the preceding results ca*
*n be
combined to the following:
Theorem 2.11. Let F and K be fields. Let ae : G ,! GL(n, F) be a permutation
representation of a finite group G and consider the "same" representation over *
*the
field K. Assume both rings of invariants, F[V ]G and K[V ]G , are Cohen-Macaula*
*y.
Then
F[V ]G = F[s11, . .,.sr|Br|, f1, . .,.fl]
if and only if
K[V ]G = K[s11, . .,.sr|Br|, f1, . .,.fl].
Proof.
From this description we find the following degree bound for the algebra gene*
*r-
ators.
Corollary 2.12. Assume that G acts by permutations and Fp[V ]G is Cohen-
Macaulay. Then
fi(Fp[V ]G ) = max{fi(Q[V ]G ), ffiV },
where ffiV is the maximal degree of an elementary symmetric function in the orb*
*it
elements of B1, . .,.Br that is not redundant as algebra generator.
Proof.This is immediate from the description given in Theorem 2.11.
SCHMID'S RESULT 9
3. Applications to general invariant theory
We apply the results of the preceding sections to several problems of degree
bounds in invariant theory of arbitrary representations of finite groups.
First we note that we can prove Killius' Conjecture for permutation represent*
*a-
tions, see [4] or Problem 7.3 in [8].
Theorem 3.1 (Killius' Conjecture for Permutation Invariants). Let ae : G ,!
GL(n, F) be a permutation representation of a finite group G. Assume that F[V ]G
is Cohen-Macaulay. Then
fi(F[V ]G ) |G|.
Proof.Since the length of an orbit Bi does not exceed the group order this resu*
*lt
follows from Corollary 2.12.
Theorem 3.2. Let ae : G ,! GL(n, F) be a faithful representation of a finite gr*
*oup.
Let V = Fn be a projective FG-module and let F[nFG]G be Cohen-Macaulay. Then
fi(F[V ]G ) fi(F[FG]G ).
Proof.Since V is a projective FG-module, the Noether map
jGG: F[nFG]G -! F[V ]G
is surjective, see Proposition 3.1 in [10]. Thus fi(F[nFG]G ) fi(F[V ]G ), cf*
*. proof
of Proposition 4.1 in [10].
By Corollary 2.12 we have
fi(F[nFG]G ) = max{fi(Q[nQG]G ), ffinFG} == max{fi(C[nCG]G ), ffinFG}.
By Weyl's Theorem the invariants of the n-fold regular representation are gener*
*ated
by polarizations and hence
fi(C[nCG]G ) = fi(C[CG]G ).
Since FG ,! nFG is a direct summand we have by Theorem 1.2 in [7] that
CM - defect(F[FG]G ) CM - defect(F[nFG]G ) = 0.
Thus F[FG]G is Cohen-Macaulay, and a second application of Corollary 2.12 gives
fi(F[FG]G ) = max{fi(Q[QG]G ), ffiFG} = max{fi(C[CG]G ), ffiFG}.
Combining these inequalities gives
fi(F[V ]G )fi(F[nFG]G ) = max{fi(C[nCG]G ), ffinFG}
= max{fi(C[CG]G ), ffinFG}
(1)
max{fi(C[CG]G ), ffiFG} = fi(F[FG]G ),
where (1) follows since an elementary symmetric function is irredundant as an
algebra generator for F[FG]G if it is so for F[nFG]G .
Theorem 3.3 (Schmid's Inequality). Let ae : G ,! GL(n, F) be a faithful represe*
*n-
tation of a finite group G. Assume that the ground field F has characteristic z*
*ero
or p 6 ||G|. Then fi(F[V ]G ) fi(F[FG]G ).
Proof.Since ae is a nonmodular representation the FG-module V = Fn is projectiv*
*e,
and F[nFG]G is Cohen-Macaulay. Thus the Theorem 3.2 applies.
10 MARA D. NEUSEL
Remark 3.4. We note that the argument used in the preceding proof shows that
rings of invariants of sums of permutation representations that are Cohen-Macau*
*lay
are generated by polarizations independent of the ground field, see [3], [5], a*
*nd [14]
for recent progress on polarizations.
Remark 3.5. Finally, note that the preceding results yield also that the relati*
*ve
Noether bound holds for Cohen-Macaulay permutation invariants, cf. [8].
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*00), 2199-
2201.
Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lub-
bock, Texas 79409
E-mail address: Mara.D.Neusel@ttu.edu