THE INVARIANTS OF MODULAR INDECOMPOSABLE
REPRESENTATIONS OF Zp2
MARA D. NEUSEL AND M"UF`IT SEZER
Abstract.We consider the invariant ring for an indecomposable representa-
tion of a cyclic group of order p2over a field F of characteristic p. We*
* describe
a set of F-algebra generators of this ring of invariants, and thus deriv*
*e an upper
bound for the largest degree of an element in a minimal generating set f*
*or the
ring of invariants. This bound, as a polynomial in p, is of degree two.
Introduction
Let ae : G ,! GL(n, F) be a faithful representation of a finite group G. Deno*
*te
by V = Fn the n-dimensional vector space over F. Then G acts via ae on V , which
in turn induces an action on the dual space V *. This extends to the symmetric
algebra S(V *) = F[V ]. The algebra of invariant polynomials
F[V ]G = {f 2 F[V ] | g(f) = f, 8g 2 G} F[V ]
is a graded connected commutative Noetherian subalgebra of F[V ], see [11] for a
general treatment of the subject. Let
fi(F[V ]G)
denote the smallest integer d such that F[V ]G is generated as an F-algebra by
homogeneous polynomials of degree at most d. In the nonmodular case, i.e., |G| 2
Fx, we have that
fi(F[V ]G) |G|,
see Theorem 2.3.3 in [11] and the references there. This bound does not remain
valid in the modular case, i.e., when |G| 0 2 F. Indeed, Richmann constructed
modular representations V with arbitrarily large fi(F[V ]G), see [12]. In other*
* words,
there cannot be a degree bound for fi(F[V ]G) that depends only on the group, s*
*ee
[10] for an overview in these matters.
In this paper we want to study rings of invariants of cyclic p-groups Zprof o*
*rder
pr over a field F of finite characteristic p. There are exactly pr indecomposa*
*ble
Zpr-modules, which we denote by V1, V2, . .,.Vpr, see Chapter II in [1], where *
*Vn
has dimension n as a vector space over F.
We note that G"obel's bound gives, of course, a bound on the degrees of a gen-
erating set of F[Vpr]Zprfor any p and r, see Corollary 3.4.4 in [11]. In this c*
*ase we
have ` r'
fi(F[Vpr]Zpr) max{pr, p2 }.
This bound depends on the dimension of the representation which coincides in th*
*is
case with the order of the group.
___________
Date: August 15, 2007, 15 h 2 min.
1
2 M. D. NEUSEL AND M. SEZER
If r = 1 and G = Zp is the cyclic group of prime order, then a general degree
bound for a minimal generating set of the ring of invariants for any Zp-module V
was given in in [5]. This bound is sharp, as the case of the regular representa*
*tion
of Z3 shows.
For the case r = 2 much less is known: In [9] we find an explicit description*
* of
the ring of invariants F[V3]Z4. This was generalized to F[Vp+1]Zp2in [13]. Furt*
*her-
more, in [8] we find an explicit description of the ring of invariants of the r*
*egular
representation of Z4.
We want to extend this study and find an upper bound for fi(F[Vn]Zp2) for any
indecomposable Zp2-module Vn. In Section 1 we derive an upper bound for the top
degree of the coinvariant ring. In Section 2 we describe a set of F-algebra gen*
*erators
for F[Vn]Zp2. This description yields an upper bound for fi(F[Vn]Zp2). This bou*
*nd
transpires to be quadratic in p. We postpone some technical calculations to Sec*
*tion
3.
For the remainder of the paper, we assume that G ~=Zp2and that H ~=Zp is
the non-trivial subgroup.
We choose a basis x1, . .,.xn for Vn*and write
F[Vn] ~=F[x1, . .,.xn].
Next, we choose a generator oe for the group G. Then
(
oexi= x1 for i = 1, and
xi+ xi-1 for 2 i n.
Set = oe - 1. Then we have
(
(xi) = 0 for i = 1, and
xi-1 for 2 i n.
The various transfer maps involved are given by the following formulae
p2-1X p-1X p-1X
TrG = oei, TrH = oeip, and TrGH= oei.
i=0 i=0 i=0
We use the graded reverse lexicographic order with xi> xi-1for i = 2, . .,.n.
1.An Upper Bound for fi(F[V ]G)
_Since_G is a finite group, the extension F[V ]G ,! F[V ] is finite. Denote by
(F[V ]G) F[V ] the Hilbert ideal, i.e., the ideal generated by the invariants*
* of
positive degree. Thus the coinvariants
_____
F[V ]G = F[V ]=(F[V ]G)
form a finite-dimensional vector space over F. Thus its Hilbert series is a pol*
*yno-
mial. In this section we want to derive an upper bound_on_its degree.
Note that the Hilbert series of the Hilbert ideal (F[V_]G)_ F[V ] coincides *
*with
the Hilbert series of the ideal I of leading terms of (F[V ]G), see Theorem 15.*
*26 in
[3]. Thus it suffices to find an upper degree bound for F[V ]=I.
If n p then Vn is an indecomposable G=H ~=Zp-module and thus F[Vn]G ~=
F[Vn]G=H. Therefore we restrict our attention to the case n > p in what follows.
We need two somewhat technical constructions:
INVARIANTS OF MODULAR INDECOMPOSABLE REPRESENTATIONS 3
Let r be a positive integer with max{n - 2p, 1} r n - p. Choose a monomial
m = u1u2. .u.2p-2of degree 2p - 2 in F[xd, . .,.xr], where d = max{1, r - p + 1*
*}.
We may assume that u1 u2 . . .u2p-2. We define wi,0by
(
ui= (wi,0) if 1 i p - 1, and
p(wi,0) if p i 2p - 2,
and set
wi,j= j(wi,0) 1 i 2p - 2, j 2 N0.
For a 2p - 2-tuple ff = [ff(1), ff(2), . .,.ff(2p - 2)] 2 N2p-2of natural numbe*
*rs we
define
2p-2Y
wff= wi,ff(i).
i=1
Thus we can write
p-1Y 2p-2Y
m = u1u2. .u.2p-2= wi,1 wi,p= wff0,
i=1 i=p
where ff0(i) = 1 if 1 i p - 1 and ff0(i) = p if p i 2p - 2.
Let S {1, 2, . .,.2p - 2} be a subset and set
Y
XS = wi,0.
i2S
We consider the following polynomial
X
T1(m) = (-1)|S|XS0TrG(XS),
S {1,...,2p-2}
where S0denotes the complement of S in {1, 2, . .,.2p - 2}.
Proposition 1. The leading term of T1(m) is m.
Proof.The proof of this result is postponed to Section 3.
_____
The polynomials T1(m) are by construction in the Hilbert ideal (F[V ]G) F[V*
* ].
Thus the preceding result tells us that any monomial divisible by some m is in *
*the
ideal I of leading terms of the Hilbert ideal.
We need another, similar, construction. Since n > p, the G-module Vn*decom-
poses into a direct summand of p indecomposable H-modules:
Vn*= Vn*n-p+1 . . .Vn*n.
Moreover, Vn*jis generated as a H-module by xj for j = n, . .,.n - p + 1.
For each i = n - p + 1, . .,.n, we define the H-norms
Y
NHi= oexi.
oe2H
Note that every NHihas degree p and coincides with the respective top orbit Che*
*rn
classes if i p.
Choose a monomial
Y
M = NHij2 F[NHd, . .,.NHn-1]
1 j p-1
4 M. D. NEUSEL AND M. SEZER
of degree p-1 as a polynomial in these norms. For 1 j p-1 define Wj = NHij+*
*1.
Let S {1, . .,.p - 1} be a subsetQand S0 its complement. Then similarly to the
contruction of T1(m) we set XS = j2SWj, and obtain a polynomial T2(M) as
follows. X
T2(M) = (-1)|S|XS0TrGH(XS).
S {1,...,p-1}
Proposition 2. The leading monomial of T2(M) is the leading monomial of M.
Proof.The proof of this result is postponed to Section 3.
As for T1(m) the polynomials T2(M) lie in the Hilbert ideal associated to F[V*
* ]G.
Thus the preceding result shows that any monomial divisible by the leading term
of some M is contained in the ideal I of leading terms of the Hilbert ideal.
This enables us to prove the desired result:
Theorem 3. Let n = tp + r > p, where 1 t p and 0 r < p are integers.
Then the top degree of F[Vn]G is bounded above by 3p2+ (2t - 4)p - 3t.
_____
Proof.The Hilbert series of the Hilbert ideal (F[V ]G)___F[V_] coincides with t*
*he
Hilbert series of the ideal, I, of leading terms of (F[V ]G). Thus in order to *
*find
a bound on the degrees of the coinvariants it suffices to find a degree bound f*
*or
F[V ]=I.
To that end, let m1m2xlnbe a monomial that is not in the lead term ideal of t*
*he
Hilbert ideal. Without loss of generality we assume that m1 2 F[x1, . .,.xn-p] *
*and
m2 2 F[xn-p+1, . .,.xn-1].
Let max{n-2p, 1} r n-p and m a monomial of degree 2p-2 in F[xd, . .,.xr],
where d = max{1, r - p + 1}. Then Proposition 1 shows that m appears as leading
term of some T1(m). Since T1(m) is contained in the Hilbert ideal it follows th*
*at
the degree of m1 is at most t(2p - 3).
Similary, the polynomials T2(M) are in the Hilbert ideal and thus by Proposit*
*ion
2, m2 is not divisible by the lead term of a product of p - 1 norms NHi, where
d i n - 1. Therefore the degree of m2 is at mostQ(p - 2)p + (p - 1)2.
Finally xp2nis the leading term of the norm NGn= oe2Goexn. Therefore l p2-*
*1.
Hence
deg(m1m2xln) t(2p - 3) + (p - 2)p + (p - 1)2+ p2- 1 = 3p2+ (2t - 4)p - 3t
as claimed.
Corollary 4. Let n > p. Then the image of the transfer Im(TrG) F[V ]G is
generated by forms of degree at most 3p2+ (2t - 4)p - 3t.
Proof.We write the ring of polynomials as a module over the ring of invariants *
*as
follows X
F[V ] = F[V ]Ghi.
finite
We note that by construction the hi's form a basis of F[V ]G. Since |G| = p2 0
mod p, we have that TrG(F[V ]G) = 0. Thus the image of the transfer is generat*
*ed
by the TrG(hi)'s, and the result follows from Theorem 3.
INVARIANTS OF MODULAR INDECOMPOSABLE REPRESENTATIONS 5
2.Generators for Rings of Invariants
We apply the results found in the previous section to rings of invariants. We
start with an explicit calculation for the regular representation.
Example 5. Consider the regular representation of Zp2. Its ring of invariants *
*is
generated by forms of degree at most 5p2- 7p. This can be seen as follows:
By Theorem 3.3 in [4], F[Vp2]G=ImTrG ' F[Vp]H , where the isomorphism scales
the degrees by 1_p. It is shown in [5] that F[Vp]H is generated by invariants o*
*f degree
2p - 3. Hence F[Vp2]G=ImTrG is generated by classes of degree at most (2p - 3)p.
On the other hand, Corollary 4 tells us that Im(TrG) is generated by invariants*
* of
degree at most 5p2- 7p. Hence
fi(F[Vp2]G) max{(2p - 3)p, 5p2- 7p} = 5p2- 7p
as claimed.
We proceed to the general case. As in Section 1, let n > p and
Vn*= Vn*n-p+1 . . .Vn*n
be an H-module decomposition. For i 2 {n-p+1, . .,.n} we have that xigenerates
Vn*ias H-module.
Lemma 6. The image of the relative transfer, ImTrGH, is generated by ImTrG and
G-invariants of degree at most 3p2- 3p.
Proof.Let f 2 F[Vn]H . By Lemma 2.12 the ring F[Vn]H is generated as a module
over F[NHd, . .,.NHn] by invariants of degree at most p2 - n and the image of t*
*he
transfer TrH. Thus f can be written as
X X
(O) f = pi(NHd, . .,.NHn)bi+ qj(NHd, . .,.NHn)TrH(gj)
for some polynomials pi, qj 2 F[NHd, . .,.NHn], H-invariants bi of degree at mo*
*st
p2- n and suitable gj 2 F[Vn]. Since
X X
qj(NHd, . .,.NHn)TrH(gj) = TrH( qj(NHd, . .,.NHn)gj)
we find that
X X
TrGH( qj(NHd, . .,.NHn)TrH(gj)) = TrG( qj(NHd, . .,.NHn)gj)
is in the image of the transfer TrG. Thus we need to take care of the first sum*
*mand
and assume without lost of generality that
X
(O) f = pi(NHd, . .,.NHn)bi
We sort (O) by monomials in the norms and obtain
X
f = bJNHJ,
J
where bJ is a sum of suitable bi's and thus is still an H-invariant of degree a*
*t most
p2- n.
We claim that the degree of NHJas a polynomial in NHnis at most p-1. Otherwise
set U = (NHn)p. Then
H bJNH
TrGH(bJNJ_UNGn) = NGnTrGH(___J_U)
6 M. D. NEUSEL AND M. SEZER
can be written in termsHof G-invariants of strictly smaller degree. On the oth*
*er
hand LM (bJNHJ- bJNJ_UNGn) < LM (bJNHJ). Therefore
H bJNH
TrGH(bJNHJ) = TrGH(bJNHJ- bJNJ_UNGn) + TrGH(___J_UNGn)
yields that TrGH(bJNHJ) can be eliminated from a generating set for ImTrGH.
Similarly, we claim that the degree of the bJNHJ's as a monomial in {NHi|i =
d, . .,.n - 1} is strictly less than p - 1.QAssume the contrary and let Uj 2 {N*
*Hi|i =
d, . .,.n - 1} for 1 j p - 1. Set U = 1 j p-1Uj. Then we have
H bJNH X
TrGH(bJNJ_UT2(U1. .U.p-1))= TrGH___J_U (-1)|S|XS0TrGH(XS)
S {1,...,p-1}
X bJNH
= TrGH(XS)TrGH(___J_(-1)|S|XS0).
S {1,...,p-1} U
H
Hence, TrGH(bJNJ_UT2(U1. .U.p-1)) can be written in terms of G-invariants of
H
smaller degree. By Proposition 2 we have that LM (bJNHJ- bJNJ_UT2(U1. .U.p-1)) <
LM (bJNHJ). Therefore the equation
H bJNH
TrGH(bJNHJ) = TrGH(bJNHJ- bJNJ_UT2(U1. .U.p-1)) + TrGH(___J_UT2(U1. .U.p-1))
yields that TrGH(bJNHJ) can be eliminated from a generating set for ImTrGH.
Thus, for any multi-index J, the degree (in the x's) of bJNHJis bunded above *
*by
p2- n + (p - 2)p + p(p - 1) = 3p2- 3p - n < 3p2- 3p
as claimed.
Theorem 7. Let Vn be an indecomposable G-module. Let n = tp + r > p, where
1 t p and 0 r < p are integers. Then
fi(Vn) max{3p2+ (2t - 4)p - 3t, 3p2- 3p}.
Proof.By the periodicity result of TheoremQ1.2 in [14], F[Vn] is modulo the FH-
projective submodules generated by NGn= oe2Goex and invariants of degree less
than p2. Thus F[Vn]G is generated by the G-norm NGn, invariants of degree less *
*than
p2 and image ImTrGHof the relative transfer, since the fixed pointed of project*
*ive
modules are in the image of the relative transfer.
By the previous lemma ImTrGHis generated by invariants of degree at most
3p2- 3p together with ImTrG. Therefore it follows from Corollary 4 that
fi(Vn) max{3p2+ (2t - 4)p - 3t, 3p2- 3p},
as desired.
Remark 8. We note that for n p the representation
ae : G -! GL(n, F)
has kernel Zp. Thus F[V ]G ~=F[V ]H . Hence this ring of invariants is generate*
*d by
forms of degree at most 2p - 3 by [5].
INVARIANTS OF MODULAR INDECOMPOSABLE REPRESENTATIONS 7
Remark 9. Furthermore, if n = p + 1 we find in [13] an explicit generating set *
*of
the ring of invariants and we read off
fi(F[Vp+1]G) 2p2- 2p - 1.
For p = 3 the authors of [13] refer to a Magma calculation and for fi(F[V4]G) =*
* 9.
For p = 2 we find fi(F[V3]G) = 4 by [9]. We note that
p2 2p2- 2p - 1 3p2- 3p max{3p2+ (2t - 4)p - 3t, 3p2- 3p}.
Note carefully that the degree bound given above is polynomial in p of degree*
* 2.
We thus state the following problem.
Conjecture 10. Let V be an indecomposable Zpr-module. Then fi(F[V ]Zpr) is
bounded above by a polynomial in p of degree r.
3.The Leading Terms of T1(m) and T2(M)
In this section we want to identify the leading terms of the polynomials T1(m)
and T2(M) as described in Propositions 1 and 2. We start by identifying the
coefficients of monomials that appear in T1(m).
P
Lemma 11. The coefficients of T1(m) = ff2N2p-2cffwffare given by
X 2p-2Y`l '
cff= .
0 l p2-1i=1 ff(i)
Proof.Since oelis an algebra automorphism we have that
2p-2Y X
(z ) (wi,0- oel(wi,0)) = (-1)|S|XS0oel(XS).
i=1 S {1,...,2p-2}
Thus summing over 0 l p2- 1 yields
X 2p-2Y
T1(m) = (wi,0- oel(wi,0)).
0 l p2-1i=1
Since we have1 ` ' ` ' ` '
(wi,0- oel(wi,0)) = -lwi,1- l2wi,2- l3wi,3- . .-. llwi,l,
the desired equality follows.
___________
1This equation can be easily verified by induction on l 0. If l = 0 the equ*
*ation is trivial. For
l = 1 we have
wi,0- oewi,0= wi,0= wi,1.
Assume that l > 1. Then by induction we obtain
wi,0- oelwi,0= (wi,0- oel-1wi,0) - oel-1 wi,0
il -j1 il -j1
= -(l - 1)wi,1- 2 wi,2- . .-.l - 1wi,l-1- oel-1wi,1
il -j1 il -j1 il -j1 il *
*-j1
= -(l - 1)wi,1- 2 wi,2- . .-.l - 1wi,l-1- wi,1- 1 wi,2- . .-.l -*
* 1wi,l
ilj ilj ilj
= -lwi,1- 2 wi,2- 3 wi,3- . .-.l wi,l
as desired.
8 M. D. NEUSEL AND M. SEZER
Lemma 12. Let ff 2 N2p-2. If ff(i) > 1 for some 1 i p - 1, then wff< wff0=
m.
Proof.Since u1 u2 . . .u2p-2, it suffices to show that
Yk
wi,ff(i)< u1u2. .u.k
i=1
for some 1 k 2p - 2. Let j denote the smallest integer such that ff(j) > 1.
SinceQj p - 1, it follows that ui = wi,1for i < j and wj,ff(j)< uj. Therefore
j
i=1wi,ff(i)< u1u2. .u.jand the result follows.
Lemma 13. Let ff 2 N2p-2. If ff(i) 2p for some 1 i 2p - 2, then wff<
wff0= m.
Proof.By Lemma 12 it is enough to show the result for i p. Since ui= wi,p2
F[xr-p+1, . .,.xr], it follows that wi,ff(i)2 F[x1, . .,.xr-p]. Therefore wffco*
*ntains a
variable that is smaller than all variables that appear in m.
Lemma 14. Let ff, fi be two elements in N2p-2such that ff(i) fi(i) for 1 i
2p - 2. Then
(1)wff wfi, and
(2)wff= wfiif and only if ff = fi.
Proof.Since wi,ff(i) wi,fi(i)for 1 i 2p - 2, we have
2p-2Y 2p-2Y
wff= wi,ff(i) wi,fi(i)= wfi.
i=1 i=1
For the second assertion observe that if ff(i) < fi(i) for some 1 i 2p - 2,*
* then
wi,ff(i)< wi,fi(i). Hence
2p-2Y 2p-2Y
wff= wi,ff(i)< wi,fi(i)= wfi
i=1 i=1
as desired.
Lemma 15. The coefficient of cff0of the monomial wff0in T1(m) is 1.
P lp-1
Proof.By Lemma 11, we have cff0= 0 l p2-1lp-1 p . For 0 l p2 - 1,
write l = l1p + l2, where 0 l1, l2 < p. Then we find
X `l'p-1 X `l1p + l2'p-1
lp-1 = (l1p + l2)p-1
0 l p2-1 p 0 l1,l2 p-1 p
(1) X p-1 p-1
l2 l1 mod p
0 l1,l2 p-1
(2)
1 mod p,
where (1) follows from
` ' ` '` '
(?) st a1a b1 mod p
2b2
INVARIANTS OF MODULAR INDECOMPOSABLE REPRESENTATIONS 9
( for any two integers 0 s, t < p2 with s = a1p + b1 and t = a2p + b2, where
0 ai, bi< p), see [2], and (2) from
(
X -1 mod p ifp - 1| c;
(o) lc
0 l p-1 0 mod p otherwise,
(for any natural number c), see Theorem 119 in [6].
We are now able to proof Proposition 1:
Proposition 16. The leading term of T1(m) is wff0, and thus LM (T1(m)) = m =
wff0.
Proof.The second statement follows from the first because cff0= 1 by Lemma 15.
We proceed by showing that wff0 wffand cff6= 0 implies ff = ff0.
By Lemmas 12 and 13 we may assume ff(i) = 1 for 1 i p - 1 and ff(i) < 2p
for p i 2p - 2. From Lemma 11 we have
X 2p-2Y`l ' X 2p-2Y`l '
cff= = lp-1 .
0 l p2-1i=1 ff(i) 0 l p2-1 i=p ff(i)
For p i 2p - 2 write ff(i) = aip + bi with 0 bi < p and 0 ai 1. Set
l = l1p + l2 with 0 l1, l2 < p.
X 2p-2Y`l1p +'l2
cff= (l1p + l2)p-1
0 l1,l2 p-1 i=p aip + bi
X p-12p-2Y`l1'`l2'
l2
0 l1,l2 p-1 i=p ai bi
(P p-1 p-2Q 2p-2l
l2 l1 i=p b2 if ai= 1 for all,i
P 0 l1,l2ip-1p-1Q2p-2l2P i k
0 l2 p-1l2 i=p bi( 0 l1 p-1l1) 0otherwise,
where the last equation follows since k an integer not divisible by p - 1. Thus*
* we
may assume that ai= 1 for p i 2p - 2. It follows that ff(i) = p + bi p = f*
*f0(i)
for p i 2p-2. Moreover ff(i) = ff0(i) = 1 for 1 i p-1. Now ff = ff0foll*
*ows
from Lemma 14.
From this Proposition 2 can be easily derived, cf. Lemmas 3.2 and 3.3 in [5].
Proposition 17. The leading monomial of T2(M) is the leading monomial of M.
Proof.Let M = U1. .U.p-1for Uj 2 {NHd, . .,.NHn-1}. Recall from Equation (z )
that
p-1Y X
(Wj- oel(Wj)) = (-1)|S|XS0oel(XS).
j=1 S {1,...,p-1}
Summing over 0 l p - 1 yields
X p-1Y X
(Wj- oel(Wj)) = (-1)|S|XS0TrGH(XS).
0 l p-1j=1 S {1,...,p-1}
10 M. D. NEUSEL AND M. SEZER
TheQleading term of (Wj - oel(Wj)) is -l . LM (Uj). Thus the leading term of
p-1 l p-1
j=1(Wj - oe (Wj)) is (-l) . LM (U1. .U.p-1). Hence the result follows from
Equation (o).
4.Acknowledgement
The second author wishes to thank Jim Shank for bringing Theorem 3.3 of [4]
to his attention.
References
[1]J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced *
*Mathmatics
11, Cambridge University Press, Cambridge 1986.
[2]N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1*
*947), 589-592.
[3]D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Gra*
*duate Texts
in Mathematics 150, Springer-Verlag, New York 1995.
[4]P. Fleischmann, G. Kemper, R. J. Shank, On the depth of cohomology modules,*
* Q. J. Math.
55 no.2 (2004), 167-184.
[5]P. Fleischmann, M. Sezer, R. J. Shank, C. F. Woodcock, The Noether numbers *
*for cyclic
groups of prime order, Adv. Math. 207 (2006), 149-155.
[6]G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edit*
*ion, Oxford
Science Publications, Oxford University Press, Oxford 1979.
[7]I. Hughes, G. Kemper, Symmetric power of modular representations, Hilbert s*
*eries and degree
bounds, Commun. Algebra 28 (2000), 2059-2089.
[8]M. D. Neusel, The Transfer in the Invariant Theory of Modular Permutation R*
*epresentations,
Pacific J. of Math. 199 (2001), 121-136.
[9]M. D. Neusel, Invariants of some Abelian p-Groups in Characteristic p, Proc*
*eedings of the
AMS 125 (1997), 1921-1931.
[10]M. D. Neusel, Degree Bounds. An Invitation to postmodern Invariant Theory, *
*Topology and
its Applications 154 (2007), 792-814.
[11]M. D. Neusel, L. Smith, Invariant theory of finite groups, Math. Surveys an*
*d Monographs,
Volume 94, Amer. Math. Soc., Providence RI, 2002.
[12]D. Richman, Invariants of finite groups over fields of characteristic p, Ad*
*v. Math. 124 (1996),
25-48.
[13]R. J. Shank, D. L. Wehlau, Decomposing symmetric powers of certai*
*n mod-
ular representations of cyclic groups, IMS Technical Report UKC/IMS/*
*05/13;
http://www.kent.ac.uk/IMS/personal/rjs/.
[14]P. Symonds, Cyclic group actions on polynomial rings, Bull. London Math. So*
*c.39 (2007),
181-188.
Department of Math. and Stats., Texas Tech University, MS 1042 Lubbock, TX
79409, USA
E-mail address: Mara.D.Neusel@ttu.edu
Department of Mathematics, Bilkent University, Ankara 06800, Turkey
E-mail address: mufit.sezer@boun.edu.tr