THE INVARIANTS OF MODULAR INDECOMPOSABLE REPRESENTATIONS OF ℤ_{p^{2}}

MARA D. NEUSEL AND MÜFİT SEZER

Abstract. We consider the invariant ring for an indecomposable representation of a cyclic group of order p^{2} over a field 𝔽 of characteristic p. We describe a set of 𝔽-algebra generators of this ring of invariants, and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants. This bound, as a polynomial in p, is of degree two.

Introduction

Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group G. Denote by V = 𝔽^{n} the n-dimensional vector space over 𝔽. Then G acts via ρ 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 ∈ 𝔽[V] | g(f) = f, ∀g ∈ G} ⊆ 𝔽[V]

is a graded connected commutative Noetherian subalgebra of 𝔽[V], see [11] for a general treatment of the subject. Let

β(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| ∈ 𝔽^{×}, we have that

β(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 ∈ 𝔽. Indeed, Richmann constructed modular representations V with arbitrarily large β(𝔽[V]^{G}), see [12]. In other words, there cannot be a degree bound for β(𝔽[V]^{G}) that depends only on the group, see [10] for an overview in these matters.

In this paper we want to study rings of invariants of cyclic p-groups ℤ_{p^{r}} of order p^{r} over a field 𝔽 of finite characteristic p. There are exactly p^{r} indecomposable ℤ_{p^{r}}-modules, which we denote by V_{1}, V_{2}, …, V_{p^{r}}, see Chapter II in [1], where V_{n} has dimension n as a vector space over 𝔽.

We note that Göbel’s bound gives, of course, a bound on the degrees of a generating set of 𝔽[V_{p^{r}}]^{ℤ_{p^{r}}} for any p and r, see Corollary 3.4.4 in [11]. In this case we have

β(𝔽[V_{p^{r}}]^{ℤ_{p^{r}}}) ≤ max\{p^{r}, \binom{p^{r}}{2}\}.

This bound depends on the dimension of the representation which coincides in this case with the order of the group.

Date: September 11, 2007, 15 h 30 min.

If r = 1 and G = ℤ_{p} is the cyclic group of prime order, then a general degree bound for a minimal generating set of the ring of invariants for any ℤ_{p}-module V was given in in [5]. This bound is sharp, as the case of the regular representation of ℤ_{3} shows.

For the case r = 2 much less is known: In [9] we find an explicit description of the ring of invariants 𝔽[V_{3}]^{ℤ_{4}}. This was generalized to 𝔽[V_{p+1}]^{ℤ_{p^{2}}} in [13]. Furthermore, in [8] we find an explicit description of the ring of invariants of the regular representation of ℤ_{4}.

We want to extend this study and find an upper bound for β(𝔽[V_{n}]^{ℤ_{p^{2}}} ) for any indecomposable ℤ_{p^{2}}-module V_{n}. 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 generators for 𝔽[V_{n}]^{ℤ_{p^{2}}} . This description yields an upper bound for β(𝔽[V_{n}]^{ℤ_{p^{2}}} ). This bound transpires to be quadratic in p. We postpone some technical calculations to Section 3.

For the remainder of the paper, we assume that G ≅ ℤ_{p^{2}} and that H ≅ ℤ_{p} is the non-trivial subgroup.

We choose a basis x_{1}, …, x_{n} for V_{n}^{∗} and write

𝔽[V_{n}] ≅ 𝔽[x_{1}, …, x_{n}].

Next, we choose a generator σ for the group G. Then

σx_{i} = \begin{cases}x_{1} & for i = 1, and \\ x_{i} + x_{i−1} & for 2 ≤ i ≤ n. \end{cases}

Set Δ = σ − 1. Then we have

Δ(x_{i}) = \begin{cases}0 & for i = 1, and \\ x_{i−1} & for 2 ≤ i ≤ n. \end{cases}

The various transfer maps involved are given by the following formulae

Tr^{G} = ∑_{i=0}^{p^{2}−1} σ^{i}, Tr^{H} = ∑_{i=0}^{p−1} σ^{ip}, and Tr_{H}^{G} = ∑_{i=0}^{p−1} σ^{i}.

We use the graded reverse lexicographic order with x_{i} > x_{i−1} for i = 2, …, n.

1. An Upper Bound for β(𝔽[V]_{G})

Since G is a finite group, the extension 𝔽[V]^{G} ↪ 𝔽[V] is finite. Denote by (\overline{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]/(\overline{𝔽[V]^{G}})

form a finite-dimensional vector space over 𝔽. Thus its Hilbert series is a polynomial. In this section we want to derive an upper bound on its degree.

Note that the Hilbert series of the Hilbert ideal (\overline{𝔽[V]^{G}}) ⊆ 𝔽[V] coincides with the Hilbert series of the ideal I of leading terms of (\overline{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 V_{n} is an indecomposable G/H ≅ ℤ_{p}-module and thus 𝔽[V_{n}]^{G} ≅ 𝔽[V_{n}]^{G/H}. Therefore we restrict our attention to the case n > p in what follows.

We need two somewhat technical constructions:

Let r be a positive integer with max{n − 2p, 1} ≤ r ≤ n − p. Choose a monomial m = u_{1}u_{2} ⋯ u_{2p−2} of degree 2p − 2 in 𝔽[x_{d}, …, x_{r}], where d = max{1, r − p + 1}. We may assume that u_{1} ≤ u_{2} ≤ ⋯ ≤ u_{2p−2}. We define w_{i,0} by

u_{i} = \begin{cases}Δ(w_{i,0}) & if 1 ≤ i ≤ p − 1, and \\ Δ^{p}(w_{i,0}) & if p ≤ i ≤ 2p − 2, \end{cases}

and set

w_{i,j} = Δ^{j}(w_{i,0}) 1 ≤ i ≤ 2p − 2, j ∈ ℕ_{0}.

For a 2p − 2-tuple α = [α(1), α(2), …, α(2p − 2)] ∈ ℕ^{2p−2} of natural numbers we define

w_{α} = ∏_{i=1}^{2p−2} w_{i,α(i)}.

Thus we can write

m = u_{1}u_{2} ⋯ u_{2p−2} = ∏_{i=1}^{p−1} w_{i,1} ∏_{i=p}^{2p−2} w_{i,p} = w_{α′},

where α′(i) = 1 if 1 ≤ i ≤ p − 1 and α′(i) = p if p ≤ i ≤ 2p − 2.

Let S ⊆ {1, 2, …, 2p − 2} be a subset and set

X_{S} = ∏_{i∈S} w_{i,0}.

We consider the following polynomial

T_{1}(m) = ∑_{S⊆{1,...,2p−2}}(−1)^{|S|}X_{S′}Tr^{G}(X_{S}),

where S′ denotes the complement of S in {1, 2, …, 2p − 2}.

Proposition 1. The leading term of T_{1}(m) is m.

Proof. The proof of this result is postponed to Section 3. □

The polynomials T_{1}(m) are by construction in the Hilbert ideal (\overline{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 V_{n}^{∗} decomposes into a direct summand of p indecomposable H-modules:

V_{n}^{∗} = V_{n_{n−p+1}}^{∗} ⊕ ⋯ ⊕ V_{n_{n}}^{∗}.

Moreover, V_{n_{j}}^{∗} is generated as a H-module by x_{j} for j = n, …, n − p + 1. For each i = n − p + 1, …, n, we define the H-norms

N_{i}^{H} = ∏_{σ∈H} σx_{i}.

Note that every N_{i}^{H} has degree p and coincides with the respective top orbit Chern classes if i ≥ p.

Choose a monomial

M = ∏_{1≤j≤p−1} N_{i_{j}}^{H} ∈ 𝔽[N_{d}^{H}, …, N_{n−1}^{H}]

of degree p−1 as a polynomial in these norms. For 1 ≤ j ≤ p−1 define W_{j} = N_{i_{j}+1}^{H}. Let S ⊆ {1, …, p − 1} be a subset and S′ its complement. Then similarly to the contruction of T_{1}(m) we set X_{S} = ∏_{j∈S} W_{j}, and obtain a polynomial T_{2}(M) as follows.

T_{2}(M) = ∑_{S⊆{1,...,p−1}}(−1)^{|S|}X_{S′}Tr_{H}^{G}(X_{S}).

Proposition 2. The leading monomial of T_{2}(M) is the leading monomial of M.

Proof. The proof of this result is postponed to Section 3. □

As for T_{1}(m) the polynomials T_{2}(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[V_{n}]_{G} is bounded above by 3p^{2} + (2t − 4)p − 3t.

Proof. The Hilbert series of the Hilbert ideal (\overline{𝔽[V]^{G}}) ⊆ 𝔽[V] coincides with the Hilbert series of the ideal, I, of leading terms of (\overline{𝔽[V]^{G}}). Thus in order to find a bound on the degrees of the coinvariants it suffices to find a degree bound for 𝔽[V]/I.

To that end, let m_{1}m_{2}x_{n}^{l} be a monomial that is not in the lead term ideal of the Hilbert ideal. Without loss of generality we assume that m_{1} ∈ 𝔽[x_{1}, …, x_{n−p}] and m_{2} ∈ 𝔽[x_{n−p+1}, …, x_{n−1}].

Let max{n−2p, 1} ≤ r ≤ n−p and m a monomial of degree 2p−2 in 𝔽[x_{d}, …, x_{r}], where d = max{1, r − p + 1}. Then Proposition 1 shows that m appears as leading term of some T_{1}(m). Since T_{1}(m) is contained in the Hilbert ideal it follows that the degree of m_{1} is at most t(2p − 3).

Similary, the polynomials T_{2}(M) are in the Hilbert ideal and thus by Proposition 2, m_{2} is not divisible by the lead term of a product of p − 1 norms N_{i}^{H}, where d ≤ i ≤ n − 1. Therefore the degree of m_{2} is at most (p − 2)p + (p − 1)^{2}.

Finally x_{n}^{p^{2}} is the leading term of the norm N_{n}^{G} = ∏_{σ∈G} σx_{n}. Therefore l ≤ p^{2}−1. Hence

deg(m_{1}m_{2}x_{n}^{l}) ≤ t(2p − 3) + (p − 2)p + (p − 1)^{2} + p^{2} − 1 = 3p^{2} + (2t − 4)p − 3t

as claimed. □

Corollary 4. Let n > p. Then the image of the transfer Im(Tr^{G}) ⊆ 𝔽[V]^{G} is generated by forms of degree at most 3p^{2} + (2t − 4)p − 3t.

Proof. We write the ring of polynomials as a module over the ring of invariants as follows

𝔽[V] = ∑_{finite} 𝔽[V]^{G}h_{i}.

We note that by construction the h_{i}’s form a basis of 𝔽[V]_{G}. Since |G| = p^{2} ≡ 0 mod p, we have that Tr^{G}(𝔽[V]^{G}) = 0. Thus the image of the transfer is generated by the Tr^{G}(h_{i})’s, and the result follows from Theorem 3. □

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 ℤ_{p^{2}}. Its ring of invariants is generated by forms of degree at most 5p^{2} − 7p. This can be seen as follows:

By Theorem 3.3 in [4], 𝔽[V_{p^{2}}]^{G}/ImTr^{G} ≃ 𝔽[V_{p}]^{H}, where the isomorphism scales the degrees by \frac{1}{p}. It is shown in [5] that 𝔽[V_{p}]^{H} is generated by invariants of degree 2p − 3. Hence 𝔽[V_{p^{2}}]^{G}/ImTr^{G} is generated by classes of degree at most (2p − 3)p. On the other hand, Corollary 4 tells us that Im(Tr^{G}) is generated by invariants of degree at most 5p^{2} − 7p. Hence

β(𝔽[V_{p^{2}}]^{G}) ≤ max{(2p − 3)p, 5p^{2} − 7p} = 5p^{2} − 7p

as claimed.

We proceed to the general case. As in Section 1, let n > p and

V_{n}^{∗} = V_{n_{n−p+1}}^{∗} ⊕ ⋯ ⊕ V_{n_{n}}^{∗}

be an H-module decomposition. For i ∈ {n−p+1, …, n} we have that x_{i} generates V_{n_{i}}^{∗} as H-module.

Lemma 6. The image of the relative transfer, ImTr_{H}^{G}, is generated by ImTr^{G} and G-invariants of degree at most 3p^{2} − 3p.

Proof. Let f ∈ 𝔽[V_{n}]^{H}. By Lemma 2.12 the ring 𝔽[V_{n}]^{H} is generated as a module over 𝔽[N_{d}^{H}, …, N_{n}^{H}] by invariants of degree at most p^{2} − n and the image of the transfer Tr^{H}. Thus f can be written as

(∘)  f = ∑ p_{i}(N_{d}^{H}, …, N_{n}^{H})b_{i} + ∑ q_{j}(N_{d}^{H}, …, N_{n}^{H})Tr^{H}(g_{j})

for some polynomials p_{i}, q_{j} ∈ 𝔽[N_{d}^{H}, …, N_{n}^{H}], H-invariants b_{i} of degree at most p^{2} − n and suitable g_{j} ∈ 𝔽[V_{n}]. Since

∑ q_{j}(N_{d}^{H}, …, N_{n}^{H})Tr^{H}(g_{j}) = Tr^{H}(∑ q_{j}(N_{d}^{H}, …, N_{n}^{H})g_{j})

we find that

Tr_{H}^{G}(∑ q_{j}(N_{d}^{H}, …, N_{n}^{H})Tr^{H}(g_{j})) = Tr^{G}(∑ q_{j}(N_{d}^{H}, …, N_{n}^{H})g_{j})

is in the image of the transfer Tr^{G}. Thus we need to take care of the first summand and assume without lost of generality that

(∘)  f = ∑ p_{i}(N_{d}^{H}, …, N_{n}^{H})b_{i}

We sort (∘) by monomials in the norms and obtain

f = ∑_{J} b_{J}N_{J}^{H},

where b_{J} is a sum of suitable b_{i}’s and thus is still an H-invariant of degree at most p^{2} − n.

We claim that the degree of N_{J}^{H} as a polynomial in N_{n}^{H} is at most p−1. Otherwise set U = (N_{n}^{H})^{p}. Then

Tr_{H}^{G}(\frac{b_{J}N_{J}^{H}}{U} N_{n}^{G}) = N_{n}^{G}Tr_{H}^{G}(\frac{b_{J}N_{J}^{H}}{U})

can be written in terms of G-invariants of strictly smaller degree. On the other hand LM(b_{J}N_{J}^{H} − \frac{b_{J} N_{J}^{H}}{U} N_{n}^{G}) < LM(b_{J}N_{J}^{H}). Therefore

Tr_{H}^{G}(b_{J}N_{J}^{H}) = Tr_{H}^{G}(b_{J}N_{J}^{H} − \frac{b_{J}N_{J}^{H}}{U} N_{n}^{G}) + Tr_{H}^{G}(\frac{b_{J}N_{J}^{H}}{U} N_{n}^{G})

yields that Tr_{H}^{G}(b_{J}N_{J}^{H}) can be eliminated from a generating set for ImTr_{H}^{G}.

Similarly, we claim that the degree of the b_{J}N_{J}^{H}’s as a monomial in {N_{i}^{H}|i = d, …, n − 1} is strictly less than p − 1. Assume the contrary and let U_{j} ∈ {N_{i}^{H}|i = d, …, n − 1} for 1 ≤ j ≤ p − 1. Set U = ∏_{1≤j≤p−1} U_{j}. Then we have

Tr_{H}^{G}(\frac{b_{J}N_{J}^{H}}{U} T_{2}(U_{1} ⋯ U_{p−1})) = Tr_{H}^{G}(\frac{b_{J}N_{J}^{H}}{U} ∑_{S⊆\{1,...,p−1\}}(−1)^{|S|}X_{S′}Tr_{H}^{G}(X_{S}))

= ∑_{S⊆\{1,...,p−1\}} Tr_{H}^{G}(X_{S})Tr_{H}^{G}(\frac{b_{J}N_{J}^{H}}{U} (−1)^{|S|}X_{S′}).

Hence, Tr_{H}^{G}(\frac{b_{J} N_{J}^{H}}{U} T_{2}(U_{1} ⋯ U_{p−1})) can be written in terms of G-invariants of

smaller degree. By Proposition 2 we have that LM(b_{J}N_{J}^{H} − \frac{b_{J} N_{J}^{H}}{U} T_{2}(U_{1} ⋯ U_{p−1})) < LM(b_{J}N_{J}^{H}). Therefore the equation

Tr_{H}^{G}(b_{J}N_{J}^{H}) = Tr_{H}^{G}(b_{J}N_{J}^{H} − \frac{b_{J}N_{J}^{H}}{U} T_{2}(U_{1} ⋯ U_{p−1})) + Tr_{H}^{G}(\frac{b_{J}N_{J}^{H}}{U} T_{2}(U_{1} ⋯ U_{p−1}))

yields that Tr_{H}^{G}(b_{J}N_{J}^{H}) can be eliminated from a generating set for ImTr_{H}^{G}. Thus, for any multi-index J, the degree (in the x’s) of b_{J}N_{J}^{H} is bunded above by

p^{2} − n + (p − 2)p + p(p − 1) = 3p^{2} − 3p − n < 3p^{2} − 3p

as claimed. □

Theorem 7. Let V_{n} be an indecomposable G-module. Let n = tp + r > p, where 1 ≤ t ≤ p and 0 ≤ r < p are integers. Then

β(V_{n}) ≤ max{3p^{2} + (2t − 4)p − 3t, 3p^{2} − 3p}.

Proof. By the periodicity result of Theorem 1.2 in [14], 𝔽[V_{n}] is modulo the 𝔽Hprojective submodules generated by N_{n}^{G} = ∏_{σ∈G} σx and invariants of degree less than p^{2}. Thus 𝔽[V_{n}]^{G} is generated by the G-norm N_{n}^{G}, invariants of degree less than p^{2} and image ImTr_{H}^{G} of the relative transfer, since the fixed pointed of projective modules are in the image of the relative transfer.

By the previous lemma ImTr_{H}^{G} is generated by invariants of degree at most 3p^{2} − 3p together with ImTr^{G}. Therefore it follows from Corollary 4 that

β(V_{n}) ≤ max{3p^{2} + (2t − 4)p − 3t, 3p^{2} − 3p},

as desired. □

Remark 8. We note that for n ≤ p the representation

ρ : G ⟶ GL(n, 𝔽)

has kernel ℤ_{p}. Thus 𝔽[V]^{G} ≅ 𝔽[V]^{H}. Hence this ring of invariants is generated by forms of degree at most 2p − 3 by [5].

Remark 9. Furthermore, if n = p + 1 we find in [13] an explicit generating set of the ring of invariants and we read off

β(𝔽[V_{p+1}]^{G}) ≤ 2p^{2} − 2p − 1.

For p = 3 the authors of [13] refer to a Magma calculation and for β(𝔽[V_{4}]^{G}) = 9. For p = 2 we find β(𝔽[V_{3}]^{G}) = 4 by [9]. We note that

p^{2} ≤ 2p^{2} − 2p − 1 ≤ 3p^{2} − 3p ≤ max{3p^{2} + (2t − 4)p − 3t, 3p^{2} − 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 ℤ_{p^{r}}-module. Then β(𝔽[V]^{ℤ_{p^{r}}}) is bounded above by a polynomial in p of degree r.

3. The Leading Terms of T_{1}(m) and T_{2}(M)

In this section we want to identify the leading terms of the polynomials T_{1}(m) and T_{2}(M) as described in Propositions 1 and 2. We start by identifying the coefficients of monomials that appear in T_{1}(m).

Lemma 11. The coefficients of T_{1}(m) = ∑_{α∈ℕ^{2p−2}} c_{α}w_{α} are given by

c_{α} = ∑_{0≤l≤p^{2}−1} ∏_{i=1}^{2p−2} \binom{l}{α(i)}.

Proof. Since σ^{l} is an algebra automorphism we have that

(✠)  ∏_{i=1}^{2p−2}(w_{i,0} − σ^{l}(w_{i,0})) = ∑_{S⊆{1,...,2p−2}}(−1)^{|S|}X_{S′}σ^{l}(X_{S}).

Thus summing over 0 ≤ l ≤ p^{2} − 1 yields

T_{1}(m) = ∑_{0≤l≤p^{2}−1} ∏_{i=1}^{2p−2}(w_{i,0} − σ^{l}(w_{i,0})).

Since we have^{1}

(w_{i,0} − σ^{l}(w_{i,0})) = −lw_{i,1} − \binom{l}{2}w_{i,2} − \binom{l}{3}w_{i,3} − ⋯ − \binom{l}{l}w_{i,l},

the desired equality follows. □

w_{i,0} − σw_{i,0} = Δw_{i,0} = w_{i,1}.

= −(l − 1)w_{i,1} − \binom{l − 1}{2}w_{i,2} − ⋯ − \binom{l − 1}{l − 1}w_{i,l−1} − σ^{l−1}w_{i,1}

= −(l − 1)w_{i,1} − \binom{l − 1}{2}w_{i,2} − ⋯ − \binom{l − 1}{l − 1}w_{i,l−1} − w_{i,1} − \binom{l − 1}{1}w_{i,2} − ⋯ − \binom{l − 1}{l − 1}w_{i,l}

= −lw_{i,1} − \binom{l}{2}w_{i,2} − \binom{l}{3}w_{i,3} − ⋯ − \binom{l}{l}w_{i,l}

1This equation can be easily verified by induction on l ≥ 0. If l = 0 the equation is trivial. For l = 1 we have Assume that l > 1. Then by induction we obtain w_{i,0} − σ^{l}w_{i,0} = (w_{i,0} − σ^{l−1}w_{i,0}) − σ^{l−1}Δw_{i,0} as desired.

Lemma 12. Let α ∈ ℕ^{2p−2}. If α(i) > 1 for some 1 ≤ i ≤ p − 1, then w_{α} < w_{α′} = m.

Proof. Since u_{1} ≤ u_{2} ≤ ⋯ ≤ u_{2p−2}, it suffices to show that

∏_{i=1}^{k} w_{i,α(i)} < u_{1}u_{2} … u_{k}

for some 1 ≤ k ≤ 2p − 2. Let j denote the smallest integer such that α(j) > 1. Since j ≤ p − 1, it follows that u_{i} = w_{i,1} for i < j and w_{j,α(j)} < u_{j}. Therefore ∏_{i=1}^{j} w_{i,α(i)} < u_{1}u_{2} … u_{j} and the result follows. □

Lemma 13. Let α ∈ ℕ^{2p−2}. If α(i) ≥ 2p for some 1 ≤ i ≤ 2p − 2, then w_{α} < w_{α′} = m.

Proof. By Lemma 12 it is enough to show the result for i ≥ p. Since u_{i} = w_{i,p} ∈ 𝔽[x_{r−p+1}, …, x_{r}], it follows that w_{i,α(i)} ∈ 𝔽[x_{1}, …, x_{r−p}]. Therefore w_{α} contains a variable that is smaller than all variables that appear in m. □

Lemma 14. Let α, β be two elements in ℕ^{2p−2} such that α(i) ≥ β(i) for 1 ≤ i ≤ 2p − 2. Then

(1) w_{α} ≤ w_{β}, and

(2) w_{α} = w_{β} if and only if α = β.

Proof. Since w_{i,α(i)} ≤ w_{i,β(i)} for 1 ≤ i ≤ 2p − 2, we have

w_{α} = ∏_{i=1}^{2p−2} w_{i,α(i)} ≤ ∏_{i=1}^{2p−2} w_{i,β(i)} = w_{β}.

For the second assertion observe that if α(i) < β(i) for some 1 ≤ i ≤ 2p − 2, then w_{i,α(i)} < w_{i,β(i)}. Hence

w_{α} = ∏_{i=1}^{2p−2} w_{i,α(i)} < ∏_{i=1}^{2p−2} w_{i,β(i)} = w_{β}

as desired. □

Lemma 15. The coefficient of c_{α′} of the monomial w_{α′} in T_{1}(m) is 1.

Proof. By Lemma 11, we have c_{α′} = ∑_{0≤l≤p^{2}−1} l^{p−1}\binom{l}{p}^{p−1}. For 0 ≤ l ≤ p^{2} − 1, write l = l_{1}p + l_{2}, where 0 ≤ l_{1}, l_{2} < p. Then we find

∑_{0≤l≤p^{2}−1} l^{p−1}\binom{l}{p}^{p−1} = ∑_{0≤l_{1},l_{2}≤p−1}(l_{1}p + l_{2})^{p−1}\binom{l_{1}p + l_{2}}{p}^{p−1}

≡^{(1)} ∑_{0≤l_{1},l_{2}≤p−1} l_{2}^{p−1}l_{1}^{p−1} mod p

≡^{(2)} 1 mod p,

where (1) follows from

(⋆)  \binom{s}{t} ≡ \binom{a_{1}}{a_{2}}\binom{b_{1}}{b_{2}} mod p

( for any two integers 0 ≤ s, t < p^{2} with s = a_{1}p + b_{1} and t = a_{2}p + b_{2}, where 0 ≤ a_{i}, b_{i} < p), see [2], and (2) from

(•)  ∑_{0≤l≤p−1} l^{c} ≡ \begin{cases}−1 mod p if p − 1| c; \\ 0 mod p otherwise, \end{cases}

(for any natural number c), see Theorem 119 in [6]. □

We are now able to proof Proposition 1:

Proposition 16. The leading term of T_{1}(m) is w_{α′}, and thus LM(T_{1}(m)) = m = w_{α′}.

Proof. The second statement follows from the first because c_{α′} = 1 by Lemma 15. We proceed by showing that w_{α′} ≤ w_{α} and c_{α} ≠ 0 implies α = α′.

By Lemmas 12 and 13 we may assume α(i) = 1 for 1 ≤ i ≤ p − 1 and α(i) < 2p for p ≤ i ≤ 2p − 2. From Lemma 11 we have

c_{α} = ∑_{0≤l≤p^{2}−1} ∏_{i=1}^{2p−2} \binom{l}{α(i)} = ∑_{0≤l≤p^{2}−1} l^{p−1} ∏_{i=p}^{2p−2} \binom{l}{α(i)}.

For p ≤ i ≤ 2p − 2 write α(i) = a_{i}p + b_{i} with 0 ≤ b_{i} < p and 0 ≤ a_{i} ≤ 1. Set l = l_{1}p + l_{2} with 0 ≤ l_{1}, l_{2} < p.

c_{α} = ∑_{0≤l_{1},l_{2}≤p−1}(l_{1}p + l_{2})^{p−1} ∏_{i=p}^{2p−2} \binom{l_{1}p + l_{2}}{a_{i}p + b_{i}}

≡ ∑_{0≤l_{1},l_{2}≤p−1} l_{2}^{p−1} ∏_{i=p}^{2p−2} \binom{l_{1}}{a_{i}}\binom{l_{2}}{b_{i}}

≡ \begin{cases}∑_{0≤l_{1},l_{2}≤p−1} l_{2}^{p−1}l_{1}^{p−2} ∏_{i=p}^{2p−2} \binom{l_{2}}{b_{i}} & if a_{i} = 1 for all i, \\ ∑_{0≤l_{2}≤p−1} (l_{2}^{p−1} ∏_{i=p}^{2p−2} \binom{l_{2}}{b_{i}}(∑_{0≤l_{1}≤p−1} l_{1}^{k})) ≡ 0 & otherwise, \end{cases}

where the last equation follows since k an integer not divisible by p − 1. Thus we may assume that a_{i} = 1 for p ≤ i ≤ 2p − 2. It follows that α(i) = p + b_{i} ≥ p = α′(i) for p ≤ i ≤ 2p − 2. Moreover α(i) = α′(i) = 1 for 1 ≤ i ≤ p − 1. Now α = α′ follows 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 T_{2}(M) is the leading monomial of M.

Proof. Let M = U_{1} ⋯ U_{p−1} for U_{j} ∈ {N_{d}^{H}, …, N_{n−1}^{H}}. Recall from Equation (✠) that

∏_{j=1}^{p−1}(W_{j} − σ^{l}(W_{j})) = ∑_{S⊆{1,...,p−1}}(−1)^{|S|}X_{S′}σ^{l}(X_{S}).

Summing over 0 ≤ l ≤ p − 1 yields

∑_{0≤l≤p−1} ∏_{j=1}^{p−1}(W_{j} − σ^{l}(W_{j})) = ∑_{S⊆{1,...,p−1}}(−1)^{|S|}X_{S′}Tr_{H}^{G}(X_{S}).

The leading term of (W_{j} − σ^{l}(W_{j})) is −l · LM(U_{j}). Thus the leading term of ∏_{j=1}^{p−1}(W_{j} − σ^{l}(W_{j})) is (−l)^{p−1} · LM(U_{1} ⋯ U_{p−1}). Hence the result follows from Equation (•). □

4. Acknowledgement

The second author wishes to thank Jim Shank for bringing Theorem 3.3 of [4] to his attention.

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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