(revised version)

THE FIRST OCCURRENCE FOR THE

IRREDUCIBLE MODULES OF GENERAL LINEAR

GROUPS IN THE POLYNOMIAL ALGEBRA

Pham Anh Minh*and Ton That Tri

Department of Mathematics College of Sciences University of Hue Dai hoc Khoa hoc Hue, Vietnam

Abstract. Let p be a prime number and let GL_{n} be the group of all invertible matrices over the prime field 𝔽_{p}. It is known that every irreducible GL_{n}-module can be occurred as a submodule of P, the polynomial algebra with n variables over 𝔽_{p}. Given an irreducible GL_{n}-module ρ, the purpose of this paper is to find out the first value of the degree d of which ρ occurs as a submodule of P_{d}, the subset of P consisting of homogeneous polynomials of degree d. This generalizes Schwartz-Tri’s result to the case of any prime p.

1. Introduction

Let p be a prime number and let GL_{n} = GL(n, 𝔽_{p}) be the group of all invertible matrices over the prime field 𝔽_{p}. Denote by P = 𝔽_{p}[x_{1}, …, x_{n}] the polynomial algebra with variables x_{1}, …, x_{n} over 𝔽_{p}. There is then an action of GL_{n} on P given by

\begin{aligned}^{σ}x_{i} & = ∑_{j=1}^{n} a_{ji}x_{j}, 1 ≤ i ≤ n, \\ ^{σ}f(x_{1}, …, x_{n}) & = f(^{σ}x_{1}, …,^{σ} x_{n}) \end{aligned}

with σ = (a_{ij})_{1≤i,j≤n} ∈ GL_{n}, f(x_{1}, …, x_{n}) ∈ P.

For an irreducible GL_{n}-module ρ, it is known (see e.g. [1]) that ρ is isomorphic to a submodule of P. One may ask about the first value of the degree d for which ρ occurs as a submodule of P_{d}, the subset of P consisting of homogeneous polynomials of degree d. This first occurrence problem also arises

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20.

*Corresponding author. Email: paminh@bdvn.vnd.net

from topology, as it is related to the description of the cohomology of certain spaces as modules over the Steenrod algebra (see e.g. [7]). For the prime 2, the problem has been solved independently by Schwartz [4] and Tri [6]. The purpose of this paper is to solve this problem by generalizing the argument used in [6] to the case of any prime p.

Let us recall (see e.g. [2, Chapter 8]) that the irreducible GL_{n}-modules are indexed, up to isomorphism, by column regular partitions, i.e., by sequences of non-negative integers

α_{1}, …, α_{n}

satisfying 0 ≤ α_{i}−α_{i+1} ≤ p−1 for 1 ≤ i ≤ n−1 and 0 ≤ α_{n} ≤ p−1. Namely, the irreducible GL_{n}-modules are indexed by column regular partitions α_{1}, …, α_{n} satisfying α_{n} ≤ p − 2. Recently, in [5], they were also obtained by means of modular invariants, as follows. For 1 ≤ i ≤ n, let

V_{i} = ∏_{λ_{j}∈𝔽_{p}}(λ_{1}x_{1} + ⋯ + λ_{i−1}x_{i−1} + x_{i})

be the Mui invariant and let

L_{i} = V_{1} · . . . · V_{i}

be the Dickson invariant. It is clear that

L_{n} = | \begin{matrix}x_{1} & x_{2} & … & x_{n} \\ x_{1}^{p} & x_{2}^{p} & … & x_{n}^{p} \\ . & . & … & . \\ x_{1}^{p^{n−1}} & x_{2}^{p^{n−1}} & … & x_{n}^{p^{n−1}} \end{matrix} |

and ^{σ}L_{n} = det σ · L_{n} for σ ∈ GL_{n}. Let β = (β_{1}, …, β_{n}) with 0 ≤ β_{i} ≤ p − 1 and set L^{β} = ∏_{i=1}^{n} L_{i}^{β_{i}} ∈ P. Denote by H_{β} = H_{β}(GL_{n}) the GL_{n}-module generated by L^{β} (so H_{β} is nothing but the 𝔽_{p}-vector space generated by the set {^{σ}L^{β}|σ ∈ GL_{n}}). We have

Theorem A ([5; Corollary 1.2]). {H_{β}|β = (β_{1}, …, β_{n}), 0 ≤ β_{i} ≤ p − 1, β_{n} ≠ 0} is a complete set of (p − 1)p^{n−1} distinct irreducible modules for the algebra 𝔽_{p}[GL_{n}].

By noting that H_{(β_{1},... ,β_{n−1},p−1)} ≅ H_{(β_{1},... ,β_{n−1},0)}, we can restate the theorem as follows.

Theorem B (compare [1, Proposition 1.3]). {H_{β}|β = (β_{1}, …, β_{n}), 0 ≤ β_{i} ≤ p − 1, β_{n} ≠ p − 1} is a complete set of (p − 1)p^{n−1} distinct irreducible modules for the algebra 𝔽_{p}[GL_{n}].

We should note that, in terms of column regular partitions, for 0 ≤ β_{i} ≤ p − 1, β_{n} ≠ p − 1, H_{β} corresponds to the sequence λ_{1}, …, λ_{n} with λ_{i} = β_{i} + ⋯ + β_{n}, 1 ≤ i ≤ n.

Generalizing Schwartz-Tri’s result ([4], [6]) to the case of any prime, we prove Theorem C. With β given in Theorem B, the first occurence of H_{β} as a submodule of P_{d} is for d = deg L^{β}.

2. Preliminaries

Let T_{n} be the group consisting of upper triangular matrices with 1 on the diagonal. It follows from [3] that

P^{T_{n}} = 𝔽_{p}[V_{1}, …, V_{n}].

We shall use the following notation. Given g = g(x_{1}, …, x_{n}) ∈ P, λ ∈ 𝔽_{p} and 1 ≤ i, k ≤ n, then ^{σ_{i,k,λ}}g (resp. ^{η_{i,λ}}g, ^{τ_{i,k}}g) denotes the polynomial obtained from g by replacing x_{i} by x_{i} + λx_{k} (resp. by replacing x_{i} by λx_{i}, by interchanging x_{i} and x_{k}). Consider a homogeneous element f = f(x_{1}, …, x_{n}) of P^{T_{n}}, it is clear that V_{i} is a factor of f if and only if so is x_{i}. We have

Lemma 1. If ^{η_{i,λ}}f = λ^{ℓ}f with 1 ≤ ℓ < p − 1 and λ ≠ 0, 1, then f contains x_{i} as a factor.

Proof. Write f = ∑_{r=0}^{m} x_{i}^{r}f_{r}(x_{1}, …, x_{n}) with f_{r} free of x_{i}. It follows that ∑_{r=0}^{m} x_{i}^{r}(λ^{ℓ}f_{r} − λ^{r}f_{r}) = 0. So λ^{ℓ}f_{0} − f_{0} = 0, which implies that f_{0} = 0. Therefore f contains x_{i} as a factor. The lemma follows. □

The case p = 2 in the following lemma was treated in [6]. However, our proof given here, for any prime p, is much simpler.

Lemma 2. If i < k, then f contains x_{i} as a factor provided that one of the following conditions is satisfied:

(i) ∑_{r=0}^{p−1} ^{σ_{i,k,r}}f = − ^{τ_{i,k}}f and ^{η_{i,μ}}f = f for every μ ≠ 0; (ii) f contains x_{k} as a factor and ∑_{r=0}^{p−1} ^{σ_{i,k,r}}f = α · ^{τ_{i,k}}f with α ∈

{0, 1, −1}.

Proof. For 1 ≤ r ≤ p − 1, since f is T_{n}-invariant, we have

f(x_{1}, …, x_{i−1}, rx_{k}, x_{i+1}, …, x_{k}, …, x_{n})
=f(x_{1}, …, x_{i−1}, rx_{k}, x_{i+1}, …, x_{k} − x_{k}, …, x_{n})
=f(x_{1}, …, x_{i−1}, rx_{k}, x_{i+1}, …, 0, …, x_{n})
=\{ \begin{matrix}f(x_{1}, …, x_{i−1}, x_{k}, x_{i+1}, …, 0, …, x_{n}) & if ^{η_{i,r}}f = f \\ 0 & if f contains x_{k} as a factor. \end{matrix}

Consider ∑_{r=0}^{p−1} ^{σ_{i,k,r}}f and set x_{i} = 0. (i) and the above equality imply that

∑_{r=0}^{p−1} ^{σ_{i,k,r}}f = f(x_{1}, …, x_{i−1}, 0, x_{i+1}, …, x_{k}, …, x_{n})
− f(x_{1}, …, x_{i−1}, x_{k}, x_{i+1}, …, 0, …, x_{n})
= −f(x_{1}, …, x_{i−1}, x_{k}, x_{i+1}, …, 0, …, x_{n}),

so f(x_{1}, …, x_{i−1}, 0, x_{i+1}, …, x_{k}, …, x_{n}) = 0; on the other hand, (ii) implies that

∑_{r=0}^{p−1} ^{σ_{i,k,r}}f = f(x_{1}, …, x_{i−1}, 0, x_{i+1}, …, x_{k}, …, x_{n})

= αf(x_{1}, …, x_{i−1}, x_{k}, x_{i+1}, …, 0, …, x_{n})

= 0 since f contains x_{k} as a factor.

In any case, f contains x_{i} as a factor. The lemma follows. □

3. Proof of Theorem C

We proceed by induction on n. The proof is trivial for n = 1. Assume that the theorem holds for n − 1.

Let W be an irreducible GL_{n}-module of homogeneous polynomials. There exist then β = (β_{1}, …, β_{n}) with 0 ≤ β_{i} ≤ p − 1, β_{n} ≠ p − 1 and an GL_{n}- isomorphism ζ : W → H_{β}(GL_{n}). Set f = f(x_{1}, …, x_{n}) = ζ^{−1}(L^{β}). We need to prove that deg f ≥ deg L^{β}. It is clear that f is invariant under the action of T_{n} and W is the GL_{n}-module generated by f. For convenience, write W = W_{f}(GL_{n}).

Denote by (β_{n_{1}}, …, β_{n_{k}}) the subsequence of β consisting of all non-zero elements, so L^{β} = L_{n_{1}}^{β_{1}} · . . . · L_{n_{k}}^{β_{n_{k}}}. We first prove that f contains L_{n_{k}} as a factor (note that β_{n_{k}} < p − 1 if n_{k} = n). Let m be an integer satisfying n_{k−1} < m ≤ n_{k}. For every μ ≠ 0, as ^{η_{m,μ}}L^{β} = μ^{β_{n_{k}}} L^{β}, it follows that η_{m,μ}f = μ^{β_{n_{k}}} f. If β_{n_{k}} < p − 1, Lemma 1 implies that f contains x_{m}, therefore V_{m}, as a factor. If β_{n_{k}} = p − 1 (which implies n_{k} < n), then ^{η_{m,μ}}f = f and

σ_{m,n,r} L_{n_{k}} = L_{n_{k}}(x_{1}, …, x_{m−1}, x_{m} + rx_{n}, x_{m+1}, …, x_{n_{k}})
= L_{n_{k}}(x_{1}, …, x_{m−1}, x_{m}, x_{m+1}, …, x_{n_{k}})
+ rL_{n_{k}}(x_{1}, …, x_{m−1}, x_{n}, x_{m+1}, …, x_{n_{k}})

for r ∈ 𝔽_{p}. Since

∑_{r=0}^{p−1}(L_{n_{k}}(x_{1}, …, x_{m−1}, x_{m}, x_{m+1}, …, x_{n_{k}})
+ rL_{n_{k}}(x_{1}, …, x_{m−1}, x_{n}, x_{m+1}, …, x_{n_{k}}))^{λ}

= \{ \begin{matrix}0 & if λ < p − 1 \\ −L_{n_{k}}^{p−1}(x_{1}, …, x_{m−1}, x_{n}, x_{m+1}, …, x_{n_{k}}) & if λ = p − 1, \end{matrix}  (∗)

we have ∑_{r=0}^{p−1} ^{σ_{m,n,r}}L_{n_{k}}^{p−1} = −L_{n_{k}}^{p−1}(x_{1}, …, x_{m−1}, x_{n}, x_{m+1}, …, x_{n_{k}}) which implies that ∑_{r=0}^{p−1} ^{σ_{m,n,r}}L^{β} = −^{τ_{m,n}}L^{β}. Hence ∑_{r=0}^{p−1} ^{σ_{m,n,r}}f = −^{τ_{m,n}}f. It follows from Lemma 2 that f contains x_{m}, hence V_{m} as a factor.

Set n_{0} = 0. Suppose that f contains x_{t+1}, …, x_{n_{k}} as factors with n_{ℓ−1} < t ≤ n_{ℓ} < n_{k}. It follows from (∗) that

∑_{r=0}^{p−1} ^{σ_{t,n_{ℓ}+1,r}}L_{n_{ℓ}}^{β_{n_{ℓ}}} = \{ \begin{matrix}0 & if β_{n_{ℓ}} < p − 1 \\ −^{τ_{t,n_{ℓ}+1}}L_{n_{ℓ}}^{p−1} & if β_{n_{ℓ}} = p − 1. \end{matrix}

Therefore

∑_{r=0}^{p−1} ^{σ_{t,n_{ℓ}+1,r}}L^{β} = \{ \begin{matrix}0 & if β_{n_{ℓ}} < p − 1 \\ α^{τ_{t,n_{ℓ}+1}}L^{β} & if β_{n_{ℓ}} = p − 1, \end{matrix}

with α = ±1. So

∑_{r=0}^{p−1} ^{σ_{t,n_{ℓ}+1,r}}f = \{ \begin{matrix}0 & if β_{n_{ℓ}} < p − 1 \\ α^{τ_{t,n_{ℓ}+1}}f & if β_{n_{ℓ}} = p − 1. \end{matrix}

By Lemma 2, f contains x_{t}, hence V_{t} as a factor. Therefore f contains L_{n_{k}} as a factor.

Write f = L_{n_{k}}g with g = g(x_{1}, …, x_{n}) ∈ P. So g ∈ P^{T_{n}}. Consider GL_{n_{k}} as a subgroup of GL_{n} by the usual way. H_{β}(GL_{n_{k}}) and W = W_{f}(GL_{n_{k}}) are

then GL_{n_{k}}-modules and H_{β}(GL_{n_{k}}) ≅^{ζ} W_{f}(GL_{n_{k}}) ≅ det ⊗W_{g}(GL_{n_{k}}). Set β′ = (β_{1}, …, β_{n_{k}} − 1) and β′′ = (β_{1}, …, β_{n_{k−1}}). As GL_{n_{k}}-modules, H_{β}(GL_{n_{k}}) ≅ det ⊗H_{β′}(GL_{n_{k}}), so W_{g}(GL_{n_{k}}) ≅ H_{β′}(GL_{n_{k}}). If n_{k} < n, it follows from the inductive hypothesis that deg g ≥ deg L^{β′}, so deg f ≥ deg L^{β}. Suppose that n_{k} = n. The argument used above shows that g contains L_{n} as a factor, hence f contains L_{n}^{β_{n}} as a factor. By writing f = L_{n}^{β_{n}}h, we have W_{h}(GL_{n}) ≅ H_{β′′}(GL_{n}). By replacing f by h, arguing as in the case n_{k} < n yields deg h ≥ deg L^{β′′}, hence deg f ≥ deg L^{β}. The theorem is proved.

Acknowledgments. We thank Grant Walker who introduced us to the study of this first occurrence problem. Part of the result was obtained while the first author visited the University of Manchester in Autumn 1997; he thanks Robert Sandling for making the visit possible.

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