lower bounds for degrees of irreducible brauer characters of finite general linear

groups

jonathan brundan and alexander kleshchev ^{∗}

Introduction

Let G denote the finite general linear group GL_{n}(𝔽_{q}) over the finite field with q elements. Associated to each partition λ of n, there is an irreducible unipotent complex character χ_{λ} of G. The degree of χ_{λ} is a polynomial in q given by Green’s hook formula [8, p.444]; the polynomial is monic of degree b(λ′) where λ′ is the transpose of λ and, for a partition μ = (m_{1} ≥ m_{2} ≥ ⋯ ≥ m_{h} > 0) of n, b(μ) denotes \frac{n(n+1)}{2} − ∑_{i=1}^{h} im_{i}. An easy consequence of Green’s formula is that

χ_{λ}(1) ≥ q^{b(λ′)}.

The purpose of this note is to prove similar lower bounds for the degrees of the irreducible p-modular Brauer characters of G when p is coprime to q.

We first state a special case of our main result. Let F be an algebraically closed field of characteristic p > 0 not dividing q. Then, for each partition λ of n, there is an associated irreducible unipotent F G-module L(1, λ) (see [3, §3.5], or [11] where it is denoted D_{λ}). For an integer N ≥ 1, we say that a partition is N-regular if it does not have N or more non-zero parts that are equal; in particular, the only 1-regular partition is the zero partition. Then we show:

Theorem A. Let λ be an e-regular partition of n, where e is minimal such that 1 + q + ⋯ + q^{e−1} ≡ 0 (mod p). Then, dim L(1, λ) ≥ q^{b(λ′)}.

Thus, for e-regular partitions, the same lower bound as in characteristic 0 can be used. If the regularity assumption is dropped, it is easy to find examples where the bound fails (e.g. take λ = (1^{e})).

To formulate our main theorem in the general case, we need to recall a parametrization of the irreducible F G-modules. First, take σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q}. Define ℓ(d) to be the order of the image of q^{d} in F^{×}. A partition λ will be called σ-regular if either σ is a p′-element and λ is ℓ(d)-regular, or σ has order divisible by p and λ is p-regular. For k ≥ 1 and each (not necessarily regular) partition λ of k, there is an associated irreducible F GL_{dk}(q)-module L(σ, λ) (see [3, §3.5], or [12] where it is denoted D(σ, λ)).

1991 subject classification: 20C20, 20C33, 20G05, 17B37.

Note that the precise labelling of L(σ, λ) is not canonical: in the approach of [3], it depends ultimately on the choice of an embedding of 𝔽̄_{q}^{×} into the group of units of the algebraic closure of the p-adic field.

Choose a set Φ_{0} of orbit representatives for the action of Gal(𝔽̄_{q}/𝔽_{q}) on 𝔽̄_{q}^{×}. If σ ∈ Φ_{0} is a p′-element of degree d and i ≥ 0, we say that τ is an ith twist of σ if τ ∈ Φ_{0} has degree dℓ(d)p^{i}, order divisible by p and p′-part conjugate to σ. Now let Φ_{p} be some subset of Φ_{0} containing all p′-elements of Φ_{0}, and exactly one ith twist of every p′-element of Φ_{0} for every i ≥ 0 (such a subset exists by [3, (2.1a)]). We should point out that L(σ, λ) ≅ L(τ, λ) whenever σ, τ ∈ 𝔽̄_{q}^{×} have the same degree and conjugate p′-parts, which ensures that the parametrization of irreducibles described in the next paragraph is really independent of this choice of Φ_{p}.

We also need the Harish-Chandra operator ⋄ as introduced originally by Green [8, p.403]; see [3, §5.2, §2.2] for its precise definition and properties in the modular case. Then, every irreducible F G-module L can be written as

L ≅ L(σ_{1}, λ_{1}) ⋄ L(σ_{2}, λ_{2}) ⋄ … ⋄ L(σ_{a}, λ_{a})

for some a ≥ 1, distinct elements σ_{1}, …, σ_{a} ∈ Φ_{p} of degrees d_{1}, …, d_{a}, and σ_{i}-regular partitions λ_{i} of integers k_{i} ≥ 1 (so n necessarily equals n_{1}k_{1} + ⋯ + n_{a}k_{a}). This labelling of L is unique up to reordering of the terms in the Harish-Chandra product, and the resulting parametrization of the irreducible F G-modules is the one that arises naturally from Harish-Chandra theory, see e.g. [4]. Its relationship to the more usual parametrization used in [12] and [3, §4.4] was originally explained in [6]; it is best understood as an application of the non-defining characteristic tensor product theorem, see [5] or [3, §4.3].

Our main result is as follows.

Theorem B. Given an irreducible F G-module L ≅ L(σ_{1}, λ_{1}) ⋄ … ⋄ L(σ_{a}, λ_{a}) for a ≥ 1, distinct elements σ_{1}, …, σ_{a} ∈ Φ_{p} of degrees d_{1}, …, d_{a}, and σ_{i}-regular partitions λ_{i} of integers k_{i} ≥ 1 for i = 1, …, a, we have that

dim L ≥ | GL_{n}(q) : GL_{k_{1}}(q^{d_{1}}) × ⋯ × GL_{k_{a}}(q^{d_{a}})|′q^{d_{1}b(λ′_{1})+···+d_{a}b(λ′_{a})},

where for an integer N, N′ denotes its largest divisor that is coprime to q.

We remark that the lower bound in Theorem B is exact if and only if each partition λ_{i} has either just one row (“trivial”) or just one column (“Steinberg”). Moreover, the lower bound is always a polynomial in q whose leading term is the same as the leading term in the generic degree for the ordinary irreducible character with the same labelling.

The remainder of the article is organized as follows. In §1, we prove the key auxiliary result, namely, an analogue of the Premet-Suprunenko theorem [16, 17] for quantum linear groups. Our proof of this follows the original arguments of [16, 17] closely. The main result is proved in §2, ultimately as a consequence of the Premet-Suprunenko theorem and [3, Theorem 5.5d]. Finally, in §3 we apply the theorem to improve on results of Guralnick and Tiep [9] determining the irreducible F G-modules of small dimension. In particular, in Theorem 3.4, we list all irreducible F G-modules of dimensions ≤ q^{3n−9} explicitly.

1 A q-analogue of the Premet-Suprunenko theorem

In this section, F denotes an arbitrary algebraically closed field and q is a primitive ℓth root of unity in F. For convenience, we exclude the possibility that q = 1, since the main result below is already known [16, 17] in the classical case. Choose a square root v of q in F so that if ℓ is odd, then v is also a primitive ℓth root of unity. We are concerned with the divided power version of the quantized enveloping algebra of 𝔤𝔩_{n} over F at the parameter v, as defined originally by Lusztig [14, 15] and Du [7, §2] (who extended Lusztig’s construction from 𝔰𝔩_{n} to 𝔤𝔩_{n}). We also cite [1, 2] as general references for the rational representation theory of quantum groups at roots of unity.

To recall some definitions, let t be an indeterminate. Then, the quantized enveloping algebra U_{ℚ(t)} associated to 𝔤𝔩_{n} is the ℚ(t)-algebra with generators {E_{i}, F_{i}, K_{j}^{±1} | 1 ≤ i < n, 1 ≤ j ≤ n} subject to the relations

\begin{matrix}K_{i}K_{j} = K_{j}K_{i}, & K_{i}K_{i}^{−1} = K_{i}^{−1}K_{i} = 1, \\ K_{i}E_{j} = t^{δ_{i,j}−δ_{i,j+1}}E_{j}K_{i}, & K_{i}F_{j} = t^{δ_{i,j+1}−δ_{i,j}}F_{j}K_{i}, \end{matrix}

E_{i}F_{j} − F_{j}E_{i} = δ_{i,j} \frac{K_{i,i+1} − K_{i,i+1}^{−1}}{t − t^{−1}} ,

E_{i}E_{j} = E_{j}E_{i}, F_{i}F_{j} = F_{j}F_{i}  if |i − j| > 1,

\begin{matrix}E_{i}^{2}E_{j} − (t + t^{−1})E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0, \\ F_{i}^{2}F_{j} − (t + t^{−1})F_{i}F_{j}F_{i} + F_{j}F_{i}^{2} = 0 & if |i − j| = 1. \end{matrix}

Here, for any 1 ≤ i < j ≤ n, K_{i,j} denotes K_{i}K_{j}^{−1}. For a, b ∈ ℕ, X ∈ U_{ℚ(t)} and 1 ≤ j ≤ n, define

[a]! := ∏_{c=1}^{a} \frac{t^{c} − t^{−c}}{t − t^{−1}} ,  [ \begin{matrix}a \\ b \end{matrix} ] := ∏_{c=1}^{b} \frac{t^{a−c+1} − t^{−a+c−1}}{t^{c} − t^{−c}} ,

X^{(a)} := \frac{X^{a}}{[a]!},  [ \begin{matrix}K_{j} \\ b \end{matrix} ] := ∏_{c=1}^{b} \frac{K_{j}t^{−c+1} − K_{j}^{−1}t^{c−1}}{t^{c} − t^{−c}} .

Let U_{ℤ[t,t^{−1}]} be the ℤ[t, t^{−1}]-subalgebra of U_{ℚ(t)} generated by the elements E_{i}^{(a)}, F_{i}^{(a)}, K_{j}^{±1}

and [ \begin{matrix}K_{j} \\ a \end{matrix} ] for a ≥ 0, 1 ≤ i < n, 1 ≤ j ≤ n. We then obtain the F -algebra

U := F ⊗_{ℤ[t,t^{−1}]} U_{ℤ[t,t^{−1}]}

on change of rings, where we are regarding F as a ℤ[t, t^{−1}]-module by letting t ∈ ℤ[t, t^{−1}] act on F by multiplication by v ∈ F. ¿From now on, we only work with U, so can

denote the images of F_{i}^{(a)}, E_{i}^{(a)}, K_{i}^{±1}, [ \begin{matrix}K_{i} \\ b \end{matrix} ] ∈ U_{ℤ[t,t^{−1}]} in U by the same names with-

out confusion. Finally, let U^{−}, U^{+} and U^{0} denote the subalgebras of U generated by

\{F_{i}^{(a)} | 1 ≤ i < n, a ≥ 0\} , \{E_{i}^{(a)} | 1 ≤ i < n, a ≥ 0\} and \{K_{i}^{±1}, [ \begin{matrix}K_{i} \\ b \end{matrix} ] | 1 ≤ i ≤ n, b ≥ 0\}

respectively.

Let E denote the Euclidean space with orthonormal basis ε_{1}, …, ε_{n}. For 0 ≠ λ ∈ E, λ^{∨} denotes 2λ/⟨λ, λ⟩, where ⟨., .⟩ is the inner product on E. The root system of type A can be identified with the subset {ε_{i} − ε_{j} | 1 ≤ i, j ≤ n, i ≠ j} of E, and a set of simple roots is given by the α_{i} := ε_{i} − ε_{i+1} for i = 1, …, n − 1. The weight lattice is then the ℤ-submodule X of E generated by ε_{1}, …, ε_{n}, where we identify the element λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X with the unique algebra homomorphism λ : U^{0} → F such that

K_{j} ↦ v^{λ_{j}}, [ \begin{matrix}K_{j} \\ a \end{matrix} ] ↦ [ \begin{matrix}λ_{j} \\ a \end{matrix} ] for 1 ≤ j ≤ n, a ≥ 0.

A weight λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X is dominant, relative to our choice of simple roots, if and only if λ_{1} ≥ ⋯ ≥ λ_{n}; we let X^{+} ⊆ X denote the set of all such dominant weights. We also have the usual dominance ordering ≤ on X: λ ≤ μ if and only if μ − λ is a sum of simple roots.

We say a vector v in a U-module V has weight λ if Kv = λ(K)v for all K ∈ U^{0}, and is a highest weight vector if E_{i}^{(a)}v = 0 for all 1 ≤ i < n and a ≥ 1. For each λ ∈ X^{+}, there is a unique irreducible U-module L(λ) generated by a non-zero highest vector v_{λ} of weight λ. We also recall that L(λ) decomposes as the direct sum of its weight spaces, i.e. L(λ) = ⨁_{μ∈X} L(λ)_{μ} where L(λ)_{μ} = {v ∈ L(λ) | v has weight μ}. For λ ∈ X^{+}, Ω(λ) denotes the set of all weights μ ∈ X appearing with non-zero multiplicity in Weyl’s character formula for the irreducible 𝔤𝔩_{n}(ℂ)-module of highest weight λ. So, μ ∈ Ω(λ) if and only if μ and all its conjugates under the Weyl group are ≤ λ in the dominance order. Call λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X^{+} ℓ-restricted if ⟨λ, α_{i}^{∨}⟩ = λ_{i} − λ_{i+1} < ℓ for all i = 1, …, n − 1. The first lemma is well known.

1.1. Lemma. Let λ ∈ X^{+} be ℓ-restricted and 0 ≠ v ∈ L(λ)_{μ} for some μ ∈ X. If E_{i}v = 0 for all 1 ≤ i < n, then μ = λ.

Proof. This is equivalent to the fact that for ℓ-restricted λ, the module L(λ) is irreducible over the (thickened) Frobenius kernel, i.e. the subalgebra of U generated by U^{0} and {E_{i}, F_{i}}_{1≤i<n} (see e.g. [2, 1.9]).

1.2. Lemma. Let λ ∈ X^{+} be ℓ-restricted and fix some 1 ≤ i < n. Given μ = λ − ∑_{j≠i} m_{j}α_{j} for integers m_{j} ≥ 0, the restriction of the operator F_{i}^{(a)} to the weight space L(λ)_{μ} is injective for all 0 ≤ a ≤ ⟨μ, α_{i}^{∨}⟩.

Proof. We use induction on M := ∑_{j≠i} m_{i}. Suppose first that M = 0, when μ = λ and

we need to prove that F_{i}^{(a)}v_{λ} ≠ 0 for all 0 ≤ a ≤ ⟨λ, α_{i}^{∨}⟩. Using well known commutation relations (see e.g. [15, 6.5(a2)], [1, Lemma 1.1]),

E_{i}^{(a)}F_{i}^{(a)}v_{λ} = [ \begin{matrix}⟨λ,α_{i}^{∨}⟩ \\ a \end{matrix} ] v_{λ}

which is non-zero as a ≤ ⟨λ, α_{i}^{∨}⟩ < ℓ. So certainly F_{i}^{(a)}v_{λ} ≠ 0.

Now suppose that M > 0 and that the lemma has been proved for all smaller M. Take 0 ≠ v ∈ L(λ)_{μ}. We need to show that F_{i}^{(a)}v ≠ 0 for all 0 ≤ a ≤ ⟨μ, α_{i}^{∨}⟩. Noting that E_{i}v = 0 by weights, Lemma 1.1 implies that for some h ≠ i, E_{h}v ≠ 0. By induction, F_{i}^{(a)}E_{h}v ≠ 0 for all 0 ≤ a ≤ ⟨λ + α_{h} − ∑_{j≠i} m_{j}α_{j}, α_{i}^{∨}⟩. So, since F_{i}^{(a)} and E_{h} commute,

we deduce that F_{i}^{(a)}v ≠ 0 for all 0 ≤ a < ⟨λ − ∑_{j≠i} m_{j}α_{j}, α_{i}^{∨}⟩. Now consider the case a = ⟨λ − ∑_{j≠i} m_{j}α_{j}, α_{i}^{∨}⟩. Here,

E_{i}^{(a)}F_{i}^{(a)}v = [ \begin{matrix}a \\ a \end{matrix} ] v = v

so F_{i}^{(a)}v ≠ 0 as required

1.3. Lemma ([17, Lemma 3],[16, Lemma 3]). Let λ ∈ X^{+} and μ = λ − ∑_{j} m_{j}α_{j} for integers m_{j} ≥ 0 such that μ is dominant. Choose i such that m_{i} is minimal among all the m_{j}. Then, λ − ∑_{j≠i} m_{j}α_{j} is a weight in Ω(λ), and m_{i} ≤ ⟨λ − ∑_{j≠i} m_{j}α_{j}, α_{i}^{∨}⟩.

Proof. We claim more: for any d ≤ m_{i} and any root α, the weight μ + dα lies in Ω(λ). To see this, let α_{0} = α_{1} + ⋯ + α_{n−1} be the highest root and take w in the Weyl group such that w(μ + dα) is dominant. Then, wμ + wdα ≤ wμ + dα_{0} ≤ μ + dα_{0} ≤ λ by choice of d. So, w(μ + dα) ≤ λ, as required to show that μ + dα ∈ Ω(λ). The second statement of the lemma now follows easily on considering the α_{i}-string through λ − ∑_{j≠i} m_{j}α_{j}.

Now we can prove the main result of this section. The theorem asserts that for ℓrestricted λ, the set of non-zero weights of L(λ) is the same as the set Ω(λ) of non-zero weights of the corresponding Weyl module. Combining the theorem with the q-analogue of Steinberg’s tensor product theorem, it allows one to determine the non-zero weights of L(λ) for arbitrary λ ∈ X^{+}. The proof given here is essentially identical to Suprunenko’s proof in the classical case [17].

1.4. Theorem. Let λ ∈ X^{+} be ℓ-restricted. Then, dim L(λ)_{μ} ≥ 1 for all μ ∈ Ω(λ).

Proof. The theorem is clear in the case n = 1, so suppose that n > 1 and that the result has been proved for all smaller n. For a fixed ℓ-restricted λ ∈ X^{+}, we proceed by downward induction on the dominance order on Ω(λ), the result being clear for μ = λ. It suffices to consider the case that μ ∈ Ω(λ) is dominant. Write μ = λ − ∑_{j} m_{j}α_{j} and choose i such that m_{i} is minimal.

Suppose first that m_{i} = 0. Then, we pass to the Levi subalgebra U′ of U generated by U^{0} and all E_{j}^{(a)}, F_{j}^{(a)} for j ≠ i. The vector v_{λ} generates a U′-submodule of L(λ) which is a highest weight U′-module of highest weight λ (actually, the irreducible U′-module of highest weight λ though we do not need this much). By the hypothesis on n, this submodule has non-zero μ-weight space, as required.

If m_{i} > 0, λ − ∑_{j≠i} m_{j}α_{j} > μ also lies in Ω(λ) by Lemma 1.3. So, by the induction hypothesis, L(λ)_{λ−∑_{j≠i} m_{j}α_{j}} ≠ 0. By Lemma 1.3 again, m_{i} ≤ ⟨λ − ∑_{j≠i} m_{j}α_{j}, α_{i}^{∨}⟩. So

by Lemma 1.2, the operator F_{i}^{(m_{i})} gives an injection of L(λ)_{λ−∑_{j≠i} m_{j}α_{j}} into the required weight space L(λ)_{μ}. Hence L(λ)_{μ} is non-zero too.

2 Main results

Let n ≥ 1 and μ = (m_{1}, m_{2}, …, m_{h}) be any composition of n (i.e. m_{1}, …, m_{h} are non-negative integers summing to n) with m_{h} ≠ 0. We say that μ has no gaps if m_{i} > 0 for all i = 1, 2, …, h. If μ has no gaps, we write ν ∼ μ if ν is a composition with no gaps obtained from μ by reordering its non-zero parts. We denote by μ^{+} the unique partition obtained by ordering the parts of μ in decreasing order. Also, μ^{op} denotes the composition with no gaps obtained by reversing the order of the parts (e.g. (2, 5, 7)^{op} = (7, 5, 2)).

For a composition μ = (m_{1}, …, m_{h}) of n, we define

Int(μ) := {1, 2, …, n} \ {m_{1}, m_{1} + m_{2}, …, m_{1} + ⋯ + m_{h−1}, m_{1} + ⋯ + m_{h}}.

If in addition λ is a partition of n and t is an indeterminate, the following polynomials in ℤ[t] were defined in [3, §5.5] by:

R_{μ}(t) := ∏_{i∈Int(μ)}(t^{i} − 1), S_{λ}(t) := ∑_{μ∼λ} R_{μ}(t).

We also let ≤ denote the dominance order on partitions as in the previous section, and for a partition λ we define the integer b(λ) as in the introduction.

2.1. Lemma. Let λ be a partition of n. Then R_{λ^{op}}(t) is a monic polynomial of degree b(λ), and deg R_{μ}(t) < deg R_{λ^{op}}(t) for all μ ∼ λ different from λ^{op}. In particular, S_{λ}(t) is monic of degree b(λ).

Proof. Follows from the definitions.

2.2. Lemma. If μ < λ are distinct partitions of n then b(λ) < b(μ).

Proof. Let λ = (l_{1}, …, l_{h}) and μ = (m_{1}, …, m_{k}) be the parts of λ and μ. Then, we have ∑_{i=1}^{j} l_{i} ≥ ∑_{i=1}^{j} m_{i} for all j, with strict inequality for at least one j. Hence ∑_{j=1}^{n} ∑_{i=1}^{j} l_{i} > ∑_{j=1}^{n} ∑_{i=1}^{j} m_{i} or ∑_{i=1}^{n} il_{i} > ∑_{i=1}^{n} im_{i}, which implies the result.

2.3. Lemma. Let μ be a composition of n with no gaps, M := Int(μ^{op}), and N be any subset of M. Then there exists a composition ν of n with no gaps such that ν^{+} ≤ μ^{+} and Int(ν) = N.

Proof. Let M = {i_{1}, …, i_{r}} and 1 ≤ j ≤ r. It suffices to prove the result for N = {i_{1}, …, î_{j}, …, i_{r}}. Let us label the boxes of the Young diagram of μ with 1, 2, …, n from left to right along the rows starting from the bottom row, as in the following picture:

7 8 4 5 6 2 3 1 Then i ∈ M if and only if i is not the last entry in a row. Now, let i_{j} be contained in row k and column l. Set ν := (μ_{1}, …, μ_{k−1}, l, μ_{k} − l, μ_{k+1}, …, μ_{h}). For instance, if i_{j} = 5 in the example above, then

ν =

It is clear that Int(ν) = {i_{1}, …, î_{j}, …, i_{r}} and ν^{+} ≤ μ^{+}.

If M ⊆ {1, …, n}, we set

φ_{M}(t) := ∏_{i∈M}(t^{i} − 1).

In particular, R_{ν}(t) = φ_{Int(ν)}(t). Also set σ(M) = ∑_{i∈M} i.

2.4. Lemma. ∑_{N⊆M} φ_{N}(t) = t^{σ(M)}.

Proof. ∑_{N⊆M} ∏_{i∈N}(t^{i} − 1) = ∏_{i∈M}((t^{i} − 1) + 1) = ∏_{i∈M} t^{i} = t^{σ(M)}.

2.5. Theorem. For a positive integer q and a partition λ of n,

∑_{μ≤λ} S_{μ}(q) ≥ q^{b(λ)},

summing over partitions μ of n, with equality if and only if λ = (n) or (1^{n}).

Proof. For every N ⊆ Int(λ^{op}) pick ν_{N} with (ν_{N})^{+} ≤ λ and Int(ν_{N}) = N. This is possible in view of Lemma 2.3. Then, we have

∑_{μ≤λ} S_{μ}(q) = ∑_{N⊆Int(λ^{op})} R_{ν_{N}} (q) + (∗)

where (∗) stands for a sum of certain terms of the form R_{ν}(q). Observe by the definition of R_{ν}(q), (∗) is a non-negative integer. Moreover, using Lemma 2.4 we get

∑_{N⊆Int(λ^{op})} R_{ν_{N}} (q) = ∑_{N⊆Int(λ^{op})} φ_{Int(ν_{N})}(q) = ∑_{N⊆Int(λ^{op})} φ_{N}(q) = q^{σ(Int(λ^{op}))}

which equals q^{b(λ)}. This proves the inequality. To see that it is strict if λ ≠ (n), (1^{n}), observe that then there is some 1 ≤ m < n not contained in Int(λ^{op}). Set ν = (1^{m−1}, 2, 1^{n−m−1}); then, ν^{+} ≤ λ and Int(ν) = {m} which is not contained in Int(λ^{op}). So the term R_{ν}(q) gives a non-zero contribution to (∗) which makes the inequality strict. Finally, for λ = (n) or (1^{n}), one easily sees that the term (∗) is zero so equality holds in these cases.

Now we can prove Theorem B from the introduction. For partitions λ, μ of n and d ≥ 1 define

m_{λ,μ}(q^{d}) := dim L(λ)_{μ},

the dimension of the μ-weight space of the irreducible highest weight module L(λ) over the quantum 𝔤𝔩_{n} over F as in §1 but with the image of q^{d} in F replacing the parameter q there; all other notation is as in the introduction. We state [3, Theorem 5.5d]:

2.6. Lemma. For σ ∈ Φ_{p} of degree d and a partition λ of k ≥ 1,

dim L(σ, λ′) = | GL_{dk}(q) : GL_{k}(q^{d})|′ ∑_{μ≤λ} m_{λ,μ}(q^{d})S_{μ}(q^{d})

where the sum is over partitions μ of n.

Combining this with Theorem 1.4 (or [16, 17] if q^{d} ≡ 1 (mod p)) and Theorem 2.5, we deduce immediately that:

2.7. Theorem. For σ ∈ Φ_{p} of degree d and a σ-regular partition λ of k ≥ 1,

dim L(σ, λ′) ≥ | GL_{dk}(q) : GL_{k}(q^{d})|′q^{db(λ)}

with equality if and only if λ = (k) or (1^{k}).

Theorem B in the introduction now follows immediately from this and the definition of the Harish-Chandra operator ⋄. To deduce Theorem A, it is obviously a special case of Theorem B unless q ≡ 1 (mod p), in which case there exists a 0th twist σ of 1, also of degree 1 over 𝔽_{q}. Then, L(1, λ) ≅ L(σ, λ) and using this observation, the statement in Theorem A again follows as a special case of Theorem B.

3 Application: low-dimensional representations

Now we illustrate the usefulness of Theorem B by applying it to list all irreducible F Gmodules of dimension ≤ q^{3n−9}. For simplicity, we only consider n ≥ 5 (for n < 5 it is an easy matter to explicitly list all irreducible F G-modules and their dimensions using [13]).

Given non-negative integers n_{1}, …, n_{a} summing to n, define

\{n\}! := \frac{(q^{n} − 1)(q^{n−1} − 1) … (q − 1)}{(q − 1)^{n}}, \{ \begin{matrix}n \\ n_{1},...,n_{a} \end{matrix} \} := \frac{\{n\}!}{\{n_{1}\}!\{n_{2}\}! … \{n_{a}\}!},

\{k|d\} := \frac{∏_{i=1}^{dk}(q^{i} − 1)}{∏_{i=1}^{k}(q^{di} − 1)}.

Note that the index | GL_{n}(q) : GL_{n_{1}}(q)×⋯×GL_{n_{a}}(q)|′ equals \{ \begin{matrix}n \\ n_{1},...,n_{a} \end{matrix} \} and the index

| GL_{dk}(q) : GL_{k}(q^{d})|′ = {k|d}.

3.1. Lemma. For a ≥ 2 and n_{1}, …, n_{a} ≥ 1 with n = n_{1} + ⋯ + n_{a},

\{ \begin{matrix}n \\ n_{1},...,n_{a} \end{matrix} \} > q^{∑_{1≤i<j≤a} n_{i}n_{j}}.

Proof. Noting that \frac{q^{n}−1}{q^{m}−1} > q^{n−m} for n > m, we have for a = 2 that

\{ \begin{matrix}n \\ n_{1},n_{2} \end{matrix} \} = \frac{(q^{n} − 1) … (q^{n−n_{1}+1} − 1)}{(q^{n_{1}} − 1) … (q − 1)} > (q^{n−n_{1}})^{n_{1}} = q^{n_{1}n_{2}}.

Since \{ \begin{matrix}n \\ n_{1},...,n_{a} \end{matrix} \} = \{ \begin{matrix}n \\ n_{1}+n_{2},n_{3},...,n_{a} \end{matrix} \} \{ \begin{matrix}n_{1}+n_{2} \\ n_{1},n_{2} \end{matrix} \} , the general case a > 2 now follows easily

by induction.

3.2. Lemma. Suppose that n = kd for integers k ≥ 1, d ≥ 2. Then,

\{k|d\} > q^{\frac{1}{2} n(n−k)−1−δ_{q,2}} ≥ q^{\frac{n^{2}}{4} −2}.

Proof. Dividing both sides by q^{\frac{1}{2} n(n−k)}, the inequality is equivalent to proving that

∏_{1≤i≤n,d∤i}(1 − \frac{1}{q^{i}} ) > \frac{1}{q^{1+δ_{q,2}}} .

The left hand side is certainly greater than the infinite product ∏_{i≥1} (1 − \frac{1}{q^{i}} ) . Using a theorem of Euler [10, Theorem 353], this is bounded below by

1 − \frac{1}{q} − \frac{1}{q^{2}} = \frac{q^{2} − q − 1}{q^{2}} ≥ \frac{1}{q^{1+δ_{q,2}}} ,

which completes the proof of the first inequality. The second is obvious.

3.3. Lemma. Suppose that n ≥ 5, a ≥ 2, 1 ≤ n_{1} ≤ ⋯ ≤ n_{a} and n = n_{1} + ⋯ + n_{a}. If

\{ \begin{matrix}n \\ n_{1},...,n_{a} \end{matrix} \} ≤ q^{3n−9} then (n_{1}, …, n_{a}) is equal to either (r, n−r) with r ≤ 2 or (1, 1, n−2).

Proof. First suppose that a = 2. It is enough to show that \{ \begin{matrix}n \\ 3,n−3 \end{matrix} \} > q^{3n−9}, which

follows from Lemma 3.1. Now suppose that a = 3. Since \{ \begin{matrix}n \\ n_{1},n_{2},n_{3} \end{matrix} \} ≥ \{ \begin{matrix}n \\ n_{1}+n_{2},n_{3} \end{matrix} \}, one can only have (n_{1}, n_{2}, n_{3}) = (1, 1, n − 2) by what we have proved so far. Similarly, a > 3 cannot occur.

Now we prove the main result of the section:

3.4. Theorem. Let n ≥ 5 and L be an irreducible F G-module. Then,

dim L ≤ q^{3n−9}

if and only if L is isomorphic to one of the modules in the table below. In the table, δ_{n}, ε are defined by

δ_{n} = \{ \begin{matrix}1 & if e|n, \\ 0 & otherwise, \end{matrix} ε = \{ \begin{matrix}1 & if e > 2 and e|n − 1, or e = 2 and 2p|(n − 1), \\ −1 & if e = 2, 2|n and 2p ∤ (n − 2), \\ 0 & otherwise \end{matrix}

where e is the smallest positive integer such that 1 + q + ⋯ + q^{e−1} ≡ 0 (mod p).

L  dim L  Conditions
L(σ, (n))  1  deg(σ)=1
L(σ, (n−1, 1))  \frac{q^{n}−q}{q−1} − δ_{n}  deg(σ)=1
L(σ, (n−2, 2))  \frac{(q^{n}−1)(q^{n−1}−q^{2})}{(q^{2}−1)(q−1)} −δ_{n−2} \frac{q^{n}−q}{q−1} −ε  deg(σ)=1, n > 5,
(n, q) ≠ (6, 2)
L(σ, (n−2, 1^{2})) \frac{(q^{n}−q^{2})(q^{n}−q)}{(q^{2}−1)(q−1)} − δ_{n} \frac{q^{n}−2q+1}{q−1} deg(σ)=1, n > 6,
(n, q) ≠ (7, 2)
L(σ, (3))  (q − 1)(q^{3} − 1)(q^{5} − 1)  deg(σ)=2, n=6
L(σ, (4))  27559  deg(σ)=2, n=8, q =2
L(σ_{1}, (1)) ⋄ L(σ_{2}, (n−1))  \frac{q^{n}−1}{q−1}  deg(σ_{i})=1, σ_{1} ≠σ_{2}
L(σ_{1}, (1)) ⋄ L(σ_{2}, (n−2, 1))  \frac{q^{n}−1}{q−1} (\frac{q^{n−1}−q}{q−1} − δ_{n−1})  deg(σ_{i})=1, σ_{1} ≠σ_{2}, n > 6,
(n, q) ≠ (7, 2)
L(σ_{1}, (2)) ⋄ L(σ_{2}, (n−2)) \frac{(q^{n}−1)(q^{n−1}−1)}{(q^{2}−1)(q−1)} deg(σ_{i})=1, σ_{1} ≠σ_{2}, n > 5,
(n, q) ≠ (6, 2)
L(σ_{1}, (1^{2})) ⋄ L(σ_{2}, (n−2)) \frac{(q^{n}−1)(q^{n−1}−1)}{(q^{2}−1)(q−1)}(q − δ_{2}) deg(σ_{i})=1, σ_{1} ≠σ_{2}, n > 6,

if e ∤ 2 then (n, q) ≠ (7, 2)

L(σ_{1}, (1)) ⋄ L(σ_{2}, (n−2))  \frac{(q^{n}−1)(q^{n−1}−1)}{(q^{2}−1)}  deg(σ_{1})=2, deg(σ_{2})=1, n > 6,
L(σ_{1}, (1)) ⋄ L(σ_{2}, (1))  \frac{(q^{n}−1)(q^{n−1}−1)}{(q−1)^{2}}  deg(σ_{i})=1, σ_{i} ≠σ_{j}, n > 6,
⋄L(σ_{3}, (n−2))  (n, q) ≠ (7, 2)

Proof. First, we show that if dim L ≤ q^{3n−9}, then L is one of the modules in the table. Take

L ≅ L(σ_{1}, λ_{1}) ⋄ … ⋄ L(σ_{a}, λ_{a})

where the σ_{i} of degree d_{i} and the σ_{i}-regular partitions λ_{i} of k_{i} are as in Theorem B. Also let x_{i} be the number of nodes in the diagram of λ_{i} outside of the first row and set n_{i} = d_{i}k_{i}, for each i = 1, …, a. By Theorem B and Lemma 3.3, we may assume that (n_{1}, …, n_{a}) = (n), (r, n−r) with r ≤ 2 or (1, 1, n−2). Consider the case (1, 1, n−2), when

we need to show that d_{3} = 1 and x_{3} = 0. Well, if d_{3} > 1 then dim L > q^{2n−3}q^{\frac{(n−2)^{2}}{4} −2} using Lemmas 3.1–3.2, which for n ≥ 5 is > q^{3n−9}. So, d_{3} = 1. Now suppose x_{3} > 0. Then, by Lemma 3.1 and Theorem B, dim L > q^{2n−3}q^{n−3} > q^{3n−9}, so x_{3} = 0. For the case (2, n − 2), the same argument forces d_{2} = 1, x_{2} = 0. For (1, n − 1), first suppose that d_{2} > 1. Then, dim L > q^{n−1}q^{\frac{(n−1)^{2}}{4} −2} which is easily checked to be ≥ q^{3n−9} for n ≥ 5. So d_{2} = 1. Also if x_{2} > 1, then dim L > q^{n−1}q^{2n−6} > q^{3n−9} using Lemma 3.1 and Theorem B. So x_{2} ≤ 1. Finally, consider the case a = 1. If d_{1} ≥ 2, we get dim L > q^{\frac{n^{2}}{4} −2} which easily forces n ≤ 8. A little further direct calculation reveals the only possibilities are then d_{1} = 2 and n = 6, or d_{1} = 2, n = 8 and q = 2. Finally, if d_{1} = 1, and x_{1} ≥ 3, then dim L > q^{3n−9} applying Theorem 2.7 (note the equality there is certainly strict for x_{1} ≥ 3).

To calculate the dimensions of the entries in the table, one uses the hook formula and the following fragment of the unipotent decomposition matrix of GL_{n}(𝔽_{q}) (in the notation of [13]):

(n) (n − 1, 1) (n − 2, 2) (n − 2, 1^{2})
(n)  1 0  0  0
(n − 1, 1) δ_{n} 1  0  0
(n − 2, 2) κ δ_{n−2}  1  0
(n − 2, 1^{2}) 0 δ_{n}  0  1

where

κ = \{ \begin{matrix}1 & if e = 2 and n ≡ 1 or 2 (mod 2p), \\ δ_{n−1} & if e > 2, \\ 0 & otherwise. \end{matrix}

The entries in this decomposition matrix follow at once from [13, Theorem 6.22] (for hook partitions) or [11, Theorem 20.6] (for two row partitions).

Finally, we need to determine which of the remaining possibilities do indeed satisfy the bound. Certainly, the largest dimension of any of our modules is

\frac{(q^{n} − 1)(q^{n−1} − 1)}{(q − 1)^{2}} < q^{2n−2+δ_{q,2}},

so for n ≥ 7 + δ_{q,2}, all entries do definitely satisfy the bound dim L ≤ q^{3n−9}. Further calculation for n < 7 + δ_{q,2} gives rise to some extra exclusions for small n and q, as in the table.

3.5. Remark. There are several irreducible F G-modules with q^{3n−9} < dim L ≤ q^{3n−8}. The smallest of these is the module L(σ, (n − 3, 3)) for σ of degree 1, whose dimension generically is \frac{(q^{n}−1)(q^{n−1}−q)(q^{n−2}−q^{2})}{(q^{3}−1)(q^{2}−1)(q−1)}.

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brundan@darkwing.uoregon.edu, klesh@math.uoregon.edu

Department of Mathematics, University of Oregon, Eugene, Oregon, U.S.A.