Generalized Blocks of Unipotent Characters in the Finite General Linear Group

Jean-Baptiste Gramain

Department of Mathematical Sciences University of Aberdeen, Scotland

August, 2006

Abstract In a paper of 2003, B. Külshammer, J. B. Olsson and G. R. Robinson defined ℓ-blocks for the symmetric groups, where ℓ > 1 is an arbitrary integer, and proved that they satisfy an analogue of the Nakayama Conjecture. Inspired by this work and the definitions of generalized blocks and sections given by the authors, we give in this paper a definition of d-sections in the finite general linear group, and construct d-blocks of unipotent characters, where d ≥ 1 is an arbitrary integer. We prove that they satisfy one direction of an analogue of the Nakayama Conjecture, and, in some cases, prove the other direction. We also prove that they satisfy an analogue of Brauer’s Second Main Theorem.

1 Introduction, generalized blocks

Throughout this paper, we let n be a positive integer and q be a power of a prime p. We let V be an n-dimensional vector space over a finite field F_{q} with q elements. We will work in the finite general linear group G = GL(V). It will sometimes be convenient to choose a basis for V, and then identify G with the group GL(n, q) of invertible n by n matrices with entries in F_{q}. The irreducible complex characters of G have been described by J. A. Green in [6], using deep combinatorial arguments. Then, using in particular the Deligne-Lusztig theory, P. Fong and B. Srinivasan have classified the blocks of G (cf [5]). The unipotent characters of G are parametrized by the partitions of n. It turns out that, if r is a prime not dividing q, then two unipotent characters belong to the same r-block of G if and only if the partitions labeling them have the same e-core, where e is the multiplicative order of q modulo r. This result is shown using analogues of the Murnaghan-Nakayama rule for irreducible characters of G. Our aim is to use these analogues to obtain properties of some generalized blocks we define in G, and which don’t depend on any prime. We construct unipotent blocks for G which satisfy one direction of an analogue of the Nakayama Conjecture. In some cases, we also prove that they satisfy both directions.

The concept of generalized blocks was introduced by B. Külshammer, J. B. Olsson and G. R. Robinson (cf [8]). Take any finite group H, and take a union C of conjugacy classes of H containing the identity. We can consider the restriction to C of the ordinary scalar product on characters of H. We denote by Irr(H) the set of complex irreducible characters of H. For χ, ψ ∈ Irr(H), we let

< χ, ψ >_{C}:= \frac{1}{|H|} ∑_{h∈C} χ(h)ψ(h^{−1}).

Then χ and ψ are said to be directly C-linked if < χ, ψ >_{C}≠ 0, and orthogonal across C otherwise. Then direct C-linking is a reflexive (since 1 ∈ C) and symmetric binary relation on Irr(H). Extending it by transitivity, we obtain an equivalence relation (called C-linking) on Irr(H) whose equivalence classes are called the C-blocks. Note that, since they are orthogonal on the whole of H, two distinct irreducible characters are directly linked across C if and only if they are directly linked across H \ C. Note also that, if we take C to be the set of r-regular elements of H, for some prime r, then the C-blocks are just the r-blocks (cf e.g. [10]).

In [8], the authors have defined ℓ-blocks for the symmetric groups, where ℓ ≥ 2 is any integer. To obtain this, they take C to be the set of ℓ-regular elements, i.e. none of whose cycle has length divisible by ℓ (in particular, if ℓ is a prime r, then the ℓ-blocks are just the r-blocks). The ℓ-blocks thus obtained satisfy an analogue of the Nakayama Conjecture: two irreducible characters φ_{λ} and φ_{μ} of the symmetric group S_{n} (where λ and μ are partitions of n) belong to the same ℓ-block if and only if λ and μ have the same ℓ-core.

Following this work, A. Maróti studied generalized blocks in the alternating groups, and proved that, if ℓ is 2 or any odd integer greater than 1, then the ℓ-blocks of the alternating groups also satisfy an analogue of the Nakayama Conjecture (cf [9]).

In the case of symmetric groups, the argument for one of the directions goes as follows. Any element of a symmetric group can be written uniquely as a disjoint product of an ℓ-regular element and an ℓ-element (i.e. all of whose cycles have length divisible by ℓ). Two elements are said to belong to the same ℓ-section if their ℓ-parts are conjugate. The ℓ-section of the identity is thus the set of ℓ-regular elements. The ℓ-sections have properties very similar to those of ordinary prime sections. Even though we don’t need sections in order to construct generalized blocks, they are important when one wants to discuss properties of blocks such as analogues of Brauer’s Second Main Theorem.

If x is an h-cycle of the symmetric group S_{n} and y ∈ S_{n} fixes the points moved by x (and can thus be seen as an element of S_{n−h}), then the Murnaghan-Nakayama Rule allows us the write the value of a character φ_{λ} of S_{n} at xy as a linear combination of the φ_{μ}(y)’s, where μ runs through the set of partitions of n − h which can be obtained by removing an h-hook from λ.

Repeated use of this rule allows us to remove all of the ℓ-part of an element by removing a sequence of ℓ-hooks from the partition labeling the character we consider. An induction argument then proves that, if two characters of S_{n} are linked across any ℓ-section, then the partitions labeling them must have the same ℓ-core.

Now, in G = GL(V), we don’t have such a nice cycle structure anymore. However, we can use the rational canonical form, which is based on the elementary divisors of a given element g. These are polynomials over F_{q}, and they correspond to the cyclic subspaces of V under the action of g. Then, very much like in the symmetric group, the Murnaghan-Nakayama Rule for unipotent characters allows us to relate the ”removal” of a cyclic subspace associated to a polynomial of degree d to the removal of d-hooks from the partition labeling the unipotent character we consider.

The definitions of d-element and d-regular element we give are such that we can apply the Murnaghan-Nakayama Rule and show by an induction argument that, if two unipotent characters are linked across any d-section, then the partitions labeling them have the same d-core.

In section 2, we present the conjugacy classes of GL(V) and the primary decomposition, which is the basis for the definitions we introduce. In section 3, we define the d-sections we work on, and the generalized blocks they define. Section 4 is devoted to the analogue of the Nakayama Conjecture for unipotent d-blocks. We prove that one direction is true in general, and that the other is true for d = 1 and, for any d > 0, for blocks of weight at most (d + 1)/2. We conjecture that, in fact, these restrictions are not necessary. Finally, in section 5, we discuss an analogue of Brauer’s Second Main Theorem for unipotent dblocks.

2 Conjugacy classes, primary decomposition

2.1 Rational canonical form

We first introduce the theory of elementary divisors and the rational canonical form in G, which gives a parametrization of the conjugacy classes. For the results we give in this section, we refer to Green [6].

For any non-negative integer k, we will write ν ⊢ k to say that ν is a partition of k, and we will write |ν| = k. We denote by P the set of all partitions. We write F = {f_{i}, i ∈ I} the set of irreducible, monic polynomials distinct from X and of degree at most n in F_{q}[X] (so that I is finite). For f ∈ F, we write δ(f) for the degree of f. Then the conjugacy classes of G are parametrized by the set of partition-valued functions ζ: F ⟶ P such that ∑_{f∈F} |ζ(f)|δ(f) = n. For such a ζ, the corresponding conjugacy class of G will be written (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}), where {f_{1}, …, f_{r}} = {f ∈ F, |ζ(f)| ≠ 0}, and, for 1 ≤ i ≤ r, ν_{i} = ζ(f_{i}). We will sometimes also use the notation (f^{ζ(f)})_{f∈F} (thus allowing |ζ(f)| to be 0).

Take a conjugacy class c = (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}) of G, and g ∈ c. Then, the characteristic polynomial of g over F_{q} is

Char(g) = ∏_{i=1}^{r} f_{i}^{|ν_{i}|}.

Writing, for each i, ν_{i} = (λ_{i,1} ≥ λ_{i,2} ≥ … ≥ λ_{i,s}) (where s can be chosen to be big enough for all f′_{i}s appearing in Char(g) (and even for all g ∈ G), by taking for example s = n), the minimal polynomial of g over F_{q} is

Min(g) = ∏_{i=1}^{r} f_{i}^{λ_{i,1}}.

For k = 1, …, s, the polynomial ED_{k}(g) = ∏_{i=1}^{r} f_{i}^{∑_{j=1}^{k} λ_{i,j}} is the k-th elementary divisor of g over F_{q}.

For any irreducible monic polynomial f over F_{q}, we let U(f) be the companion matrix of f, i.e. if f(X) = X^{d} − a_{d−1}X^{d−1} − ⋯ − a_{0}, then

U(f) = U_{1}(f) = ( \begin{matrix}0 & 1 \\ ⋮ & ... & ... & (0) \\ ⋮ & & ... & ... \\ 0 & ⋯ & ⋯ & 0 & 1 \\ a_{0} & a_{1} & ⋯ & ⋯ & a_{d−1} \end{matrix} ) .

For any positive integer λ, we write

U_{λ}(f) = ( \begin{matrix}U(f) & I_{d} \\ & U(f) & I_{d} \\ & & ... & ... \\ & & & ... & I_{d} \\ & & & & U(f) \end{matrix} ) ,

where I_{d} is the d by d identity matrix, and there are λ blocks U(f) on the diagonal. Note that U_{λ}(f) is equivalent over F_{q} to the companion matrix of f^{λ}. Finally, for ν = (λ_{1} ≥ … ≥ λ_{s}) any partition (of an integer k say), we write

U_{ν}(f) = ( \begin{matrix}U_{λ_{1}}(f) \\ & . & . & . \\ & & & & U_{λ_{s}}(f) \end{matrix} ) ,

where U_{0}(f) has to be seen as the empty matrix. If g ∈ c = (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}), then, in any matrix representation of G, g is equivalent over F_{q} to

U_{(f_{1}^{ν_{1}} , ..., f_{r}^{νr})} = ( \begin{matrix}U_{ν_{1}}(f_{1}) \\ & . & . & . \\ & & & & U_{ν_{r}}(f_{r}) \end{matrix} ) .

This is the rational canonical form of g over F_{q}.

2.2 The Jordan decomposition

An element g ∈ G is semisimple if and only if it is diagonalizable over an algebraic extension of F_{q} (i.e. some F_{q^{k}}, k ≥ 1), i.e. Min(g) splits over F_{q^{k}} and has only simple roots. Equivalently, g ∈ GL(V) is semisimple if and only if there is a basis of V consisting of eigenvectors of g. Let c be the conjugacy class of g. If c = (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}), then g is semisimple if and only if, for all 1 ≤ i ≤ r, we have ν_{i} = (1, …, 1).

Choose some basis for V. We write Δ((q_{i})) for the diagonal matrix with diagonal blocks the q_{i}’s. Suppose g = Δ((U_{ν_{i}}(f_{i}))). Then g = g_{S} + g_{N} = Δ((Uν̃_{i}(f_{i}))) + Δ((Ĩ_{ν_{i}}(f_{i}))), where ν̃_{i} = (1, …, 1) ⊢ |ν_{i}| = k_{i} and

Ĩ_{ν_{i}}(f_{i}) = ( \begin{matrix}(0) & I_{d_{i}} \\ & (0) & I_{d_{i}} \\ & & ... & ... \\ & & & ... & I_{d_{i}} \\ & & & & (0) \end{matrix} ) ,

with |ν_{i}| = k_{i} diagonal d_{i} by d_{i} blocks, where f_{i} has degree d_{i}.

Thus g_{S} = Δ((U_{k_{i}}(f_{i}))) is the semisimple part of g, and the unipotent part of g is g_{U} = I_{n} + g_{S}^{−1}g_{N}.

Now, if g̃ = hgh^{−1} for some h ∈ G, then g̃ = (hg_{S}h^{−1})(hg_{U}h^{−1}), and we see that the semisimple and unipotent parts of g̃ are hg_{S}h^{−1} and hg_{U}h^{−1} respectively.

2.3 Primary decomposition

We first mention a fact about centralizers in GL(V). Take g ∈ GL(V). If, in some matrix representation corresponding to the decomposition V = V_{1} ⊕ V_{2}, we have

g = ( \begin{matrix}g_{1} \\ & g_{2} \end{matrix} )

with, for i ∈ {1, 2}, g_{i} ∈ G_{i} = GL(V_{i}), and gcd(Min(g_{1}), Min(g_{2})) = 1, then

C_{G}(g) = \{( \begin{matrix}a \\ & b \end{matrix} ) , a ∈ C_{G_{1}}(g_{1}), b ∈ C_{G_{2}}(g_{2})\} ,

and thus C_{G}(g) ≅ C_{G_{1}}(g_{1}) × C_{G_{2}}(g_{2}) via h ⟼ (h|_{V_{1}}, h|_{V_{2}}) (which is independant on the matrix representation).

We also give the following lemma

Lemma 2.1. Let g ∈ G = GL(V). Suppose V_{1} and V_{2} are g-stable subspaces of V such that V = V_{1} ⊕ V_{2}. If g|_{V_{1}} ∈ (f^{ζ_{1}(f)})_{f∈F} and g|_{V_{2}} ∈ (f^{ζ_{2}(f)})_{f∈F} for some ζ_{1}, ζ_{2} : F ⟶ P, then g ∈ (f^{ζ_{1}(f)∪ζ_{2}(f)})_{f∈F}, where, by ∪, we denote the concatenation of partitions (i.e., for partitions μ and ν, the components of the partition μ ∪ ν are those of μ together with those of ν. If μ ⊢ k and ν ⊢ ℓ, then μ ∪ ν ⊢ k + ℓ.)

Proof. Taking any bases for V_{1} and V_{2} (which then add to give a basis of V), we can write

g = ( \begin{matrix}g_{1} \\ & g_{2} \end{matrix} )

(where g_{1} (resp. g_{2}) is the matrix of g|_{V_{1}} (resp. g|_{V_{2}})). The idea is that we can obtain the rational canonical form of g by reducing to this form g_{1} and g_{2}. There exist h_{1} ∈ GL(V_{1}) and h_{2} ∈ GL(V_{2}) such that

g = ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} )^{−1} ( \begin{matrix}U_{(f^{ζ_{1}(f)})_{f∈F}} \\ & U_{(f^{ζ_{2}(f)})_{f∈F}} \end{matrix} ) ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} ) .

Thus, for some permutation matrix P,

g = ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} )^{−1} P^{−1}U_{(f^{ζ_{1}(f)∪ζ_{2}(f)})_{f∈F}} P ( \begin{matrix}h_{1} \\ & h_{2} \end{matrix} ) ,

so that g ∈ (f^{ζ_{1}(f)∪ζ_{2}(f)})_{f∈F}.

We now turn to the primary decomposition of elements of G. Take g ∈ GL(V) and suppose g ∈ (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}). Then there exists a unique decomposition V = V_{1} ⊕ ⋯ ⊕ V_{r}, where the V_{i}’s are g-stable subspaces of V and, for 1 ≤ i ≤ r, g|_{V_{i}} ∈ (f_{i}^{ν_{i}}) ⊂ GL(V_{i}). For each 1 ≤ i ≤ r, V_{i} is given by

V_{i} = {v ∈ V | f_{i}^{k}(g)v = 0 for some k > 0}.

We have C_{G}(g) ≅ C_{GL(V_{1})}(g|_{V_{1}})×⋯×C_{GL(V_{r})}(g|_{V_{r}}). Then there exists a unique writing g = g_{1} … g_{r}, where, for each 1 ≤ i ≤ r, g_{i} ∈ GL(V), V_{i} is g_{i}-stable and g_{i}|_{V_{j}} = 1 for all j ≠ i. Indeed, we must have g_{i}|_{V_{i}} = g|_{V_{i}} for each 1 ≤ i ≤ r, and g_{i} is uniquely determined by this and its properties listed before. Furthermore, the g_{i}’s are pairwise commuting elements. We say that g_{1} … g_{r} is the primary decomposition of g. The g_{i}’s are the primary components of g.

More generally, an element of G is said to be primary if its characteristic polynomial is divisible by at most one irreducible polynomial distinct from X−1. We have the following (cf Fong-Srinivasan [5], Proposition (1A))

Proposition 2.2. Suppose that h is a semisimple primary element of some general linear group GL(m, q), and that h ∈ (f^{ν}) for some f ≠ X − 1. Writing d the degree of f, we thus have m = kd, where ν = (1, …, 1) ⊢ k. Then

C_{GL(kd,q)}(h) ≅ GL(k, q^{d}).

¿From the primary decomposition of g ∈ G, we deduce the following: if F is the disjoint union of F_{1} and F_{2}, then there is a unique decomposition V = V_{1}⊕V_{2} where V_{1} and V_{2} are g-stable, g|_{V_{1}} ∈ (f^{ν(f)})_{f∈F_{1}} and g|_{V_{2}} ∈ (f^{ν(f)})_{f∈F_{2}}. Then g has a unique decomposition g = g_{1}g_{2} = g_{2}g_{1} where g_{1}, g_{2} ∈ GL(V), g_{1}|_{V_{2}} = 1 and g_{2}|_{V_{1}} = 1. Indeed, under these hypotheses, necessarily, for i ∈ {1, 2}, V_{i} is g_{i}-stable and g_{i}|_{V_{i}} = g|_{V_{i}}. We then have C_{G}(g) ≅ C_{GL(V_{1})}(g|_{V_{1}})×C_{GL(V_{2})}(g|_{V_{2}}).

3 Sections, blocks

3.1 Sections

The idea of the following definitions is, in the rational canonical form we gave above, to isolate blocks corresponding to irreducible polynomials whose degree is divisible by a given (positive) integer d. We let, writing δ(f) the degree of any polynomial f,

F_{d} = {f ∈ F | f ≠ X − 1 and d|δ(f)} and F_{d}^{0} = F \ F_{d}.

We define the following union of conjugacy classes of G:

X_{d} = {g ∈ (f^{ν(f)})_{f∈F_{d}∪{X−1}} | ν(X − 1) = (0) or (1, …, 1)}.

Note that we have 1 ∈ X_{d}.

For x ∈ X_{d}, we let F_{x} = F_{d} and F_{x}^{0} = F_{d}^{0}. Then, for each x ∈ X_{d}, the set F is the disjoint union of F_{x} and F_{x}^{0}. For each x ∈ X_{d}, there exists a unique decomposition V = V_{x} ⊕ V_{x}^{0} such that V_{x} is x-stable, x|_{V_{x}^{0}} = 1 and x|_{V_{x}} ∈ (f^{ν(f)})_{f∈F_{x}}. We then have C_{G}(x) ≅ C_{GL(V_{x})}(x|_{V_{x}}) × GL(V_{x}^{0}), and we write C_{G}(x) = H_{x} × H_{x}^{0}. We let

Y_{d}(x) = {y ∈ GL(V) | V_{x}^{0} is y -stable, y|_{V_{x}} = 1 and y|_{V_{x}^{0}} ∈ (f^{ν(f)})_{f∈F_{x}^{0}}}.

From the definitions, we see that Y_{d}(x) ⊂ C_{G}(x) for any x ∈ X_{d}. We also see, using the remarks we made on the primary decomposition, that, for any g ∈ G, there exist unique x ∈ X_{d} and y ∈ Y_{d}(x) such that g = xy. Indeed, if g ∈ (f_{1}^{ν_{1}}, …, f_{r}^{ν_{r}}) and if we write, as above, V = V_{1} ⊕ ⋯ ⊕ V_{r} and g = g_{1} … g_{r}, then we necessarily have V_{x} = ⊕_{i∈I}V_{i}, where I ⊂ {1, …, r} is the set of indices i such that f_{i}^{ν_{i}} has the property defining X_{d}, V_{x}^{0} = ⊕_{i∉I}V_{i}, x|_{V_{x}} = g|_{V_{x}} and y|_{V_{x}^{0}} = g|_{V_{x}^{0}} (and x ∈ X_{d} and y ∈ Y_{d}(x) are uniquely determined by these conditions). We have x = ∏_{i∈I} g_{i} and y = ∏_{i∉I} g_{i}.

These definitions allow us to define (X_{d}, Y_{d})-sections as introduced in section 2 of [8]. The following proposition shows that X_{d} and Y_{d}(x) (x ∈ X_{d}) behave in many respects like the set of r-elements (r a prime) and the sets of r-regular elements of their centralizers respectively.

Proposition 3.1. Take any integer d > 0. Then, for any x ∈ X_{d},

(i) Y_{d}(x) is a union of conjugacy classes of C_{G}(x). (ii) For all y ∈ Y_{d}(x), C_{G}(xy) ≤ C_{G}(x). (iii) For all g ∈ G, Y_{d}(x^{g}) = Y_{d}(x)^{g}. (iv) Two elements of xY_{d}(x) are G-conjugate if and only if they are C_{G}(x)- conjugate.

(v) G = ∐_{x∈X_{d}/G}{(xy)^{G}, y ∈ Y_{d}(x)/C_{G}(x)}.

Proof. (i) We have C_{G}(x) = H_{x} × H_{x}^{0}, and we see C_{G}(x) as a subgroup of GL(V_{x}) × GL(V_{x}^{0}). For y ∈ Y_{d}(x), we have y = (y|_{V_{x}}, y|_{V_{x}^{0}}) = (1, y_{x}^{0}). Then, for all h = (h_{x}, h_{x}^{0}) ∈ C_{G}(x), h^{−1}yh = (1, (h_{x}^{0})^{−1}y_{x}^{0}h_{x}^{0}) ∈ Y_{d}(x). (ii) For all y ∈ Y_{d}(x), we have C_{G}(xy) ≅ C_{GL(V_{x})}(x|_{V_{x}}) × C_{GL(V_{x}^{0})}(y|_{V_{x}^{0}}) ≤ C_{GL(V_{x})}(x|_{V_{x}}) × GL(V_{x}^{0}) ≅ C_{G}(x), and C_{G}(xy) ≤ C_{G}(x) since the isomorphism on the left is the restriction to C_{GL(V_{x})}(x|_{V_{x}})×C_{GL(V_{x}^{0})}(y|_{V_{x}^{0}}) of the isomorphism on the right. (iii) Take g ∈ G and y ∈ Y_{d}(x). Then x^{g} ∈ X_{d} and F_{x^{g}} = F_{x} (and thus F_{x^{g}}^{0} = F_{x}^{0}). We have V = g^{−1}V = g^{−1}V_{x} ⊕ g^{−1}V_{x}^{0}. Furthermore, x^{g}|_{g^{−1}V_{x}^{0}} = 1, g^{−1}V_{x} is x^{g}-stable, and x^{g}|_{g^{−1}V_{x}} ∈ (f^{ν})_{f∈F_{x}}. Thus g^{−1}V_{x} = V_{x^{g}} and g^{−1}V_{x}^{0} = V_{x^{g}}^{0}. Now, since y ∈ Y_{d}(x), we see that y^{g}|_{g^{−1}V_{x}} = 1, g^{−1}V_{x}^{0} is y^{g}-stable, and y^{g}|_{g^{−1}V_{x}^{0}} ∈ (f^{ν})_{f∈F_{x}^{0}}. Hence y^{g} ∈ Y(x^{g}), and Y_{d}(x)^{g} ⊂ Y_{d}(x^{g}) for all g ∈ G. Then, for any g ∈ G, Y_{d}(x^{g})^{g^{−1}} ⊂ Y_{d}(x) so that Y_{d}(x^{g}) ⊂ Y_{d}(x)^{g}. Hence the result. (iv) Suppose that, for some y, z ∈ Y_{d}(x), there exists h ∈ G such that xy = h^{−1}xzh. Writing g = xy, we also have g = x^{h}z^{h}, and x^{h} ∈ X_{d} (since X_{d} is a union of G-conjugacy classes) and z^{h} ∈ Y_{d}(x^{h}) (by (iii)). By the uniqueness of such a writing for g, we have x^{h} = x and z^{h} = y. Hence h ∈ C_{G}(x). In particular, xy and xz are C_{G}(x)-conjugate.

(v) For any g ∈ G, there exists a unique x ∈ X such that g ∈ xY_{d}(x). Thus

G = ∐_{x∈X_{d}} xY_{d}(x)

= ∐_{x∈X_{d}/G}(xY_{d}(x))^{G} (because of (iii))

= ∐_{x∈X /G} ⋃_{y∈Y_{d}(x)}(xy)^{G}

= ∐_{x∈X_{d}/G} ∐_{y∈Y_{d}(x)/C_{G}(x)}(xy)^{G},

this last equality being true by (iv) (if (xy)^{g} = (xz)^{g′}, then xy = xz^{g′g^{−1}} so that, by (iv), y and z are C_{G}(x)-conjugate).

Because of the analogy we mentioned before the proposition, we introduce some terminology. We will refer to the elements of X_{d} as d-elements. One set of particular importance is Y_{d}(1). This is the set of elements of G whose minimal polynomial has no irreducible factor of degree divisible by d (except possibly X −1). We call the elements of Y_{d}(1) the d-regular elements of G. The elements of G \ Y_{d}(1) are called d-singular.

Note that, with these definitions, a 1-regular element of G is an element whose minimal polynomial has no irreducible factor of degree divisible by 1, except X − 1. This means that its minimal polynomial is a power of X − 1. Hence the 1-regular elements of GL(n, q) are precisely the unipotent elements.

For any x ∈ X_{d}, we call the union of the G-conjugacy classes meeting xY_{d}(x) the Y_{d}-section of x. The above proposition implies that, for each x ∈ X_{d}, induction of complex class functions gives an isometry from the space of class functions of C_{G}(x) vanishing outside xY_{d}(x) onto the space of class functions of G vanishing outside the Y_{d}-section of x (cf [8]).

We remark that the Y_{d}-sections of G are quite different from ordinary prime sections. Take any 1 ≠ x ∈ X_{d}. Then, by definition, Y_{d}(x) ⊂ Y_{d}(1) ∩ C_{G}(x). However, if d ≠ 1, then there exists λ ∈ F_{q}^{×} such that λI_{n} ∈ Y_{d}(1) ∩ C_{G}(x) but λI_{n} ∉ Y_{d}(x), so that Y_{d}(x) ≠ Y_{d}(1) ∩ C_{G}(x) (while this equality holds when we define (X_{d}, Y_{d})-sections to be the ordinary r-sections for some prime r).

Furthermore, still supposing d ≠ 1, if x is a (non trivial) r-element of G for some prime r, then, most of the time (that is, when q − 1 is not a power of r), there exists an r-regular element λ ∈ F_{q}^{×} such that λI_{n} ∉ Y_{d}(x), so that xλI_{n} ∉ (xY(x))^{G}. But xλI_{n} belongs to the r-section of x. Hence the Y_{d}-section of x is not a union of r-sections.

3.2 Blocks

With the definition of d-regular element we have given, we can now define generalized blocks of irreducible characters, as described in the introduction, by taking C to be the set Y_{d}(1) of d-regular elements. For any positive integer d, the d-blocks of G are defined to be minimal subsets of Irr(G) which are orthogonal across d-regular elements. Recall that two characters χ, ψ ∈Irr(G) are orthogonal across d-regular elements if

< χ, ψ >_{Y_{d}(1)}:= \frac{1}{|G|} ∑_{y∈Y_{d}(1)} χ(y)\overline{ψ(y)} = 0.

Otherwise, χ and ψ are said to be directly linked across d-regular elements.

We will particularly consider the restriction of the relation of Y_{d}(1)-linking to the subset of unipotent characters. The unipotent characters of G are the irreducible components of the permutation representation of G on the cosets of a Borel subgroup (i.e. the normalizer in G of a Sylow p-subgroup of G, where p is the defining characteristic). They are labeled by the partitions of n. For λ ⊢ n, we will write χ_{λ} the unipotent (irreducible) character of G labeled by λ. We write Unip(G) = {χ_{λ}, λ ⊢ n}. The equivalence classes of Unip(G) modulo the restriction of Y_{d}(1)-linking will be called unipotent d-blocks of G. It is clear that, for any d-block B of G, B ∩ Unip(G) is a union of unipotent d-blocks.

4 Nakayama Conjecture for unipotent blocks

4.1 Murnaghan-Nakayama Rule for unipotent characters

Pick g ∈ G, and write Char(g) = ∏_{i} f_{i}^{k_{i}} and the corresponding decomposition g = ∏_{i} g_{i}. Then pick i_{0}, and write g = ρσ, where

ρ = g_{i_{0}} and σ = ∏_{i≠i_{0}} g_{i}.

Now fix some matrix representation of G, and see G as GL(n, q). Writing d the degree of f_{i_{0}}, m = k_{i_{0}}d, and l = n − m, we have, writing ∼ for equivalence of matrices over F_{q},

g ∼ ( \begin{matrix}. & . & . \\ & & & U_{ν_{i}}(f_{i}) \\ & & & & . & . & . \end{matrix} ) ,

ρ ∼ ( \begin{matrix}I_{l} \\ & U_{ν_{i_{0}}} (f_{i_{0}}) \end{matrix} )

(and we may consider that ρ ∈ G_{0} = GL(m, q)), and

σ ∼ ( \begin{matrix}() \\ & . & . & . \\ & & & & () \\ & & & & & I_{m} \end{matrix} )

(and we may consider that σ ∈ G_{1} = GL(l, q)).

Then, using the results on the Jordan decomposition given in the first section, we see that the semi-simple part ρ_{S} of ρ is equivalent to

( \begin{matrix}I_{l} \\ & U_{k_{i_{0}}} (f_{i_{0}}) \end{matrix} ) .

We have ρ_{S} ∈ GL(n, q) and Char(ρ_{S}) = f_{i_{0}}^{k_{i_{0}}} (X − 1)^{l}, and we may consider that ρ_{S} ∈ G_{0} = GL(m, q) and Char(ρ_{S}) = f_{i_{0}}^{k_{i_{0}}} .

We have C_{G}(ρ_{S}) = H = H_{0} × H_{1}, where H_{1} ≅ GL(l, q) and H_{0} = C_{GL(m,q)}(ρ_{S}) ≅ GL(k_{i_{0}}, q^{d}).

We find in [5] an analogue of the Murnaghan-Nakayama Rule which applies to unipotent characters of G. More precisely, if χ_{ν} ∈ Unip(G), then the following theorem gives us information on the values of a class function χ^{ν} which is, up to a sign, the same as χ_{ν} (it is this class function, rather than the unipotent character, which appears naturally in the Deligne-Lusztig theory).

Theorem 4.1 (Murnaghan-Nakayama Rule). ([5], Theorem (2G)) Let g ∈ G, and ρ and σ be as above, and let ν ⊢ n. Then

χ^{ν}(ρσ) = ∑_{λ∈L_{ν}} a_{νλ}^{ρ}χ^{λ}(σ),

where L_{ν} is the set of partitions λ of l which can be obtained from ν by removing k_{i_{0}} d-hooks, and a_{νλ}^{ρ} ∈ Z[q^{d}]. If L_{ν} = ∅, then χ^{ν}(ρσ) = 0. We have a_{νλ}^{ρ} ≠ 0 for λ ∈ L_{ν}. (The coefficients of a_{νλ}^{ρ} depend on the characters of the symmetric group S_{k_{i_{0}}} and the Green functions of GL(k_{i_{0}}, q^{d}) ≅ H_{0} (applied to the unipotent part ρ_{U} of ρ), and all the non-zero coefficients of a_{νλ}^{ρ} have the same sign).

Remark: it is easy to see that, if a_{νλ}^{ρ} ≠ 0, then ν and λ have the same d-core.

The idea is to use this theorem recursively so as to be able to obtain information about < χ^{λ}, χ^{μ} >_{xY_{d}(x)}, for λ, μ ⊢ n and x ∈ X_{d}. We first use it to obtain a formula for χ^{μ}(xy), where μ ⊢ n and y ∈ Y_{d}(x). We take 1 ≠ x ∈ X_{d}. Suppose x ∈ c_{x} = (f_{1}^{λ_{1}}, …, f_{s}^{λ_{s}}) where, for each 1 ≤ i ≤ s, λ_{i} ⊢ k_{i}, and d|δ(f_{i}) or f_{i} = X − 1. If f_{i} ≠ X − 1, we let δ(f_{i}) = m_{i}d. From the definition of X_{d}, we see that, in the primary decomposition of x, we may omit the component corresponding to X − 1, because it is necessarily the identity. We relabel f_{1}, …, f_{r} the f_{i}’s which are distinct from X − 1, and we write x = x_{1} … x_{r}, where each x_{i} has exactly one elementary divisor distinct from X − 1 (namely, f_{i}), and with the same multiplicity as in Char(x). We will say that x has d-type (k_{1}m_{1}, …, k_{r}m_{r}).

By repeated use of the Murnaghan-Nakayama Rule, we obtain that, for any y ∈ Y_{d}(x) and for μ ⊢ n, we have

\begin{aligned}χ^{μ}(xy) & = ∑_{μ_{1}∈L_{μ}} a_{μμ_{1}}^{x_{1}}χ^{μ_{1}}(x_{2} … x_{r}y) \\ & = ∑_{μ_{1}∈L_{μ}} a_{μμ_{1}}^{x_{1}} (∑_{μ_{2}∈L_{μ_{1}}} a_{μ_{1}μ_{2}}^{x_{2}}χ^{μ_{2}}(x_{3} … x_{r}y)) \\ & = ∑_{μ_{1}∈L_{μ}} … ∑_{μ_{r}∈L_{μ_{r−1}}} a_{μμ_{1}}^{x_{1}} … a_{μ_{r−1}μ_{r}}^{x_{r}}χ^{μ_{r}}(y) \end{aligned} ,

which can be written

χ^{μ}(xy) = ∑_{λ∈L_{(k_{1}m_{1},...,kr mr)d}^{μ}} α_{μλ}^{(x_{1},...,x_{r})}χ^{λ}(y),

where the α_{μλ}^{(x_{1},...,x_{r})}’s are integers and L_{(k_{1}m_{1},...,k_{r}m_{r})d}^{μ} is the set of partitions of n−(∑_{i} k_{i}m_{i})d which can be obtained from μ by removing k_{1} (m_{1}d)-hooks, then k_{2} (m_{2}d)-hooks, …, and finally k_{r} (m_{r}d)-hooks. We will call such a sequence of removals a (k_{1}m_{1}, …, k_{r}m_{r})d-path from μ to λ.

Note that, in this sum, each λ can appear several times, if there is more than one (k_{1}m_{1}, …, k_{r}m_{r})d-path from μ to λ. Note also that, in the right side of this equality, y has implicitely been seen as an element of GL(l, q), where l = n − (∑_{i} k_{i}m_{i})d. If L_{(k_{1}m_{1},...,k_{r}m_{r})d}^{μ} = ∅, then χ^{μ}(xy) = 0.

We have α_{μλ}^{(x_{1},...,x_{r})} ∈ Z[q^{d}], and, if we separate the possibly multiple occurences

of each λ in the sum, then, for λ ∈ L_{(k_{1}m_{1},...,k_{r}m_{r})d}^{μ}, each of the α_{μλ}^{(x_{1},...,x_{r})}’s is non-zero (these are indexed by the (k_{1}m_{1}, …, k_{r}m_{r})d-paths from μ to λ).

If α_{μλ}^{(x_{1},...,x_{r})} ≠ 0, then, since there is a (k_{1}m_{1}, …, k_{r}m_{r})d-path from μ to λ, and since the removal of a hook of length md can be obtained by the removal of a sequence of m hooks of length d, we see that μ and λ have the same d-core.

We call the α_{μλ}^{(x_{1},...,x_{r})}’s the MN-coefficients, and we will from now on write

α_{μλ}^{x} for α_{μλ}^{(x_{1},...,x_{r})}.

4.2 One direction of the Nakayama Conjecture

We are now able to prove that the unipotent d-blocks of G satisfy one direction of a generalized Nakayama Conjecture. The proof we give is an adaptation to our case of the proof given by Külshammer, Olsson and Robinson in the case of symmetric groups.

Take x ∈ X_{d} of d-type km = (k_{1}m_{1}, …, k_{r}m_{r}). We let l = n − kmd = l − (∑_{i=1}^{r} k_{i}m_{i})d. Writing G_{0} = GL(kmd, q) = ∏_{i=1}^{r} GL(k_{i}m_{i}d, q) and G_{1} = GL(l, q), we have x = (x_{0}, x_{1}) = (x_{0}, 1) ∈ G_{0} × G_{1}. Then C_{G}(x) = H_{0} × H_{1}, where H_{1} ≅ GL(l, q) and H_{0} = C_{G_{0}}(x_{0}) (note that, if x is semisimple, then H_{0} ≅ GL(k, q^{md}) ≅ ∏_{i=1}^{r} GL(k_{i}, q^{m_{i}d})).

Now take y ∈ Y_{d}(x). Then, as an element of C_{G}(x) = H_{0} × H_{1}, we have y = (y_{0}, y_{1}) = (1, y_{1}).

Writing Y_{d}^{t}(u) for Y_{d}(u) when u ∈ GL(t, q), we have that y, element of C_{G}(x), belongs to Y_{d}^{n}(x) if and only if y = (1, y_{1}) ∈ H_{0} × H_{1}, where y_{1} belongs to Y_{d}^{l}(1). Hence Y_{d}^{n}(x) is in natural one to one correspondence with Y_{d}^{l}(1). Now

we consider μ, μ′ ⊢ n, and x ∈ X_{d} of d-type km = (k_{1}m_{1}, …, k_{r}m_{r}). We have

< χ^{μ}, χ^{μ′} >_{xY_{d}(x)}= \frac{1}{|G|} ∑_{y∈Y_{d}(x)} χ^{μ}(xy)\overline{χ^{μ′}(xy)}

= \frac{1}{|G|} ∑_{y=(y_{0},y_{1})∈Y_{d}(x)} [( ∑_{λ∈L_{km}^{μ}} α_{μλ}^{x}χ^{λ}(y_{1})) ( ∑_{λ′∈L_{km}^{μ′}} α_{μ′λ′}^{x} \overline{χ^{λ′}(y_{1})})]

= \frac{1}{|G|} ∑_{y_{1}∈Y_{d}^{l}(1)} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x}χ^{λ}(y_{1})\overline{χ^{λ′}(y_{1})}

(by the above remark on Y_{d}(x)).

We write A_{μμ′}^{x} =< χ^{μ}, χ^{μ′} >_{xY(x)}. Then

A_{μμ′}^{x} = \frac{1}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x} ∑_{y_{1}∈Y_{d}^{l}(1)} χ^{λ}(y_{1})\overline{χ^{λ′}(y_{1})}

= \frac{|H_{1}|}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x} < χ^{λ}χ^{λ′} >_{Y_{d}^{l}(1)}

i.e.

A_{μμ′}^{x} = \frac{|H_{1}|}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x}A_{λλ′}^{1}.

We use induction on n to prove that, if A_{μμ′}^{x} ≠ 0, then μ and μ′ have the same d-core. We may assume that μ ≠ μ′.

If n < d, then each partition is its own d-core. Furthermore, in this case, X_{d} = {1} and Y_{d}(1) = G. Thus, for all x ∈ X_{d}, we have A_{μμ′}^{x} = A_{μμ′}^{1}, and A_{μμ′}^{1} =< χ^{μ}, χ^{μ′} >_{G}= 0 (since μ ≠ μ′). Hence the result is true in this case. Thus, we suppose n ≥ d. First suppose x ≠ 1 and A_{μμ′}^{x} ≠ 0. Then

A_{μμ′}^{x} = \frac{|H_{1}|}{|G|} ∑_{λ∈L_{km}^{μ}, λ′∈L_{km}^{μ′}} α_{μλ}^{x}α_{μ′λ′}^{x}A_{λλ′}^{1} ≠ 0.

Thus there exist λ ∈ L_{km}^{μ} and λ′ ∈ L_{km}^{μ′} such that α_{μλ}^{x}α_{μ′λ′}^{x}A_{λλ′}^{1} ≠ 0. Then α_{μλ}^{x} ≠ 0 implies that μ and λ have the same d-core, and α_{μ′λ′}^{x} ≠ 0 implies that μ′ and λ′ have the same d-core. And, by the induction hypothesis (applied to n − kmd < n), A_{λλ′}^{1} ≠ 0 implies that λ and λ′ have the same d-core.

Now, if x = 1, we see, by the existence and uniqueness of the decomposition we introduced, that

0 =< χ^{μ}, χ^{μ′} >_{G}= ∑_{x∈X_{d}} < χ^{μ}, χ^{μ′} >_{xY_{d}(x)}= ∑_{x∈X_{d}} A_{μμ′}^{x}.

Hence, if A_{μμ′}^{1} ≠ 0, then there exists an x′ ∈ X_{d} \ {1} such that A_{μμ′}^{x′} ≠ 0. This in turn implies, by the previous case, that μ and μ′ have the same d-core. Skipping back from class functions to irreducible characters, we see that we have proved the following

Theorem 4.2. Let d > 0 be any integer. If two unipotent (irreducible) characters χ_{μ} and χ_{μ′} of G = GL(V) are directly linked across some xY_{d}(x), where x ∈ X_{d}, then μ and μ′ have the same d-core (and this is true in particular for x = 1).

Extending by transitivity the relation of direct Y_{d}(1)-linking, we obtain

Theorem 4.3. If two unipotent characters χ_{μ} and χ_{μ′} of G are in the same unipotent d-block of G, then μ and μ′ have the same d-core.

Each unipotent d-block of G is therefore associated to a d-core. For each given d-core γ, we can consider the union of the (possibly many) unipotent dblocks associated to γ. The (a priori) bigger blocks obtained in this way are parametrized by the set of d-cores of partitions of n, and they satisfy an analogue of the Nakayama Conjecture. In accordance with the terminology used in [8], we call them combinatorial unipotent d-blocks.

We would like to prove that, in fact, the notions of unipotent d-block and combinatorial unipotent d-block coincide. Unfortunately, it seems hard to prove (of infirm) in general. In the following sections, we give examples where the computations are easier, so that the result can be established.

4.3 The case d = 1

In this section, we consider the extreme case d = 1. Recall that the 1-regular elements of G = GL(V) are just the unipotent elements. Note that all partitions of n have the same 1-core (the only 1-core there is, that is the empty one). In order to prove an analogue of the Nakayama Conjecture for unipotent 1-blocks, we therefore need to prove that any two unipotent characters belong to the same 1-block. We actually prove more than that. Namely, any irreducible character of G is directly linked across unipotent elements to the trivial character.

Our tool to deal with this case is the Alvis-Curtis duality. D. Alvis has proved (cf [1]) that there is an isometry of order 2, D_{G}: ch(G) ⟶ ch(G), where ch(G) denotes the ring of virtual characters of G (in fact, this is true in a much more general context, that is when G is any finite group with a split (B, N)-pair of characteristic p). In particular, the image under the duality map D_{G} of any irreducible (complex) character of G is, up to a sign, an irreducible character. The case of unipotent characters has been made more precise by C. W. Curtis (cf [2]). Namely, if χ_{λ} ∈Unip(G), then D_{G}(χ_{λ}) = χ_{ελ}, where ε is the sign character of the symmetric group S_{n} (that is, χ_{ελ} is the unipotent character χ_{μ} where φ_{μ} is the irreducible character of S_{n} such that φ_{μ} = εφ_{λ}). This can be written as D_{G}(χ_{λ}) = χ_{λ^{∗}}, where λ^{∗} is the partition conjugate to λ. In particular, we have D_{G}(1_{G}) = χ_{ε} = St_{G}, the Steinberg character of G.

We define the virtual character χ_{u} of G via

χ_{u}(g) = \{ \begin{matrix}1 if g is unipotent \\ 0 otherwise \end{matrix} .

We then have the following:

Lemma 4.4. ([3], Lemme 11.1) We have χ_{u} = D_{G}(|G|_{p′}^{−1}reg_{G}), where reg_{G} is the character of the regular representation of G.

We let G_{u} be the set of unipotent elements of G. Then, for any χ ∈Irr(G), we have

< χ, 1_{G} >_{G_{u}}:= \frac{1}{|G|} ∑_{g∈G_{u}} χ(g) = \frac{1}{|G|} ∑_{g∈G} χ(g)χ_{u}(g) =< χ, χ_{u} >_{G} .

Thus, since D_{G} is an isometry, we have

< χ, 1_{G} >_{G_{u}}=< D_{G}(χ), D_{G}(χ_{u}) >_{G}= \frac{1}{|G|_{p′}} < D_{G}(χ), reg_{G} >_{G}= \frac{1}{|G|_{p′}} D_{G}(χ)(1).

Hence < χ, 1_{G} >_{G_{u}}≠ 0, i.e. χ and 1_{G} are directly linked across unipotent elements. We thus have Theorem 4.5. The only 1-block of G is Irr(G), and the only unipotent 1-block of G is Unip(G). In particular, the unipotent 1-blocks of G satisfy an analogue of the Nakayama Conjecture.

Note that, if χ = χ_{λ} ∈Unip(G), then we have < χ_{λ}, 1_{G} >_{G_{u}}= \frac{1}{|G|_{p′}} χ_{λ^{∗}}(1).

4.4 d-weight and simple partitions

¿From now on, we fix the integers n and d, d ≠ 1, and some d-core γ. Our aim is to prove that, if two partitions λ and μ of n have the same d-core γ, then the unipotent characters χ_{λ} and χ_{μ} belong to the same unipotent d-block of G = GL(V). However, if we simply write down the inner product of χ_{λ} and χ_{μ} on d-regular (or d-singular) elements of G, we obtain a huge expression which is very hard to manipulate, let alone trying to prove that it is non-zero.

We can associate to each combinatorial unipotent d-block a d-weight, the d-weight of any partition of n labeling some unipotent character in the block.

The first observation comes from the Murnaghan-Nakayama Rule: if λ has d-weight w, then χ_{λ} vanishes on any element of G whose d-part has d-type (k_{1}m_{1}, …, k_{r}m_{r}) with ∑_{i} k_{i}m_{i} > w.

This leads to the first definition and simplification. For any element g ∈ G with d-part of d-type (k_{1}m_{1}, …, k_{r}m_{r}), we define the d-weight of g to be ∑_{i} k_{i}m_{i}. Thus, for partitions of d-weight w, it is enough to consider the inner product of χ_{λ} and χ_{μ} on d-singular elements of d-weight at most w.

The Murnaghan-Nakayama Rule also implies that, if, for all m > 1, λ contains no md-hook, then χ_{λ} vanishes on any element g with d-part of d-type (k_{1}m_{1}, …, k_{r}m_{r}) when there exists i such that m_{i} > 1 (i.e. whenever there exists an irreducible factor of degree md with m > 1 in Min(g)). If λ is such a partition, we will say that λ is simple (or d-simple).

Given any partition λ of n, we can store all the information about λ on an abacus with d runners. Then the removal of one d-hook from λ corresponds to moving one bead of the abacus one place down on its runner. The removal of an md-hook, m > 1, corresponds to doing m such moves (involving one or more beads) on a single runner of the abacus. Hence, if λ is d-simple of weight w, then we see that the moves corresponding to the removals of all the d-hooks from λ must be stored on exactly w distinct runners of the abacus (so that, in particular, one must have w ≤ d), and they are all elementary, i.e. involve each a single bead which can be taken exactly once one place down.

We say that two partitions are disjoint if the possible moves on their abacus are stored on different runners. For any partition λ and integer i ≥ 0, we write L_{λ}^{i} (respectively L_{λ}^{(i)}) the set of partitions we can get by removing i d-hooks (respectively one id-hook) form λ. In particular, L_{λ}^{0} = {λ}, and, if λ has d-weight w and d-core γ, then L_{λ}^{w} = {γ} and, for all i > w, L_{λ}^{i} = ∅.

4.5 One result of direct linking

Suppose that λ and μ are partitions of n of d-weight w ≠ 0, and with the same d-core γ. Suppose furthermore that μ is simple and disjoint from λ. Then, by definition(s), we have:

• L_{μ}^{(i)} = ∅ for 1 < i ≤ w.

• L_{λ}^{i} ∩ L_{μ}^{i} = ∅ for 0 ≤ i < w.

• L_{λ}^{w} = L_{μ}^{w} = {γ}.

• |L_{μ}^{i}| = w + 1 − i for 1 ≤ i ≤ w.

These properties induce a large number of simplifications in the expression for the inner product of χ_{λ} and χ_{μ} on d-singular elements. We will prove the following:

Theorem 4.6. Suppose λ and μ are partitions of n of d-weight w ≠ 0, and with the same d-core γ. Suppose furthermore that μ is simple and disjoint from λ. Then the unipotent characters χ_{λ} and χ_{μ} are directly linked across the set of d-regular elements of G.

We write F = |F_{d}|, where F_{d} is the set of irreducible monic polynomials of degree d over F_{q}, and we suppose that F ≥ n/d (so that F ≥ w for any possible d-weight w).

We use induction on w to prove that, if λ and μ are as above, then the inner product of χ^{λ} and χ^{μ} on the set of d-regular elements of G is

< χ^{λ}, χ^{μ} >_{d−reg}= \frac{(−1)^{w}F^{w}}{w!(q^{d} − 1)^{w}} ∑_{P_{λ}∈P_{γ}^{λ}, P_{μ}∈P_{γ}^{μ}}(−1)^{L_{P_{λ}}} (−1)^{L_{Pμ}} ,

where P_{γ}^{λ} (resp. P_{γ}^{μ}) denotes the set of paths from λ (resp. μ) to γ, in the lattice of partitions, obtained by successive removals of d-hooks, and, for P_{λ} ∈ P_{γ}^{λ} (resp. P_{μ} ∈ P_{γ}^{μ}), L_{P_{λ}} (resp. L_{P_{μ}}) is the sum of the leg lengths of the hooks removed along the corresponding path.

We suppose the result established for partitions of d-weight up to w − 1 ≠ 0, and we take λ and μ of weight w as above. We order the elements of F_{d}, and write F_{d} = {f_{1}, …, f_{F} }. We write C for the set of d-singular elements of weight at most w of G. The conjugacy classes in C are parametrized by the F -tuples of (possibly empty) partitions (ν_{1}, …, ν_{F}), where ν_{i} ⊢ k_{i} for 1 ≤ i ≤ F and 1 ≤ k_{1} + ⋯ + k_{F} ≤ w. We choose a basis for V and identify G = GL(V) with GL(n, q). We will write G_{n} for GL(n, q), and similarly for smaller dimensions. Writing ∼ for conjugation, we get

C/ ∼ = ⋃_{k=1}^{w} \{( \begin{matrix}ρ \\ & σ \end{matrix} ) , ρ ∈ C_{k}/ ∼, σ ∈ G_{n−kd}, σ d − reg\} ,

where

C_{k}/ ∼ = ⋃_{k_{1}+···+k_{F} =k} ⋃_{ν_{1}⊢k_{1}, ..., ν_{F} ⊢k_{F}} \{( \begin{matrix}U_{ν_{1}}(f_{1}) \\ & . & . & . \\ & & & & U_{ν_{F}} (f_{F}) \end{matrix} )\} .

We have

< χ^{λ}, χ^{μ} >_{d−sing}=< χ^{λ}, χ^{μ} >_{C}= ∑_{k=1}^{w} < χ^{λ}, χ^{μ} >_{C_{k}}

and, for 1 ≤ k ≤ w, < χ^{λ}, χ^{μ} >_{C_{k}} is equal to

∑_{k_{1}+···+k_{F} =k} ∑_{ν_{1}⊢k_{1}, ..., ν_{F} ⊢k_{F}} ∑_{{σ∈G_{n−kd} \atop σ d−reg}} \frac{χ^{λ}(ρ^{ν_{1}} … ρ^{ν_{F}} σ)\overline{χ^{μ}(ρ^{ν_{1}} … ρ^{ν_{F}} σ)}}{|C_{n}(ρ^{ν_{1}} … ρ^{ν_{F}} σ)|},

where we write C_{n}(g) for C_{G_{n}}(g) = C_{GL(n,q)}(g) and ρ^{ν} for U_{ν}(f), f ∈ F (abuse justified since the values of unipotent characters on U_{ν}(f) and the size of its centralizer don’t depend on f, but only on ν).

Repeated use of the Murnaghan-Nakayama Rule gives us

χ^{λ}(ρ^{ν_{1}} … ρ^{ν_{F}} σ) = ∑_{λ_{1}∈L_{λ}^{k_{1}} , λ_{2}∈L_{λ_{1}}^{k_{2}} , ..., λ_{F} ∈L_{λ_{F −1}}^{k_{F}}} a_{λλ_{1}}^{ν_{1}}a_{λ_{1}λ_{2}}^{ν_{2}} … a_{λ_{F −1}λ_{F}}^{ν_{F}} χ^{λ_{F}} (σ),

and a similar expression for χ^{μ}(ρ^{ν_{1}} … ρ^{ν_{F}} σ).

We obtain

< χ^{λ}, χ^{μ} >_{d−sing}= ∑_{k=1}^{w} ∑_{k_{1}+···+k_{F} =k} A_{k_{1},...,k_{F}}^{λ,μ} ,

where

A_{k_{1},...,k_{F}}^{λ,μ} = ∑_{λ_{1}∈L_{λ}^{k_{1}} , ..., λ_{F} ∈L_{λ_{F −1}}^{k_{F}}} ∑_{μ_{1}∈L_{μ}^{k_{1}} , ..., μ_{F} ∈L_{μ_{F −1}}^{k_{F}}} B_{λ_{1},μ_{1}}^{λ,μ} … B_{λ_{F} ,μ_{F}}^{λ_{F −1},μ_{F −1}}D_{λ_{F} ,μ_{F}} ,

with

B_{λ_{1},μ_{1}}^{λ,μ} = ∑_{ν_{1}⊢k_{1}} \frac{a_{λλ_{1}}^{ν_{1}}a_{μμ_{1}}^{ν_{1}}}{|C_{k_{1}d}(ρ^{ν_{1}})|}, …, B_{λ_{F} ,μ_{F}}^{λ_{F −1},μ_{F −1}} = ∑_{ν_{F} ⊢k_{F}} \frac{a_{λ_{F −1}λ_{F}}^{ν_{F}} a_{μ_{F −1}μ_{F}}^{ν_{F}}}{|C_{k_{F} d}(ρ^{ν_{F}})|}

and

D_{λ_{F} ,μ_{F}} = ∑_{{σ∈G_{n−kd}/∼ \atop σ d−reg}} \frac{χ^{λ_{F}} (σ)\overline{χ^{μ_{F}} (σ)}}{|C_{n−kd}(σ)|}.

Now, for all 1 ≤ k ≤ w, we have

∑_{{σ∈G_{n−kd}/∼ \atop σ d−reg}} \frac{χ^{λ_{F}} (σ)\overline{χ^{μ_{F}} (σ)}}{|C_{n−kd}(σ)|} =< χ^{λ_{F}} , χ^{μ_{F}} >_{G_{n−kd}, d−reg},

which is known, by the induction hypothesis, since λ_{F} and μ_{F} have weight w − k < w (and the same d-core γ).

In order to deal with the other factors, we start with the following lemma, which doesn’t depend on the special choice of μ:

Lemma 4.7. For any 1 ≤ i ≤ F, we have, writing λ_{0} and μ_{0} for λ and μ respectively,

∑_{ν_{i}⊢k_{i}} \frac{a_{λ_{i−1}λ_{i}}^{ν_{i}}a_{μ_{i−1}μ_{i}}^{ν_{i}}}{|C_{k_{i}d}(ρ^{ν_{i}})|} = ε_{λ_{i−1}λ_{i}}ε_{μ_{i−1}μ_{i}} ∑_{α⊢k_{i}} \frac{ϕ_{λ_{i−1}|λ_{i}}(α)ϕ_{μ_{i−1}|μ_{i}}(α)}{|C_{S_{k_{i}}} (α)| |T\overline{α}|},

where:

• if α = (1^{r_{1}}, 2^{r_{2}}, …, s^{r_{s}}) ⊢ k_{i}, then \overline{α} = (d^{r_{1}}, (2d)^{r_{2}}, …, (sd)^{r_{s}}) ⊢ k_{i}d,

• T\overline{α} is a maximal torus of G_{k_{i}d} = GL(k_{i}d, q) representing the G_{k_{i}d}-conjugacy class corresponding to the S_{k_{i}}-conjugacy class of α; we have |T\overline{α}| = ∏_{i}(q^{di} − 1)^{r_{i}} (cf [5] (1.12));

• ϕ_{μ_{i−1}|μ_{i}} ∈ ZIrr(S_{k_{i}}), and, if h ∈ S_{ℓ} and g ∈ S_{k_{i}d} is of cycle type \overline{α} (α ⊢ k_{i}), then

φ_{μ_{i−1}}(gh) = ∑_{η∈L_{μ_{i−1}}^{k_{i}}} ε_{μ_{i−1}η}ϕ_{μ_{i−1}|η}(α)φ_{η}(h)

(where, for any positive integer m, we write Irr(S_{m}) = {φ_{ζ}, ζ ⊢ m}) and similarly for ϕ_{λ_{i−1}|λ_{i}} (cf [4] and [5] (2.3)).

Proof. For all ν_{i} ⊢ k_{i}, we have C_{k_{i}d}(ρ^{ν_{i}}) = C_{C_{k_{i}d}(ρ_{S}^{ν_{i}})}(ρ_{U}^{ν_{i}}), where ρ_{S}^{ν_{i}} and ρ_{U}^{ν_{i}} are the semisimple and unipotent parts of ρ^{ν_{i}} respectively, and C_{k_{i}d}(ρ_{S}^{ν_{i}}) ≅ GL(k_{i}, q^{d}). We write H_{i} for GL(k_{i}, q^{d}) (then H_{i} has Weyl group W_{H_{i}} ≅ S_{k_{i}}), so that C_{k_{i}d}(ρ^{ν_{i}}) ≅ C_{H_{i}}(ρ_{U}^{ν_{i}}).

We have, writing H_{i}^{u}/ ∼ for a set of representatives for the unipotent conjugacy classes of H_{i},

∑_{ν_{i}⊢k_{i}} \frac{a_{λ_{i−1}λ_{i}}^{ν_{i}}a_{μ_{i−1}μ_{i}}^{ν_{i}}}{|C_{k_{i}d}(ρ^{ν_{i}})|} = ∑_{ρ_{U}^{ν_{i}} ∈H_{i}^{u}/∼} \frac{a_{λ_{i−1}λ_{i}}^{ν_{i}}a_{μ_{i−1}μ_{i}}^{ν_{i}}}{|C_{H_{i}}(ρ_{U}^{ν_{i}})|} .

Now, by definition (cf [5] Theorem (2G)), we have

a_{λ_{i−1}λ_{i}}^{ν_{i}} = ε_{λ_{i−1}λ_{i}} ∑_{α⊢k_{i}} \frac{1}{|W_{H_{i}}(T\overline{α})|}Q_{T\overline{α}}^{H_{i}}(ρ_{U}^{ν_{i}})ϕ_{λ_{i−1}|λ_{i}}(α)

(and similarly for a_{μ_{i−1}μ_{i}}^{ν_{i}}), where W_{H_{i}}(T\overline{α}) is the Weyl group of T\overline{α} in H_{i}, so that |W_{H_{i}}(T\overline{α})| = |C_{S_{k_{i}}} (α)| (cf [5] (1.11)), and Q_{T\overline{α}}^{H_{i}} is the Green function, integer-valued class function defined on H_{i}^{u}, associated to T\overline{α} (cf [5]).

We obtain that ∑_{ν_{i}⊢k_{i}} \frac{a_{λ_{i−1}λ_{i}}^{ν_{i}} a_{μ_{i−1}μ_{i}}^{ν_{i}}}{|C_{k_{i}d}(ρ^{ν_{i}})|} is equal to

∑_{ρ_{U}^{ν_{i}} ∈H_{i}^{u}/∼} \frac{ε_{λ_{i−1}λ_{i}}ε_{μ_{i−1}μ_{i}}}{|C_{H_{i}}(ρ_{U}^{ν_{i}})|} ∑_{α,β⊢k_{i}} \frac{Q_{T\overline{α}}^{H_{i}}(ρ_{U}^{ν_{i}})ϕ_{λ_{i−1}|λ_{i}}(α)Q_{T\overline{β}}^{H_{i}}(ρ_{U}^{ν_{i}})ϕ_{μ_{i−1}|μ_{i}}(β)}{|W_{H_{i}}(T\overline{α})| |W_{H_{i}}(T\overline{β})|},

which, since the Green functions are class functions, can be written as

ε_{λ_{i−1}λ_{i}}ε_{μ_{i−1}μ_{i}} ∑_{α,β⊢k_{i}} \frac{ϕ_{λ_{i−1}|λ_{i}}(α)ϕ_{μ_{i−1}|μ_{i}}(β)}{|W_{H_{i}}(T\overline{α})| |W_{H_{i}}(T\overline{β})|} (\frac{1}{|H_{i}|} ∑_{ρ_{U}^{ν_{i}} ∈H_{i}^{u}} Q_{T\overline{α}}^{H_{i}}(ρ_{U}^{ν_{i}})Q_{T\overline{β}}^{H_{i}}(ρ_{U}^{ν_{i}})) .

Now, by the first orthogonality relation for Green functions in H_{i} (cf [5] (1.7)), we have

\frac{1}{|H_{i}|} ∑_{ρ_{U}^{ν_{i}} ∈H_{i}^{u}} Q_{T\overline{α}}^{H_{i}}(ρ_{U}^{ν_{i}})Q_{T\overline{β}}^{H_{i}}(ρ_{U}^{ν_{i}}) = \{ \begin{matrix}\frac{|W_{H_{i}} (T\overline{α})|}{|T\overline{α}|} & if T\overline{α} ∼_{H_{i}} T\overline{β} \\ 0 & otherwise \end{matrix} .

But T\overline{α} ∼_{H_{i}} T\overline{β} if and only if \overline{α} = \overline{β}, if and only if α = β. We thus get, since |W_{H_{i}}(T\overline{α})| = |C_{S_{k_{i}}} (α)|,

∑_{ν_{i}⊢k_{i}} \frac{a_{λ_{i−1}λ_{i}}^{ν_{i}}a_{μ_{i−1}μ_{i}}^{ν_{i}}}{|C_{k_{i}d}(ρ^{ν_{i}})|} = ε_{λ_{i−1}λ_{i}}ε_{μ_{i−1}μ_{i}} ∑_{α⊢k_{i}} \frac{ϕ_{λ_{i−1}|λ_{i}}(α)ϕ_{μ_{i−1}|μ_{i}}(α)}{|C_{S_{k_{i}}} (α)| |T\overline{α}|},

which ends the proof.

The fact that μ is simple then gives us further simplifications; we have the following

Lemma 4.8. For any 1 ≤ i ≤ F and α ∈ S_{k_{i}}, we have ϕ_{μ_{i−1}|μ_{i}}(α) = 0 unless α = (1^{k_{i}}) (the identity element of S_{k_{i}}).

Proof. Take any 1 ≤ i ≤ F. We have μ_{i−1} ⊢ m and μ_{i} ⊢ ℓ, with m = ℓ + k_{i}d. Then, by definition (cf [5] (2.3) and [4]), ϕ_{μ_{i−1}|μ_{i}} ∈ ZIrr(S_{k_{i}}), and, if h ∈ S_{ℓ} and g ∈ S_{k_{i}d} is of cycle type \overline{α} (α ⊢ k_{i}), then

φ_{μ_{i−1}}(gh) = ∑_{η∈L_{μ_{i−1}}^{k_{i}}} ε_{μ_{i−1}η}ϕ_{μ_{i−1}|η}(α)φ_{η}(h).

Now μ_{i−1} can be obtained from μ, which is d-simple, by removing a sequence of d-hooks, so that μ_{i−1} is itself simple, whence, for j > 1, there is no jd-hook in μ_{i−1}.

Thus, by the Murnaghan-Nakayama rule (in the symmetric group), if there is a cycle of length j > 1 in α (so that there is a jd-cycle in g), then

φ_{μ_{i−1}}(gh) = 0, for all h ∈ S_{ℓ}.

Hence, for all h ∈ S_{ℓ},

∑_{η∈L_{μ_{i−1}}^{k_{i}}} ε_{μ_{i−1}η}ϕ_{μ_{i−1}|η}(α)φ_{η}(h) = 0,

whence

ε_{μ_{i−1}η}ϕ_{μ_{i−1}|η}(α) = 0 for all η ∈ L_{μ_{i−1}}^{k_{i}},

in particular for η = μ_{i}.

By the previous two lemmas, we therefore get, for any 1 ≤ i ≤ F,

B_{λ_{i},μ_{i}}^{λ_{i−1},μ_{i−1}} = \frac{ε_{λ_{i−1}λ_{i}}ε_{μ_{i−1}μ_{i}}ϕ_{λ_{i−1}|λ_{i}}(1^{k_{i}})ϕ_{μ_{i−1}|μ_{i}}(1^{k_{i}})}{|C_{S_{k_{i}}} (1^{k_{i}})| |T_{(\overline{1^{k_{i}}})}|},

and we have C_{S_{k_{i}}} (1^{k_{i}}) = S_{k_{i}} of order k_{i}!, and |T_{(\overline{1^{k_{i}}})}| = |T_{(d^{k_{i}})}| = (q^{d} − 1)^{k_{i}}. Now, we have, for all h ∈ S_{ℓ},

φ_{μ_{i−1}}(\overline{1^{k_{i}}}h) = ∑_{η∈L_{μ_{i−1}}^{k_{i}}} ε_{μ_{i−1}η}ϕ_{μ_{i−1}|η}(1^{k_{i}})φ_{η}(h).

On the other hand, by repeated use of the Murnaghan-Nakayama Rule, we have, for all h ∈ S_{ℓ},

φ_{μ_{i−1}}(\overline{1^{k_{i}}}h) = ∑_{η∈L_{μ_{i−1}}^{k_{i}}} ( ∑_{P ∈P_{η}^{μ_{i−1}}}(−1)^{L_{P}} ) φ_{η}(h).

Hence, for all η ∈ L_{μ_{i−1}}^{k_{i}},

ε_{μ_{i−1}η}ϕ_{μ_{i−1}|η}(1^{k_{i}}) = ∑_{P ∈P_{η}^{μ_{i−1}}}(−1)^{L_{P}} .

Similarly, for all η ∈ L_{λ_{i−1}}^{k_{i}},

ε_{λ_{i−1}η}ϕ_{λ_{i−1}|η}(1^{k_{i}}) = ∑_{P ∈P_{η}^{λ_{i−1}}}(−1)^{L_{P}} .

We therefore have, for any 1 ≤ i ≤ F,

B_{λ_{i},μ_{i}}^{λ_{i−1},μ_{i−1}} = ∑_{P_{λ_{i}} ∈P_{λ_{i}}^{λ_{i−1}} , P_{μ_{i}} ∈P_{μ_{i}}^{μ_{i−1}}} \frac{(−1)^{L_{P_{λ_{i}}}} (−1)^{L_{Pμ_{i}}}}{k_{i}!(q^{d} − 1)^{k_{i}}} .

Coming back to our expression for < χ^{λ}, χ^{μ} >_{d−sing}, we see that, for each 1 ≤ k ≤ w, once k_{1}, …, k_{F} are chosen, then all the paths obtained by removing a sequence of k d-hooks from λ and μ appear exactly once in the sum. We get that A_{k_{1},...,k_{F}}^{λ,μ} is equal to

\frac{1}{k_{1}! … k_{F}!(q^{d} − 1)^{k}} ∑_{{λ′∈L_{λ}^{k} \atop μ′∈L_{μ}^{k}}} ∑_{{P_{λ}∈P_{λ′}^{λ} \atop P_{μ}∈P_{μ′}^{μ}}} (−1)^{L_{P_{λ}}} (−1)^{L_{Pμ}} < χ^{λ′}, χ^{μ′} >_{d−reg} .

Now, for all λ′ ∈ L_{λ}^{k} and μ′ ∈ L_{μ}^{k}, λ′ and μ′ have d-weight w − k and d-core γ, and μ′ is simple and disjoint from λ′. By the induction hypothesis, we thus have

< χ^{λ′}, χ^{μ′} >_{G_{n−kd}, d−reg}= \frac{(−1)^{w−k}F^{w−k}}{(w − k)!(q^{d} − 1)^{w−k}} ∑_{P_{λ′} ∈P_{γ}^{λ′} , P_{μ′} ∈P_{γ}^{μ′}}(−1)^{L_{P_{λ′}}} (−1)^{L_{P_{μ′}}} .

Finally, we get

< χ^{λ}, χ^{μ} >_{d−sing}= ∑_{k=1}^{w} ∑_{k_{1}+···+k_{F} =k} A_{k_{1},...,k_{F}}^{λ,μ} ,

where

A_{k_{1},...,k_{F}}^{λ,μ} = \frac{(−1)^{w−k}F^{w−k}}{k_{1}! … k_{F}!(w − k)!(q^{d} − 1)^{w}} ∑_{P_{λ}∈P_{γ}^{λ}, P_{μ}∈P_{γ}^{μ}}(−1)^{L_{P_{λ}}} (−1)^{L_{Pμ}} .

In order to prove our result, it therefore suffices to show that

∑_{k=1}^{w} ∑_{k_{1}+···+k_{F} =k} \frac{(−1)^{w−k}F^{w−k}}{k_{1}! … k_{F}!(w − k)!} = \frac{(−1)^{w+1}F^{w}}{w!} .

We rewrite the second sum in the left hand side as a sum over the partitions of k. When doing this, we “break” the ordering on the elements of F_{d}, so that we have to count how many times each given partition appears once its parts are fixed, and divide by the number of times the corresponding conjugacy class of d-elements appears. More precisely, to each partition (k^{r_{k}}, …, 1^{r_{1}}) of k with r_{1} + ⋯ + r_{k} = r parts, there are F (F − 1) ⋯ (F − r + 1) ways to associate one distinct polynomial of F_{d} to each part (which is why we supposed that F ≥ w), but (since the order of the polynomials doesn’t matter, only their multiplicity

(i.e. ( \begin{matrix}f \\ & g \end{matrix} ) and ( \begin{matrix}g \\ & f \end{matrix} ) are conjugate)) each conjugacy class of d-element

of type (k^{r_{k}}, …, 1^{r_{1}}) will appear r_{1}! … r_{k}! times in this way. The left hand side thus becomes

∑_{k=1}^{w} \frac{(−1)^{w−k}F^{w−k}}{(w − k)!} ∑_{r=1}^{k} ∑_{{(k^{r_{k}} ,...,1^{r_{1}} )⊢k \atop r_{1}+···+r_{k}=r}} \frac{F (F − 1) … (F − r + 1)}{(1!)^{r_{1}}r_{1}! … (k!)^{r_{k}}r_{k}!}.

To conclude, we just need the following:

Lemma 4.9. For any integers k ≥ 1 and F ≥ k, we have

∑_{r=1}^{k} ∑_{{(k^{r_{k}} ,...,1^{r_{1}} )⊢k \atop r_{1}+···+r_{k}=r}} \frac{F (F − 1) … (F − r + 1)}{(1!)^{r_{1}}r_{1}! … (k!)^{r_{k}}r_{k}!} = \frac{F^{k}}{k!} .

Proof. The coefficient of z^{k} in the power series of exp(z)^{F} is F^{k}/k!. Indeed,

exp(z)^{F} = exp(F z) = ∑_{n≥0} \frac{F^{n}}{n!} z^{n}.

On the other hand, we have

(∑_{n≥0} \frac{1}{n!}z^{n})^{F} = ∑_{r_{0}+r_{1}+···=F} \frac{C_{(r_{0},r_{1},...)}}{(0!)^{r_{0}}(1!)^{r_{1}} …}z^{∑_{i≥0} ir_{i}},

where C_{(r_{0},r_{1},...)} is the multinomial coefficient \frac{F!}{r_{0}!r_{1}!...}. The coefficient of z^{k} in this expression therefore corresponds to all the uplets (r_{0}, r_{1}, …) such that r_{0}+r_{1}+⋯ = F and ∑_{i≥0} ir_{i} = ∑_{i>0} ir_{i} = k. Hence (1^{r_{1}}, …) ⊢ k is a partition with ∑_{i>0} r_{i} =: r parts, r_{0} = F − ∑_{i>0} r_{i} = F − r, and C_{(r_{0},r_{1},...)} = \frac{F!}{(F −r)!r_{1}!...}. Finally, we get that the coefficient of z^{k} is

∑_{r=1}^{k} ∑_{{(k^{r_{k}} ,...,1^{r_{1}} )⊢k \atop r_{1}+···+r_{k}=r}} \frac{F (F − 1) … (F − r + 1)}{(1!)^{r_{1}}r_{1}! … (k!)^{r_{k}}r_{k}!},

which is the desired result.

Remark: In the proof, we have used the fact that F is an integer. However, the equality given for each integer k ≥ 1 in the statement of the lemma is an equality between two polynomial functions in F, satisfied for an infinite number of values of F. It is therefore in fact an equality in the polynomial ring Q[F].

Using the previous lemma, we get

∑_{k=1}^{w} ∑_{k_{1}+···+k_{F} =k} \frac{(−1)^{w−k}F^{w−k}}{k_{1}! … k_{F}!(w − k)!} = ∑_{k=1}^{w} \frac{(−1)^{w−k}F^{w−k}}{(w − k)!} \frac{F^{k}}{k!}

= \frac{F^{w}}{w!} ∑_{k=1}^{w} \frac{(−1)^{w−k}w!}{k!(w − k)!}

= \frac{(−1)^{w}F^{w}}{w!} ∑_{k=1}^{w}(−1)^{k} ( \begin{matrix}w \\ k \end{matrix} )

= \frac{(−1)^{w}F^{w}}{w!} [(1 − 1)^{w} − 1]

= \frac{(−1)^{w+1}F^{w}}{w!} .

Putting everything back together, we finally obtain

< χ^{λ}, χ^{μ} >_{d−sing}= \frac{(−1)^{w+1}F^{w}}{w!(q^{d} − 1)^{w}} ∑_{P_{λ}∈P_{γ}^{λ}, P_{μ}∈P_{γ}^{μ}} (−1)^{L_{P_{λ}}} (−1)^{L_{Pμ}} ,

which is equivalent to

< χ^{λ}, χ^{μ} >_{d−reg}= \frac{(−1)^{w}F^{w}}{w!(q^{d} − 1)^{w}} ∑_{P_{λ}∈P_{γ}^{λ}, P_{μ}∈P_{γ}^{μ}} (−1)^{L_{P_{λ}}} (−1)^{L_{Pμ}} .

To conclude by induction, it only remains to study the case w = 1. This case can in fact be treated without the simplicity hypothesis on μ.

We take λ and μ, distinct partitions of n of d-weight 1, and with the same d-core γ. Then L_{λ}^{1} = L_{μ}^{1} = {γ}, and L_{λ}^{i} = L_{μ}^{i} = ∅ for i > 1, and we write P_{γ}^{λ} = {P_{λ}} and P_{γ}^{μ} = {P_{μ}}. As above, it suffices to consider the inner product of χ^{λ} and χ^{μ} on the set of d-singular elements of d-weight at most 1. The minimal polynomial of any such element has exactly one irreducible factor of degree d, and it has multiplicity one. Any d-singular element g ∈ G of weight 1 can thus be written uniquely g = ρσ = σρ, where ρ can be seen as a d-element of G_{d} = GL(d, q), and σ as a d-regular element of G_{n−d} = GL(n − d, q).

We have

< χ^{λ}, χ^{μ} >_{d−sing} = ∑_{{ρ∈G_{d}/∼ \atop ρ d−sing}} ∑_{{σ∈G_{n−d}/∼ \atop σ d−reg}} \frac{1}{|C_{G_{n}}(ρσ)|}χ^{λ}(ρσ)χ^{μ}(ρσ)

= ∑_{{ρ∈G_{d}/∼ \atop ρ d−sing}} \frac{a_{λγ}^{ρ}a_{μγ}^{ρ}}{|C_{G_{d}}(ρ)|} ∑_{{σ∈G_{n−d}/∼ \atop σ d−reg}} \frac{χ^{λ}(ρσ)χ^{μ}(ρσ)}{|C_{G_{n−d}}(σ)|}

= F ∑_{ν⊢1} \frac{a_{λγ}^{ρ^{ν}}a_{μγ}^{ρ^{ν}}}{|C_{G_{d}}(ρ^{ν})|} < χ^{γ}, χ^{γ} >_{G_{n−d},d−reg} .

By Lemma 4.7, we thus have

< χ^{λ}, χ^{μ} >_{d−sing} = F \frac{ε_{λγ}ε_{μγ}ϕ_{λ|γ}(1)ϕ_{μ|γ}(1)}{|C_{S_{1}}(1)| |T\overline{(1)}|} < χ^{γ}, χ^{γ} >_{G_{n−d},d−reg}

= F \frac{ε_{λγ}ε_{μγ}ϕ_{λ|γ}(1)ϕ_{μ|γ}(1)}{q^{d} − 1} < χ^{γ}, χ^{γ} >_{G_{n−d},d−reg}

= F \frac{ε_{λγ}ε_{μγ}ϕ_{λ|γ}(1)ϕ_{μ|γ}(1)}{q^{d} − 1},

since γ is a d-core, whence vanishes on d-singular elements of GL(n − d, q). Finally, as after Lemma 4.8, we have ε_{λγ}ϕ_{λ|γ}(1) = (−1)^{L_{P_{λ}}} and ε_{μγ}ϕ_{μ|γ}(1) = (−1)^{L_{Pμ}}, which gives us

< χ^{λ}, χ^{μ} >_{d−sing}= \frac{F}{q^{d} − 1}(−1)^{L_{P_{λ}}} (−1)^{L_{Pμ}} .

Hence the result is true for w = 1. Thus, by induction, we have that, for any w ≥ 1, if λ and μ are partitions of n of d-weight w, with the same d-core γ, and if μ is simple and disjoint from λ, then

< χ^{λ}, χ^{μ} >_{d−reg}= \frac{(−1)^{w}F^{w}}{w!(q^{d} − 1)^{w}} ∑_{P_{λ}∈P_{γ}^{λ}, P_{μ}∈P_{γ}^{μ}}(−1)^{L_{P_{λ}}} (−1)^{L_{Pμ}} .

To conclude, we just note that, in fact, (−1)^{L_{P_{λ}}} (resp. (−1)^{L_{Pμ}}) is independent on the path P_{λ} ∈ P_{γ}^{λ} (resp. P_{μ} ∈ P_{γ}^{μ}) (cf [7], Theorem 2.7.27). We therefore write (−1)^{L_{P_{λ}}} = ε_{λ} for any P_{λ} ∈ P_{γ}^{λ} (resp. (−1)^{L_{Pμ}} = ε_{μ}, for any P_{μ} ∈ P_{γ}^{μ}). We get

< χ^{λ}, χ^{μ} >_{d−reg}= \frac{(−1)^{w}F^{w}}{w!(q^{d} − 1)^{w}} |P_{γ}^{λ}| |P_{γ}^{μ}|ε_{λ}ε_{μ} ≠ 0.

This ends the proof of Theorem 4.6.

4.6 Nakayama Conjecture for blocks of small weight

We are now in position to prove the second direction of the Nakayama Conjecture for unipotent d-blocks of G, provided their weight is “small” compared to d. For partitions λ and μ of n, we will write λ ∼ μ if χ_{λ} and χ_{μ} are directly linked across d-singular elements, and λ ≡ μ if χ_{λ} and χ_{μ} belong to the same unipotent d-block of G. First, take λ and μ of weight w ≠ 0, with the same d-core, and suppose that 3w ≤ d and F ≥ w. Then there exists ν ⊢ n which is simple and disjoint from both λ and μ. Thus, by Theorem 4.6, λ ∼ ν and μ ∼ ν, so that λ ≡ μ. In fact, we can do better than that:

Theorem 4.10. Let d > 0 be an integer. Suppose w > 2 is an integer, and w ≤ F (the number of irreducible monic polynomials of degree d over F_{q}). If d ≥ 2w − 1, then the unipotent d-blocks of G of weight w satisfy the Nakayama Conjecture.

Proof. Take λ and μ partitions of n of weight w, with the same d-core γ. The proof is easy to understand on a picture of the abacus, even though it is difficult to write down.

Throughout the proof, we will use the following notations: we say that λ uses a runner of the abacus if some d-hook removal of λ is stored on this runner; we say that μ ⊂ λ if all the runners used by μ are used by λ.

First, suppose that λ and μ use each at most w − 1 runners. Then there exists ν ⊢ n which is simple and disjoint from λ, and of d-core γ. By Theorem 4.6, we have λ ∼ ν. Now

• if μ ⊄ ν, then there exists ζ ⊢ n, disjoint from ν, with d-core γ, and which uses a single runner, which runner is also used by μ. Then ν ∼ ζ. Now there exists ξ, which is simple, disjoint from μ (and thus from ζ), and with d-core γ, so that ζ ∼ ξ and ξ ∼ μ. Hence λ ≡ μ.

• if μ ⊂ ν, then there exists ζ, disjoint from ν, with d-core γ, and which uses a single runner, which runner is not used by μ. Thus ν ∼ ζ. There exists ξ, simple, disjoint from ζ, with d-core γ, and such that at least one runner used by μ is not used by ξ; we have ζ ∼ ξ. Then ξ ∼ δ, where δ has d-core γ, and uses a single runner, used by μ but not by ξ. Now δ ∼ η, simple, with d-core γ, and disjoint from μ (and thus from δ). Thus η ∼ μ, and λ ≡ μ.

Next, suppose that λ is simple (i.e. lies on w runners). If μ is disjoint from λ, then, by Theorem 4.6, λ ∼ μ, so that λ ≡ μ. Suppose thus that λ and μ share a runner.

• Suppose μ lies on at most w − 1 runners.

If μ ⊂ λ, then λ ∼ ν with d-core γ, disjoint from λ, and using a single runner. Now ν ∼ ζ, simple, with d-core γ, and such that μ ⊄ ζ. Then ζ ∼ ξ with d-core γ, using a single runner, which runner is used by μ. Then ξ ∼ η simple, with d-core γ, and disjoint from μ (and thus from ξ), so that η ∼ μ. Hence λ ≡ μ.

If μ ⊄ λ, then λ ∼ ν with d-core γ, disjoint from λ, and using a single runner, which runner is used by μ. Thus ν ∼ ζ simple, with d-core γ, and disjoint from μ (and thus from ν), so that ζ ∼ μ. Hence λ ≡ μ.

• Suppose μ is simple.

If there exists a runner which is used by neither λ nor μ, then there exists ν with d-core γ which lies on this single runner (and thus is disjoint from both λ and μ), so that λ ∼ ν and ν ∼ μ, whence λ ≡ μ.

If not, then, necessarily, d = 2w − 1, and λ and μ share a single runner. Since w > 2, there is at least one runner used by μ and not by λ. Then λ ∼ ν with d-core γ which lies on this single runner (and is thus disjoint from λ). Now ν ∼ ζ simple, with d-core γ, disjoint from ν and which doesn’t use at least one runner which is used by λ but not by μ. Then ζ ∼ ξ with d-core γ and which lies on this single runner. In particular, ξ is disjoint from μ, so that ξ ∼ μ (since μ is simple). Hence λ ≡ μ.

This concludes the proof.

Remarks:

1. The case of blocks of weight 0 is easily dealt with. A partition of d-weight 0 is its own d-core, so that the corresponding unipotent character is alone in its unipotent d-block. Hence, for any d > 0, the unipotent d-blocks of G of weight 0 satisfy the Nakayama Conjecture.

2. As we have remarked earlier, in the case w = 1, we don’t need to use simple partitions. For any d > 0 and for any two partitions λ and μ of n of d-weight 1 and with the same d-core, the unipotent characters χ_{λ} and χ_{μ} are directly linked across d-singular elements. Hence the unipotent d-blocks of weight 1 of G satisfy the Nakayama Conjecture.

3. As can be seen in the proof of Theorem 4.10, the case w = 2 is problematic only in the case d = 2w − 1, and the problem can easily be solved if we have another runner we can use. Hence the result of the theorem is also true for w = 2 if we only suppose 2w ≤ d.

Furthermore, the case w = 2 (which is fairly small) can in fact be studied on its own, and one can show that, for any d > 0, the unipotent d-blocks of G of weight 2 satisfy the Nakayama Conjecture (provided F ≥ 2).

4. Since the d-weight of a unipotent d-block of G is at most n/d, we see that, if n ≤ d(d + 1)/2, then all the unipotent d-blocks of G satisfy the Nakayama Conjecture (provided F ≥ n/d).

4.7 And then?

The proof of the Nakayama Conjecture in the general case seems to be hard. In [8], the authors prove that, if B is an ℓ-block of weight w of some symmetric group, then there is a generalized perfect isometry between B and the set of irreducible (complex) characters of the wreath product Z_{ℓ} ≀ S_{w}. This means that the restricted scalar products of characters of B across ℓ-regular elements are, up to a sign, the same as the restricted scalar products of characters of Z_{ℓ} ≀ S_{w} across some (carefully chosen) set of elements (called regular elements).

It turns out that the computations are easier to carry out in the wreath product, where it can be shown that every character is directly linked across regular elements to the trivial character. Coming back to the symmetric group, one gets that any two characters in the combinatorial ℓ-block are linked across ℓ-regular elements. This proves that the combinatorial ℓ-blocks and the ℓ-blocks are the same, or, equivalently, that the ℓ-blocks satisfy the Nakayama Conjecture.

Now, we might try and do something similar in the case of GL(V). This would involve finding the relevant wreath product, and the set of conjugacy classes we want to distinguish in it. Given a combinatorial unipotent d-block of weight w, a natural candidate for the wreath product is GL(d, q) ≀ S_{w}. However, even in easy cases (like d = 1), this doesn’t seem to contain enough information (in particular, it is hard to have powers of q appearing).

So far, we can only conjecture that, for any d > 0, the unipotent d-blocks of G satisfy an analogue of the Nakayama Conjecture.

5 Second Main Theorem

One of the very striking properties of ordinary blocks of finite groups is Brauer’s Second Main Theorem (cf e.g. [10]). In [8], the authors give an analogue of this in the context of generalized blocks. Depending on the generalized sections we define, the blocks we obtain may satisfy this analogue or not.

5.1 Domination, Second Main Theorem Property

For all the definitions and results in this section, we refer to [8]. We start by defining blocks for the centralizers of d-elements. Recall that the d-blocks of G are defined by orthogonality across the set Y_{d}(1) of d-regular elements. As a consequence, they separate Y_{d}(1) from its complement (cf [8], Corollary 1.2). For x ∈ X_{d}, we define the d-blocks of C_{G}(x) to be the smallest (non-empty) subsets of Irr(C_{G}(x)) such that irreducible characters in distinct subsets are orthogonal across xY_{d}(x). We can equally define them to be non-empty subsets of Irr(C_{G}(x)) which are minimal subject to separating Y_{d}(x) from its complement in C_{G}(x) (or, equivalently, to separating xY_{d}(x) from its complement (since x is central in C_{G}(x))).

We now turn to the notion of domination. Suppose χ ∈ Irr(G) and β is a union of d-blocks of C_{G}(x) for some x ∈ X_{d}. We define a generalized character χ^{(β)} of C_{G}(x) via

χ^{(β)} = ∑_{μ∈β} < Res_{C_{G}(x)}^{G}(χ), μ > μ.

Definition 5.1. Let x ∈ X_{d} and b be a d-block of C_{G}(x). We say that a d-block B of G dominates b if there exist χ ∈ B and y ∈ Y_{d}(x) such that χ^{(b)}(xy) ≠ 0.

We see that, for x ∈ X_{d}, if χ ∈ B for some d-block B of G, then, for each y ∈ Y_{d}(x), we have χ(xy) = ∑_{b} χ^{(b)}(xy), where b runs through the set of d-blocks of C_{G}(x) dominated by B.

Note that, for x ∈ X_{d}, each d-block of C_{G}(x) is dominated by at least one d-block of G.

Definition 5.2. We say that the d-blocks of G satisfy the Second Main Theorem Property if, for each x ∈ X_{d} and each d-block b of C_{G}(x), b is dominated by a unique d-block of G.

Note that, if, instead of X_{d}, we take the set of r-elements of G (r a prime), and, instead of Y_{d}(x), we take the set of r-regular elements of C_{G}(x), then we obtain the r-blocks of G, and they do satisfy the Second Main Theorem Property.

Using the fact that, for x ∈ X_{d}, irreducible characters of C_{G}(x) in distinct d-blocks are orthogonal across xY_{d}(x), one proves easily the following:

Proposition 5.3. ([8], Corollary 2.2) The d-blocks of G satisfy the Second Main Theorem Property if and only if, for each d-block B of G, there is, for each x ∈ X_{d}, a (possibly empty) union β(x, B) of d-blocks of C_{G}(x) such that, for each irreducible character χ ∈ B and each character μ ∈ β(x, B), we may find a complex number c_{χ,μ} such that, for each y ∈ Y_{d}(x), we have

χ(xy) = ∑_{μ∈β(x,B)} c_{χ,μ}μ(xy),

and, furthermore, β(x, B) and β(x, B′) are disjoint whenever B and B′ are distinct d-blocks of G.

The following theorem enlighten the link between the Second Main Theorem Property and Brauer’s Second Main Theorem:

Theorem 5.4. ([8], Corollary 2.3) Suppose that the d-blocks of G satisfy the Second Main Theorem property. Then:

(i) Irreducible characters of G which are in distinct d-blocks are orthogonal across each Y_{d}-section of G. (ii) If x ∈ X_{d} and ∑_{χ∈Irr(G)} a_{χ}χ is a class function which vanishes identically on the Y_{d}-section of x in G, then, for each d-block B of G, ∑_{χ∈B} a_{χ}χ also vanishes identically on the Y_{d}-section of x in G. (iii) d-blocks of G separate Y_{d}-sections of G.

5.2 Second Main Theorem Property for combinatorial unipotent d-blocks

We first study the d-blocks of the centralizers of the d-elements. If x ∈ X_{d} has d-type km, then, writing l = n − kmd, we have C_{G}(x) = H_{0} × H_{1} where H_{1} ≅ GL(l, q) and H_{0} ≤ G_{0} ≅ GL(kmd, q). Then Irr(C_{G}(x)) =Irr(H_{0})⊗Irr(H_{1}). Note that, as we noted before, we may consider x as an element of H_{0}, and then xY_{d}(x) = {(x, y) ∈ H_{0} × H_{1}, y ∈ Y_{d}^{l}(1)}.

Take χ_{0}, ψ_{0} ∈ Irr(H_{0}) and χ_{1}, ψ_{1} ∈ Irr(H_{1}). We have

< χ_{0} ⊗ χ_{1}, ψ_{0} ⊗ ψ_{1} >_{xY_{d}(x)} = \frac{1}{|C_{G}(x)|} ∑_{y∈Y_{d}(x)}(χ_{0} ⊗ χ_{1})(xy)\overline{(ψ_{0} ⊗ ψ_{1})(xy)}

= \frac{1}{|C_{G}(x)|} ∑_{y∈Y_{d}^{l}(1)} χ_{0}(x)χ_{1}(y)\overline{ψ_{0}(x)}\overline{ψ_{1}(y)}

= \frac{χ_{0}(x)\overline{ψ_{0}(x)}}{|C_{G}(x)|} ∑_{y∈Y_{d}^{l}(1)} χ_{1}(y)\overline{ψ_{1}(y)}

= \frac{χ_{0}(x)\overline{ψ_{0}(x)}}{|H_{0}|} < χ_{1}, ψ_{1} >_{Y_{d}^{l}(1)} .

Since x is central in H_{0}, we have χ_{0}(x)\overline{ψ_{0}(x)} ≠ 0, and we see that χ_{0} ⊗ χ_{1} and ψ_{0} ⊗ ψ_{1} are directly xY_{d}(x)-linked if and only if χ_{1} and ψ_{1} are directly Y_{d}^{l}(1)- linked. Extending by transitivity, we obtain that the d-blocks of C_{G}(x) are the Irr(H_{0}) ⊗ b_{i}’s, where b_{i} runs through the set of Y_{d}^{l}(1)-blocks (i.e. d-blocks) of H_{1} ≅ GL(l, q).

In analogy with this, we define the unipotent d-blocks and combinatorial unipotent d-blocks of C_{G}(x) to be the Irr(H_{0}) ⊗ b_{i}’s, where b_{i} runs through the sets of unipotent d-blocks and combinatorial unipotent d-blocks of H_{1} ≅ GL(l, q) respectively.

We can now prove that the combinatorial unipotent d-blocks of G satisfy the Second Main Theorem Property.

Take any x ∈ X_{d}, and write C_{G}(x) = H_{0} × H_{1} as above. For any combinatorial unipotent d-block B of G, labeled by the d-core γ, we set β(x, B) = Irr(H_{0}) ⊗ b, where b is the combinatorial unipotent d-block of H_{1} labeled by γ. For any χ_{μ} ∈ B and ψ_{0} ⊗ ψ_{λ} ∈ Irr(H_{0}) ⊗ b, we set

c_{χ_{μ},ψ_{0}⊗ψ_{λ}} = \{ \begin{matrix}α_{μλ}^{x} & if ψ_{0} = 1_{H_{0}} \\ 0 & otherwise \end{matrix} ,

where the α_{μλ}^{x}’s are the MN-coefficients, obtained from the Murnaghan-Nakayama rule for unipotent characters.

Then the definition of the α_{μλ}^{x}’s shows that, for each x ∈ X_{d}, the β(x, B)’s and c_{χ_{μ},ψ_{0}⊗ψ_{λ}}’s satisfy the hypotheses of Proposition 5.3. Indeed, for each combinatorial unipotent d-block B of G, for each χ_{μ} ∈ B and for each y ∈ Y_{d}(x), we have

χ_{μ}(xy) = ∑_{ψ_{0}⊗ψ_{λ}∈β(x,B)} c_{χ_{μ},ψ_{0}⊗ψ_{λ}}(ψ_{0} ⊗ ψ_{λ})(xy),

and, furthermore, β(x, B) and β(x, B′) are disjoint whenever B and B′ are distinct combinatorial unipotent d-blocks of G. This implies that, for each x ∈ X_{d}, each combinatorial unipotent d-block of C_{G}(x) is dominated by a unique combinatorial unipotent d-block of G. This proves the following:

Theorem 5.5. For any positive integers n and d, the combinatorial unipotent d-blocks of G satisfy the Second Main Theorem Property. In particular, if d = 1, then the unipotent d-blocks of G satisfy the Second Main Theorem Property, and, for any d > 0, the unipotent d-blocks of G of weight 0, 1, 2, and at most (d+1)/2 satisfy the Second Main Theorem Property.

Remark: It is easy to see that one can change the definition of d-regular element to “an element whose minimal polynomial has no irreducible factor of degree d (except maybe X − 1)” (instead of “degree divisible by d”) without affecting any of the results we have proved, apart from Theorem 4.5 (Nakayama Conjecture in the case d = 1).

Acknowledgements:

I wish to thank deeply Meinolf Geck, Radha Kessar, Attila Maróti and Geoffrey R. Robinson for the many discussions we had about this work. I also wish to thank Paul Fong and Bhama Srinivasan for their kindness and support. Finally, I thank Olivier Mathieu for his help with the proof of Lemma 4.9.

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