Jennings’ theorem for p-split groups.

Darren Semmen

University of California, Irvine, CA, USA

Abstract

Jennings [7] used the descending central series of dimension subgroups to produce a basis of the radical layers of the group ring of a p-group over a field k of characteristic p and, with it, a Hilbert polynomial for the dimensions of these layers. Later, Quillen [9] employed Lie ring methods of Lazard [8] to refine the proof of Jennings’ theorem. Analogously, Alperin [1] extended Jennings’ methods to permutation modules for p-groups, while Shalev [12] later recast this in terms of Quillen’s formulation. All of this naturally extends to p-split groups, finite groups with normal p-Sylow.

Consider the filtration of the group ring kG by the powers of its Jacobson radical. When G is a p-group, Quillen showed that the associated graded ring is isomorphic to the graded p-restricted universal enveloping algebra UJ_{∗}^{⊗k}(G) for the graded p-restricted Lie algebra generated from the filtration of G by its descending central sequence of dimension subgroups. For a p-split group G with p-Sylow P and a given maximal subgroup A of order prime to p, the associated graded ring is isomorphic to the skew group ring UJ_{∗}^{⊗k}(P)>◁A (Theorem 4), where A is concentrated in degree zero and the action of A on UJ_{∗}^{⊗k}(P) is induced from its conjugation action on P. Let H be a subgroup of G and S a semisimple kH-module. Consider the graded module associated to the filtration by the powers of the Jacobson radical of the module formed by inducing S up to P H. This graded module will be isomorphic to the Hopf algebra tensor product of S (regarded as a semisimple kP H-module) with the quotient of UJ_{∗}^{⊗k}(P) by a left ideal generated by elements corresponding to q − 1, where the q are elements of the p-Sylow of H; inducing this graded module up to G will preserve the grading (Theorem 5). Hilbert polynomials for the Brauer character of the radical layers allow explicit computation of the isomorphism classes of the radical layers of the module produced by inducing S to G.

The first two sections introduce this article’s conventions. Section 1 briefly reviews several topics of general interest whose notation is unfortunately not standardized, while section 2 follows with the specific Lie algebraic constructions of Quillen and Shalev. Section 3 elaborates on these constructions to

Email address: dsemmen@math.uci.edu (Darren Semmen). 1 Partially supported by Michael D. Fried via NSF Award #DMS-0202259.

Preprint submitted to Elsevier Science 16 December 2003

allow the descriptions made in Theorems 4 and 5. In section 4, the Hilbert polynomials for the Brauer character are derived from the identifications made in these theorems.

There is one application of these results that should be mentioned: computing the Cartan matrix of kΥ, for extensions Υ ↠ Γ of a known finite group Γ with kernel a p-group, using the main theorem of Alperin–Collins–Sibley [2]. Examples of this are done in a future paper [10], where they are of crucial importance.

1 Notation and conventions.

Let p be a fixed prime number. Throughout this article, k will be a given field of characteristic p and H a subgroup of a given p-split group G, i.e., a finite group with normal p-Sylow P. The p-Sylow Q of H will also be normal in H. Schur-Zassenhaus ensures that G has a subgroup A isomorphic to G/P such that A ∩ H ≃ H/Q; for convenience, denote A ∩ H by B. Finally, S will be a given semi-simple left kH-module and J will be the subgroup of G generated by P and B.

For the non-negative rational integers and the positive rational integers, we use the notations ℕ and ℤ^{+}, respectively.

1.1 Group rings and their modules.

Given an action of a finite group Γ on an associative algebra Λ, i.e., a homomorphism from Γ to the group of automorphisms of Λ, we may form the skew group ring Λ>◁Γ with elements and additive structure identical to the standard group ring ΛΓ, but whose multiplication is given as follows:

( ∑_{g_{1}∈Γ} λ_{g_{1}}g_{1}) ( ∑_{g_{2}∈Γ} λ′_{g_{2}}g_{2}) = ∑_{g_{1}∈Γ} ∑_{g_{2}∈Γ} λ_{g_{1}} (^{g_{1}}λ′_{g_{2}}) g_{1}g_{2}.

Note that the action of g_{1} on Λ has been written analogously to conjugation on the left in a group: ^{g}h := ghg^{−1} for all g, h ∈ Γ. The quotient of Λ>◁Γ by the two-sided ideal generated by elements of the form g − 1, where g ∈ Γ, is called the principal Λ>◁Γ-module and is denoted by Λ, as the subalgebra Λ forms a canonical set of representatives for the elements of this module.

Turning to the standard group ring, given a subgroup Υ of Γ and a ΛΥ-module M, the induced module ΛΓ ⊗_{ΛΥ} M is denoted by M↑^{ΛΓ} and the restriction from ΛΓ to ΛΥ is denoted by M↓_{ΛΥ}.

In the case of kΓ, the principal kΓ-module is a one-dimensional simple module called 1 1 1_{kΓ}; the kernel ω_{kΓ} of the natural map from kΓ onto 11 1_{kΓ} is called the augmentation ideal. The Hopf algebra tensor product M ⊗ N of two kΓmodules M and N is a kΓ-module whose set of elements is just M ⊗_{k} N; Γ acts diagonally on the tensor product, the action of the group ring given by extending this linearly.

1.2 Graded and Lie algebras.

A graded algebra Λ over k is an algebra over k with a given grading, i.e., a direct sum decomposition Λ = ⨁_{i∈ℕ} Λ_{i} into k-submodules such that Λ_{i} · Λ_{j} ⊆ Λ_{i+j} for all i and j in ℕ. An element x of a graded algebra Λ is homogeneous if it is an element of Λ_{ℓ} for some ℓ ∈ ℕ; ℓ is called the degree of x, and is denoted ϱ(x). A graded ideal of a graded algebra is simply an ideal which is generated by its homogeneous elements. If Λ is a graded associative-algebra over k and M is a Λ-module, then M is a graded Λ-module if it has a grading, i.e., a direct sum decomposition M = ⨁_{i∈ℕ} M_{i} into k-submodules such that Λ_{i} · M_{j} ⊆ M_{i+j} for all i and j in ℕ. Finally, a graded homomorphism (of algebras or modules) is a homomorphism which preserves the grading.

Recall that, for each element x of a Lie algebra L over k, there is a k-module endomorphism ad(x) of L given by ad(x)(y) = [x, y] for all y ∈ L. A prestricted Lie algebra L over k is a Lie algebra over k which has a “p-th power map” x ↦ x^{[p]} satisfying ad(x)^{p}(y) = [x^{[p]}, y] for all x and y in L; a p-restricted Lie algebra homomorphism is a Lie algebra homomorphism which commutes with the p-th power maps. Given a p-restricted Lie algebra L over k, the p-restricted universal enveloping algebra of L is the quotient of the ordinary universal enveloping algebra by the two-sided ideal generated by all elements of the form x^{p} − x^{[p]}, where x ∈ L.

The Poincaré–Birkhoff–Witt Theorem has a form tailored for p-restricted Lie algebras:

Theorem 1 (Poincaré–Birkhoff–Witt) Given an ordinal λ, if {x_{α} | α ∈ λ} is a basis of the p-restricted Lie algebra L, then {x_{α_{1}}^{i_{1}} … x_{α_{n}}^{i_{n}} | α_{1} < α_{2} < … < α_{n}; 0 ≤ i_{j} < p, ∀j} is a basis of the p-restricted universal enveloping algebra of L.

By choosing a basis of homogeneous elements, this theorem imposes a grading on the p-restricted universal enveloping algebra compatible with that of the original p-restricted Lie algebra.

1.3 Brauer characters.

Our approach to the Brauer character is similar to that of Holt and Plesken [5, Section 2.2.5] but, to make the construction canonical, we place the value of the Brauer character in the Witt ring. A more standard treatment, placing the values of the Brauer character in the complex numbers, can be found in Benson’s text [3, Section 5.3], for instance. Either way, the key property is that knowing the Brauer character of a module is equivalent to knowing the multiplicity of appearance of the simple factors in a composition series of the module.

Consider the category whose objects are the epimorphisms with range k and domain a complete rank-one discrete valuation ring with characteristic zero quotient field; a morphism ϕ from the object f_{1} to the object f_{2} is a ring homomorphism such that f_{2} ∘ ϕ = f_{1}. The initial object in this category is called the Witt ring, as is the domain W (k) of this epimorphism; see Serre [11], e.g., for details. A W (k)Γ-lattice is a W (k)Γ-module which is free as a W (k)-module.

Let Γ be a finite group, C a p′-conjugacy class of Γ, and M a kΓ-module. Let g be any element of C. Let ℘_{k} be the unique maximal ideal of W (k). By Maschke’s theorem, H^{2}(⟨g⟩, End_{k}(M↓_{k⟨g⟩})) = 0 = H^{1}(⟨g⟩, End_{k}(M↓_{k⟨g⟩})), so there is a W (k)⟨g⟩-lattice M̂_{g} such that M̂_{g}/℘_{k}M̂_{g} ≃ M ↓_{k⟨g⟩}, unique up to isomorphism [3, Theorem 3.7.7]. The Brauer character t_{C}(M) is defined to be the trace of the action of g on M̂_{g}, an algebraic integer. The Brauer character t_{C}(M) does not depend on the choice of g.

The Brauer character commutes with field extension. If σ : k → K is a field homomorphism, there is a unique ring homomorphism σ̂ : W (k) → W (K) making the following diagram commutative:

W (k) ⟶^{σ̂} W (K)
↓  ↓
k ⟶^{σ} K

Then, σ̂(t_{C}(M)) = t_{C}(KΓ ⊗_{kΓ} M), so the value of the Brauer character is preserved under extension of scalars.

The Brauer character is the characterizing invariant of the isomorphism class of a radical layer. In a p-group, since there is only one conjugacy class whose elements have order prime to p, the Brauer character reduces to just the dimension, exactly what Jennings, et al, specified in their Hilbert polynomials.

2 The constructions of Quillen and Shalev.

Let’s review Quillen’s constructions; a concise account for p-groups may be found in Benson [3, section 3.14]. Fix a p-group Γ. Define the dimension subgroups Δ_{i}(Γ) inductively by

Δ_{i}(Γ) = [Γ, Δ_{i−1}(Γ)]Δ_{⌈i/p⌉}(Γ)^{p},

where Δ_{1}(Γ) is Γ and ⌈x⌉ is the smallest rational integer larger than x. See Huppert and Blackburn [6, VIII. Theorem 1.13], for example, for details.

Regard the successive quotients as vector spaces over 𝔽_{p} and form the graded p-restricted Lie algebra

J_{∗}^{⊗k}(Γ) := k ⊗_{𝔽_{p}} ⨁_{i∈ℤ^{+}} Δ_{i}(Γ)/Δ_{i+1}(Γ)

with the Lie bracket inherited from Γ’s commutator:

[κ_{1} ⊗ g_{1}Δ_{i_{1}+1}(Γ), κ_{2} ⊗ g_{2}Δ_{i_{2}+1}(Γ)] := (κ_{1}κ_{2}) ⊗ [g_{1}, g_{2}]Δ_{i_{1}+i_{2}+1}(Γ)

where g_{j} ∈ Δ_{i_{j}}(Γ). Our p-restricted Lie algebra also inherits its p-th power map from Γ:

(κ ⊗ gΔ_{i+1}(Γ))^{[p]} := κ^{p} ⊗ g^{p}Δ_{pi+1}(Γ),

where g ∈ Δ_{i}(Γ).

The heart of Quillen’s proof is an isomorphism from the p-restricted universal enveloping algebra of J_{∗}^{⊗k}(Γ) (which we will call UJ_{∗}^{⊗k}(Γ)) to the algebra of the group ring graded by the powers of the augmentation ideal ω_{kΓ}:

gr_{∗}(kΓ) := ⨁_{i∈ℕ}ω_{kΓ}^{i}/ω_{kΓ}^{i+1}.

Sending g ∈ Δ_{i}(Γ) to g − 1, which will lie in ω_{kΓ}^{i}, produces a well-defined graded p-restricted Lie algebra monomorphism

J_{∗}^{⊗k}(Γ) →^{Φ} gr_{∗}(kΓ)  (1)
gΔ_{i+1}(Γ) ↦ (g − 1) + ω_{kΓ}^{i+1}

and this induces the desired isomorphism between UJ_{∗}^{⊗k}(Γ) and gr_{∗}(kΓ).

On to Shalev’s constructions. Consider a transitive permutation module M of Γ; this will be isomorphic to 11 1_{kΥ}↑^{kΓ} for some subgroup Υ of Γ. This is in turn naturally isomorphic to kΓ/kΓ · ω_{kΥ}. The graded module

gr_{∗}(M) := ⨁_{i∈ℕ}ω_{kΓ}^{i}M/ω_{kΓ}^{i+1}M

is then isomorphic to the quotient of gr_{∗}(kΓ) by the graded ideal

gr_{∗}(kΓ · ω_{kΥ}) := ⨁_{i∈ℕ}((ω_{kΓ}^{i} ∩ kΓ · ω_{kΥ}) + ω_{kΓ}^{i+1}) /ω_{kΓ}^{i+1}.

This ideal corresponds to the graded left ideal of UJ_{∗}^{⊗k}(Γ) generated by the following graded p-restricted Lie subalgebra of J_{∗}^{⊗k}(Γ):

J_{∗}^{⊗k}(Γ|Υ) := ⨁_{i∈ℤ}^{+}(Δ_{i}(Γ) ∩ Υ) Δ_{i+1}(Γ)/Δ_{i+1}(Γ).

A homogeneous basis of UJ_{∗}^{⊗k}(Γ)/UJ_{∗}^{⊗k}(Γ) · J_{∗}^{⊗k}(Γ|Υ) yields a Hilbert polynomial for the dimensions of the radical layers. Namely, choose a homogeneous basis {x_{m+1}, …, x_{n}} of J_{∗}^{⊗k}(Γ|Υ) and complete it to a homogeneous basis {x_{1}, …, x_{n}} of J_{∗}^{⊗k}(Γ). Apply the Poincaré–Birkhoff–Witt theorem: {x_{1}^{α_{1}} … x_{n}^{α_{n}} | 0 ≤ α_{i} < p for all i} will be a homogeneous basis of UJ_{∗}^{⊗k}(Γ) and a product x_{1}^{α_{1}} … x_{n}^{α_{n}} will lie in UJ_{∗}^{⊗k}(Γ) · J_{∗}^{⊗k}(Γ|Υ) iff α_{i} ≠ 0 for some i > m. Hence, the image of the set {x_{1}^{α_{1}} … x_{m}^{α_{m}} | 0 ≤ α_{i} < p for all i} will be a homogeneous basis of the quotient.

Theorem 2 (Jennings, Quillen, Alperin, Shalev) Let Υ be a subgroup of a p-group Γ.

(1) gr_{∗}(kΓ) ≃ UJ_{∗}^{⊗k}(Γ).

(2) gr_{∗}(11 1_{kΥ}↑^{kΓ}) ≃ UJ_{∗}^{⊗k}(Γ)/UJ_{∗}^{⊗k}(Γ) · J_{∗}^{⊗k}(Γ|Υ).

(3) If {x_{1}, …, x_{n}} is a homogeneous basis of J_{∗}^{⊗k}(Γ) such that {x_{m+1}, …, x_{n}} forms a basis of J_{∗}^{⊗k}(Γ|Υ), then dim_{k} (gr_{ℓ}(11 1_{kΥ}↑^{kΓ})) equals the coefficient of t^{ℓ} in

∏_{i=1}^{m} ∑_{j=0}^{p−1} t^{jϱ(x_{i})}.

Note that there is a typographical error in Shalev’s paper: the rank d_{m} appearing in the corollary of his theorem should be the rank of D_{m}/(H ∩ D_{m})D_{m+1}.

Let’s end by noting Hill’s [4] application of Jennings’ theory to comparing the radical and socle layers of the group ring of a p-group, and Alperin’s [1] extension of this to permutation modules:

Corollary 3 (Hill, Alperin) For a permutation module of a p-group, over a field of characteristic p, the radical and socle layers are identical.

3 Grading by the Jacobson radical.

The Jacobson radical of kG is generated by the augmentation ideal of kP: since kG/kG·ω_{kP} is isomorphic to kA, by Maschke’s theorem a semi-simple algebra, kG·ω_{kP} must contain rad(kG). Because P is normal in G, kG·ω_{kP} = ω_{kP} ·kG and so, by induction, (kG · ω_{kP})^{ℓ} = kG · ω_{kP}^{ℓ} for all ℓ. But ω_{kP} is the Jacobson radical of kP and thus is nilpotent, forcing kG · ω_{kP} to be nilpotent as well, and therefore contained in rad(kG). We have:

rad^{ℓ}(kG) = kG · ω_{kP}^{ℓ} ,
rad^{ℓ}(kJ) = kJ · ω_{kP}^{ℓ} , and
rad^{ℓ}(kH) = kH · ω_{kQ}^{ℓ}.

Consequently, rad^{ℓ}(kG) = kG · rad^{ℓ}(kJ) for all ℓ.

Grading a kG-module M by the Jacobson radical yields a graded module we denote

Gr_{∗}(M) := ⨁_{i∈ℕ}rad^{i}(M)/rad^{i+1}(M).

Since rad(kG) is a two-sided ideal, Gr_{∗}(kG) will be a graded associative algebra.

3.1 Grading the algebra.

Conjugation on the left produces an action of A on kP as a group of algebra automorphisms. The augmentation ideal ω_{kP} is stabilized by this action, as are its powers, yielding an action of A on the graded algebra gr_{∗}(kP).

By writing the elements of kG as a double sum over the elements of P and A, using the fact that any element of G can be written uniquely as a product of one element of P and one element of A, the group ring kG is seen to be naturally isomorphic to the skew group ring kP >◁A. The powers of the Jacobson radical rad^{i}(kG) = kG · ω_{kP}^{i} then identify with the ideals ω_{kP}^{i} >◁A (for any left kP -ideal I stabilized by the action of A, I>◁A is the left kP >◁Aideal generated by the sums of products of elements of I and A), and so the two graded algebras become canonically isomorphic as well:

Gr_{∗}(kG) ≃ gr_{∗}(kP)>◁A;

in this notation, A is concentrated in degree 0.

Conjugation by A both stabilizes the dimension subgroups of P, since the latter are characteristic, and commutes with commutators and p-th powers, since conjugation by an element of A is a group automorphism of P. Hence, this conjugation induces an action of A on J_{∗}^{⊗k}(P) as a group of graded prestricted Lie algebra automorphisms. The defining property of p-restricted universal enveloping algebras lifts this action to one of associative algebra automorphisms of UJ_{∗}^{⊗k}(P). Through the map Φ in equation 1, this action is identified with the one induced on gr_{∗}(kP) in the previous paragraph and we conclude that:

Theorem 4 Regarding A as concentrated in degree 0, Gr_{∗}(kG) ≃ UJ_{∗}^{⊗k}(P)>◁A.

3.2 Grading the induced module.

We will analyze the induced module S↑^{kG} by first examining the intermediate module S↑^{kJ}, and then inducing up from there: S↑^{kG}≃ S↑^{kJ}↑^{kG}.

Normal p-subgroups always act trivially on semi-simple modules over fields of characteristic p, so the semi-simple modules for kH, kJ, and kB are all in natural one-to-one correspondence. Given the semi-simple kH-module S, we let 𝕊 denote the corresponding semi-simple kJ-module.

Theorem 5 With 𝕊 regarded as being concentrated in degree 0 and UJ_{∗}^{⊗k}(P)/UJ_{∗}^{⊗k}(P) · J_{∗}^{⊗k}(P |Q) regarded as a quotient of the principal UJ_{∗}^{⊗k}(P)>◁B-module,

(1) Gr_{∗}(S↑^{kJ}) ≃ [UJ_{∗}^{⊗k}(P)/UJ_{∗}^{⊗k}(P) · J_{∗}^{⊗k}(P |Q)] ⊗ 𝕊 and

(2) Gr_{ℓ}(S↑^{kG}) ≃ Gr_{ℓ}(S↑^{kJ})↑^{kG}, for all ℓ.

PROOF. Since rad(kH) = kH · ω_{kQ}, ω_{kQ} is in the kernel of the action of kH on S, so

S↑^{kJ} = kJ ⊗_{kH} S
≃ (kJ/kJ · ω_{kQ}) ⊗_{kH} S
≃ (kJ/kJ · ω_{kQ}) ⊗_{kB} S↓_{kB} .

We wish to view kJ simultaneously as a group algebra over k and as the skew group ring kP >◁B. Rewrite the elements of kJ to explicitly see the isomorphism of kJ with the skew group ring kP >◁B:

∑_{b∈B} (∑_{x∈P} κ_{xb}x) b.

This identifies the left ideal kJ · ω_{kQ} with the left ideal (kP · ω_{kQ}) >◁B. Notice the importance here of B stabilizing the subgroup Q under conjugation, so that B stabilizes the left ideal kP · ω_{kQ} of kP; this is the reason for the intermediate step of inducing up to kJ. We can now rewrite S↑^{kJ} using the Hopf algebra tensor product of kJ:

\begin{aligned}S↑^{kJ} & ≃ & (kJ/kJ · ω_{kQ}) ⊗_{kB} S↓_{kB} \\ & ≃ & (kP/kP · ω_{kQ}) ⊗ 𝕊, \end{aligned}

where kP/kP · ω_{kQ} is a quotient of the principal kP >◁B-module. Now,

rad^{i}(kP >◁B) · (kP/kP · ω_{kQ}) ⊗ 𝕊 = ω_{kP}^{i} (kP/ (kP · ω_{kQ})) ⊗ 𝕊
= ((ω_{kP}^{i} + kP · ω_{kQ})/ (kP · ω_{kQ})) ⊗ 𝕊

and so

Gr_{∗}(S↑^{kJ}) ≃ gr_{∗}(kP/kP · ω_{kQ}) ⊗ 𝕊
≃ gr_{∗}(11 1_{kQ}↑^{kP} ) ⊗ 𝕊  (2)

as graded kP >◁B-modules, with 𝕊 concentrated in degree 0.

The isomorphism between gr_{∗}(kP) and UJ_{∗}^{⊗k}(P) commutes with the action of B. It also identifies the kernel of the quotient map from gr_{∗}(kP) to gr_{∗}(11 1_{kQ}↑^{kP} ) with UJ_{∗}^{⊗k}(P) · J_{∗}^{⊗k}(P |Q). But B stabilizes both the subset J_{∗}^{⊗k}(P) of Lie elements of UJ_{∗}^{⊗k}(P) and the subgroup Q. Since the action of B on J_{∗}^{⊗k}(P) is induced by conjugation on P, B also stabilizes J_{∗}^{⊗k}(P |Q). Therefore,

Gr_{∗}(S↑^{kJ}) ≃ [UJ_{∗}^{⊗k}(P)/UJ_{∗}^{⊗k}(P) · J_{∗}^{⊗k}(P |Q)] ⊗ 𝕊. (3)

Proceed with the induction to kG:

rad^{ℓ}(S↑^{kG}) ≃ rad^{ℓ}(kG) · S↑^{kJ}↑^{kG}
= rad^{ℓ}(kG) · (kG ⊗_{kJ} S↑^{kJ})
= rad^{ℓ}(kG) ⊗_{kJ} S↑^{kJ}
= (kG · rad^{ℓ}(kJ)) ⊗_{kJ} S↑^{kJ}
= kG ⊗_{kJ} (rad^{ℓ}(kJ) · S↑^{kJ})
= rad^{ℓ}(S↑^{kJ})↑^{kG} .

Since taking quotients commutes with induction, Gr_{ℓ}(S↑^{kG}) ≃ Gr_{ℓ}(S↑^{kJ})↑^{kG}. □

It is now easy to mimic the proof of Hill’s and Alperin’s results:

Corollary 6 The radical layers and socle layers of S↑^{kG} are the same.

PROOF. The theorem tells us that

dim_{k} (Gr_{ℓ}(S↑^{kG})) = (G : J) · dim_{k} (Gr_{ℓ}(S↑^{kJ}))
= (G : J) · dim_{k} (S) · dim_{k} (gr_{ℓ}(11 1_{kQ}↑^{kP} )) .

Let s be the radical length of S ↑^{kG}, which, by the theorem, equals that of 11 1_{kQ} ↑^{kP}. Recall that dualizing commutes with induction, i.e., (S^{∗}) ↑^{kG}≃ (S↑^{kG})^{∗}, and that taking the dual exchanges the socle and radical layers:

dim_{k} (soc^{ℓ} (S↑^{kG}) /soc^{ℓ−1} (S↑^{kG})) = dim_{k} (Gr_{ℓ−1}((S↑^{kG})^{∗}))
= dim_{k} (Gr_{ℓ−1}(S^{∗}↑^{kG}))
= (G : J) · dim_{k} (S^{∗}) · dim_{k} (gr_{ℓ−1}(11 1_{kQ}↑^{kP} ))
= (G : J) · dim_{k} (S) · dim_{k} (gr_{s−ℓ}(11 1_{kQ}↑^{kP} ))
= dim_{k} (Gr_{s−ℓ}(S↑^{kG}));

the second-to-last equality follows from corollary 3, since permutation modules are isomorphic to their dual. Now use the fact that soc^{ℓ} (S↑^{kG}) contains rad^{s−ℓ}(S↑^{kG}) and proceed by induction on ℓ to conclude

that soc^{ℓ} (S↑^{kG}) = rad^{s−ℓ}(S↑^{kG}). □

4 Producing the Hilbert polynomials.

The intermediate step of inducing to kJ also appears in explicit calculation. The Poincaré–Birkhoff–Witt theorem produces Hilbert polynomials for the Brauer character of the graded kJ-module in the first part of Theorem 5, while use of Mackey’s lemma (corollary 8) specifies what linear combination of the Hilbert polynomials in equation 4 is the Hilbert polynomial for t_{C}(Gr_{∗}(S↑^{kG})).

As extending scalars does not affect the value of the Brauer character of a module, assume k contains the |G|_{p′}-th roots of unity. (The notation |Γ|_{p′} means the largest rational integer relatively prime to p which divides the order of the finite group Γ.) Fix a p′-conjugacy class D of J and an element b ∈ D ∩ B. Choose an ordered homogeneous basis {x_{1}, …, x_{n}} of J_{∗}^{⊗k}(P) consisting of eigenvectors for the linear transformation which b induces on J_{∗}^{⊗k}(P); also mandate that \{x_{m+1}, …, x_{n}\} forms a basis of J_{∗}^{⊗k}(P |Q). Let \overline{λ_{i}} be the eigenvalue corresponding to x_{i}.

By the Poincaré–Birkhoff–Witt theorem, the products x_{1}^{α_{1}} … x_{n}^{α_{n}} form a basis of UJ_{∗}^{⊗k}(P), where 0 ≤ α_{i} < p for all i; these elements are also eigenvectors for the action of b, the eigenvalue corresponding to x_{1}^{α_{1}} … x_{n}^{α_{n}} being \overline{λ_{1}}^{α_{1}} … \overline{λ_{n}}^{α_{n}}. As in the paragraph preceding Theorem 2, the image of the set {x_{1}^{α_{1}} … x_{m}^{α_{m}} | 0 ≤ α_{i} < p for all i} will be a homogeneous basis of the quotient UJ_{∗}^{⊗k}(P)/UJ_{∗}^{⊗k}(P) · J_{∗}^{⊗k}(P |Q). Finally, let λ_{i} ∈ W (k) be the unique lift of \overline{λ_{i}} having the same multiplicative order.

Then, by the first part of Theorem 5, t_{D}(Gr_{ℓ}(S↑^{kJ})) equals the product of t_{D∩H}(S) and the following sum, a sum over α_{i} with 0 ≤ α_{i} < p:

∑_{α_{1}ϱ(x_{1})+...+α_{m}ϱ(x_{m})=ℓ} (∏_{i=1}^{m} λ_{i}^{α_{i}});

note that D ∩ H is a single conjugacy class of H. Thus, t_{D}(Gr_{ℓ}(S↑^{kJ})) is just the coefficient of t^{ℓ} in

t_{D∩H}(S) · ∏_{i=1}^{m} ∑_{j=0}^{p−1} (λ_{i}t^{ϱ(x_{i})})^{j} . (4)

Proposition 7 Let Υ be a subgroup of a finite group Γ and M a kΥ-module. Let C be a p′-conjugacy class in Γ. Then,

t_{C}(M↑^{kΓ}) = ∑_{Υ−c.c.D⊆C∩Υ} \frac{|Γ| · |D|}{|Υ| · |C|} · t_{D}(M),

where the sum is over Υ-conjugacy classes D contained in C ∩ Υ; the sum is regarded as zero if C ∩ Υ = ∅.

PROOF. Let a ∈ C. The Mackey decomposition is then:

M↑^{kΓ}↓_{k⟨a⟩}≃ ⨁_{⟨a⟩gΥ}(^{g}M)↓_{k(⟨a⟩∩^{g}Υ)}↑^{k⟨a⟩}, (5)

where ^{g}M denotes the k^{g}Υ-submodule g ⊗_{kΥ} M of kΓ ⊗_{kΥ} M.

Assume that k contains the |Γ|_{p′}-th roots of unity. Fix an element g of Γ and let j be the minimal positive integer such that ⟨a^{j}⟩ = ⟨a⟩ ∩^{g}Υ.

First assume that j ≠ 1. Let M̂_{a^{j}} be the unique W (k)⟨a^{j}⟩-lattice with reduction (^{g}M)↓_{k⟨a^{j}⟩}. Then, M̂_{a^{j}}↑^{W (k)⟨a⟩} is the unique W (k)⟨a⟩- lattice with reduction (^{g}M)↓_{k⟨a^{j}⟩}↑^{k⟨a⟩}. Choose as basis of M̂_{a^{j}}↑^{W (k)⟨a⟩} the set {a^{i} ⊗ v_{m} | 0 ≤ i < j and 1 ≤ m ≤ n}, where {v_{1}, …, v_{n}} is a basis of M̂_{a^{j}} consisting of eigenvectors for the action of a^{j}. Let λ_{m} be the eigenvalue corresponding to v_{m}. Then, with respect to the basis just given, the matrix representing the action of a on M̂_{a^{j}}↑^{W (k)⟨a⟩} will consist of blocks on the diagonal, where the m-th block is the rational canonical form of the polynomial t^{j} − λ_{m}. The trace of this matrix is zero, and so t_{{a}} ((^{g}M)↓_{k(⟨a⟩∩^{g}Υ)}↑^{k⟨a⟩}) = 0.

If j = 1, ⟨a⟩ will be a subset of ^{g}Υ, so C ∩ Υ ≠ ∅. Each double coset ⟨a⟩gΥ satisfying ⟨a⟩ ⊆ ^{g}Υ will equal gΥ, and so assigning it to the Υ-conjugacy class D ⊆ C ∩ Υ containing ^{g^{−1}}a will be a well-defined function. Conversely, given an Υ-conjugacy class D ⊆ C ∩ Υ, let g ∈ Γ be such that ^{g^{−1}}a ∈ D; D will then be the conjugacy class to which the double coset ⟨a⟩gΥ is assigned.

How many double cosets are assigned to the same Υ-conjugacy class D ⊆ C ∩ Υ? In general, more than one. For each such conjugacy class, fix a representative x_{D} ∈ D. Given a double coset ⟨a⟩gΥ assigned to D, g may be chosen so that ^{g^{−1}}a = x_{D}. Every element h ∈ Γ such that ^{h^{−1}}a = x_{D} is equal to gz, for some element z of C_{Γ}(x_{D}). So, every double coset assigned to D is of the form ⟨a⟩gzΥ = gzΥ, for some z ∈ C_{Γ}(x_{D}). Two such double cosets gz_{1}Υ and gz_{2}Υ are equal iff z_{1}^{−1}z_{2} is in Υ (and hence in C_{Υ}(x_{D})). Thus, the number of double cosets ⟨a⟩gΥ assigned to D is the index of C_{Υ}(x_{D}) in C_{Γ}(x_{D}), i.e., (|Γ| · |D|)/(|Υ| · |C|).

For any Υ-conjugacy class D ⊆ C ∩ Υ, any double coset ⟨a⟩gΥ assigned to D, and any μ in ^{g}M, we will have a · μ = ^{g^{−1}}aμ, where μ on the right of the equation is regarded as an element of M. Notice that ^{g^{−1}}a is an element of D.

The result now follows from the Mackey decomposition of equation 5. □

Corollary 8 Let C be any p′-conjugacy class of G. Then,

t_{C}(Gr_{ℓ}(S↑^{kG})) = ∑_{J−c.c.D⊆C∩J} \frac{|G| · |D|}{|J| · |C|} · t_{D} (Gr_{ℓ}(S↑^{kJ})) ,

where the sum is over J-conjugacy classes D contained in C ∩ J; the sum is regarded as zero if C ∩ J = ∅.

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