THE NON-PROJECTIVE PART OF THE TENSOR POWERS OF A MODULE

DAVE BENSON AND PETER SYMONDS

Abstract. Let M be a finite dimensional modular representation of a finite group G. We consider the generating function for the non-projective part of the tensor powers of M, and we write γ_{G}(M) for the reciprocal of the radius of convergence of this power series. We investigate the properties of the invariant γ_{G}(M), using tools from representation theory, and from the theory of commutative Banach algebras.

1. Introduction

Throughout this paper, G is a finite group and k is a field of characteristic p and we only consider finite dimensional kG-modules. If M is such a module we consider its tensor powers M^{⊗n} and ask how large the non-projective part is.

Definition 1.1. For a kG-module M, we write M = M′ ⊕(proj) where M′ has no projective direct summands and (proj) denotes a projective module. Then M′ is called the core of M and denoted core_{G}(M). We write c_{n}^{G}(M) for the dimension of core_{G}(M^{⊗n}); it is well defined by the Krull-Schmidt Theorem.

As an example, let G be the cyclic group of order five, k a field of characteristic five, and M the two dimensional indecomposable kG-module. Then the non-projective summands of M^{⊗n} follow a Fibonacci pattern. We have c_{2n}^{G}(M) ∼ τ^{2n+1} and c_{2n+1}^{G}(M) ∼ 2τ^{2n+1}, where

τ = (1 + \sqrt{5})/2 = 2 cos(π/5) ≈ 1.618

is the golden ratio.

This suggests that we should examine \sqrt[n]{c_{n}^{G}(M)} and we define the invariant γ_{G}(M) to be lim sup_{n→∞} \sqrt[n]{c_{n}^{G}(M)}. In the example above, we have γ_{G}(M) = τ. The following theorem

summarises the results of this paper.

Theorem 1.2. The invariant γ_{G}(M) has the following properties:

(i) We have γ_{G}(M) = lim_{n→∞} \sqrt[n]{c_{n}^{G}(M)} = inf_{n⩾1} \sqrt[n]{c_{n}^{G}(M)}.

(ii) We have 0 ⩽ γ_{G}(M) ⩽ dim M.

(iii) A kG-module M is p-faithful (Definition 6.1) if and only if γ_{G}(M) < dim M.

(iv) A kG-module M is projective if and only if γ_{G}(M) = 0.

(v) A kG-module M is endotrivial if and only if γ_{G}(M) = 1.

(vi) If a kG-module M is neither projective nor endotrivial then γ_{G}(M) ⩾ \sqrt{2}.

2010 Mathematics Subject Classification. 20C20, 46J99.

Key words and phrases. Projective module, core, tensor powers, modular representation, Green ring, commutative Banach algebra.

(vii) If γ_{G}(M) = \sqrt{2} then M ⊗ M^{∗} ⊗ M ≅ M ⊕ M ⊕ (proj).

(viii) We have γ_{G}(M^{∗}) = γ_{G}(M).

(ix) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of kG-modules then each

γ_{G}(M_{i}) is at most the sum of the other two.

(x) We have max{γ_{G}(M), γ_{G}(N)} ⩽ γ_{G}(M ⊕ N) ⩽ γ_{G}(M) + γ_{G}(N).

(xi) If N is isomorphic to a direct sum of m copies of M then γ_{G}(N) = m.γ_{G}(M).

(xii) We have γ_{G}(k ⊕ M) = 1 + γ_{G}(M).

(xiii) We have γ_{G}(M ⊗ N) ⩽ γ_{G}(M)γ_{G}(N).

(xiv) We have γ_{G}(M^{⊗m}) = γ_{G}(M)^{m}.

(xv) We have γ_{G}(ΩM) = γ_{G}(M).

(xvi) If H ⩽ G then we have γ_{H}(M) ⩽ γ_{G}(M). (xvii) We have γ_{G}(M) = max_{E⩽G} γ_{E}(M), where the maximum is taken over the elementary

abelian p-subgroups E ⩽ G.

The proofs of these can be found as follows: (i) in Theorem 4.4, (ii) and (iv) in Lemma 2.6, (iii) in Theorem 6.3, (v) in Theorem 7.5, (vi) in Theorem 5.8, (vii) in Theorem 12.6, (viii) in Lemma 2.7, (ix) in Corollary 5.6, (x) in Theorem 3.2, (xi) in Theorem 3.4, (xii) in Theorem 4.6, (xiii) in Theorem 5.1, (xiv) in Theorem 5.2, (xv) in Theorem 5.4, (xvi) in Lemma 2.10, and (xvii) in Theorem 7.2.

We calculate the value of γ_{G}(M) for all modules for a cyclic group of order p in Theorem 10.1.

We use the spectral theory of commutative Banach algebras to connect the invariant γ_{G}(M) to the structure of the Green ring a(G). Recall that a species of a(G) is a ring homomorphism s: a(G) → ℂ. We say that a species s is core-bounded if for all kG-modules M we have |s([M])| ⩽ dim core_{G}(M). So for example the Brauer species are not corebounded, because they do not vanish on projective modules. The proof of the following theorem can be found in Section 9.

Theorem 1.3. If M is a kG-module then γ_{G}(M) is the supremum of |s([M])|, where s runs over the core-bounded species s: a(G) → ℂ. Furthermore, there exists a core-bounded species s of a(G) such that s([M]) = γ_{G}(M).

We formulate some conjectures about the behaviour of the invariant γ_{G}(M), two of which we restate here:

Conjecture 1.4 (Conjecture 5.3). We have γ_{G}(M ⊗ M^{∗}) = γ_{G}(M)^{2}.

Conjecture 1.5 (Conjecture 13.3). For all large enough values of n, the function c_{n}^{G}(M) satisfies a homogeneous linear recurrence relation with constant coefficients.

Finally, we have included a number of examples to illustrate how to calculate the invariant γ_{G}(M). These examples were worked out using the computer algebra system Magma [6]. Note that this paper is written using left modules, while Magma uses right modules, so the matrices have been transposed.

Acknowledgment. This material is partly based on work of the first author supported by the National Science Foundation under Grant No. DMS-1440140 while he was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester, and of the second author supported by an International Academic Fellowship from the Leverhulme Trust. We wish to thank Burt Totaro for his helpful comments and for pointing out Proposition 13.5.

2. The Invariant γ_{G}(M)

We begin with some properties of tensor products.

Proposition 2.1. Let M be a kG-module.

(i) M is isomorphic to a direct summand of M ⊗ M^{∗} ⊗ M.

(ii) If M has dimension divisible by p then M⊗M^{∗}⊗M has a direct summand isomorphic to M ⊕ M.

Proof. Let m_{i} be a basis for M and f_{i} the dual basis of M^{∗}. Thus ∑_{i} f_{i}(m_{i}) = dim(M), and for m ∈ M we have m = ∑_{i} f_{i}(m)m_{i}.

For (i) we have maps M → M ⊗M^{∗}⊗M given by m ↦ ∑_{i} m⊗f_{i}⊗m_{i} and M ⊗M^{∗}⊗M → M given by m ⊗ f ⊗ m′ ↦ f(m)m′, with composite the identity on M.

For (ii) (cf. Proposition 4.9 in Auslander and Carlson [2], where this is proved with the further hypothesis that M is indecomposable, but this hypothesis is not used in the proof), we have maps M ⊕ M → M ⊗ M^{∗} ⊗ M given by

(m, m′) ↦ ∑_{i}(m ⊗ f_{i} ⊗ m_{i} + m_{i} ⊗ f_{i} ⊗ m′)

and M ⊗M^{∗} ⊗M → M ⊕M given by m⊗f ⊗m′ ↦ (f(m)m′, f(m′)m). If M has dimension divisible by p then the composite is the identity on M ⊕ M. □

Lemma 2.2. If M is a kG-module, then M ⊗M^{∗} is projective if and only if M is projective.

Proof. The tensor product of any module with a projective module is projective. So by Proposition 2.1, M projective implies M ⊗ M^{∗} projective implies M ⊗ M^{∗} ⊗ M projective implies M projective. □

Lemma 2.3. For any kG-module M, if M^{⊗n} is projective for some n ⩾ 1, then so is M.

Proof. It follows from Proposition 2.1 that M^{⊗(n−1)} is isomorphic to a direct summand of M^{⊗n} ⊗M^{∗}. So M^{⊗n} is projective if and only if M^{⊗(n−1)} is projective. The result now follows by induction on n. □

For a kG-module M, recall that we write c_{n}^{G}(M) for the dimension of core_{G}(M^{⊗n}).

Definition 2.4. We define

γ_{G}(M) = lim sup_{n→∞} \sqrt[n]{c_{n}^{G}(M)}.

Remarks 2.5. (i) The invariant γ_{G}(M) is robust, in that the dimension of core_{G}(M) may be replaced by the number of composition factors of core_{G}(M) or the number of composition factors of the socle of core_{G}(M), and so on.

(ii) An interesting invariant is γ_{G}(M)/ dim M, which we think of as the “non-projective proportion of M in the limit.” In the example of the introduction, we have γ_{G}(M) = τ. Thus γ_{G}(M)/ dim M ≈ 0.809, and so we think of M as “about 19.1% projective in the limit.”

(iii) We shall see in Section 4, using the theory of submultiplicative functions, that in fact lim_{n→∞} \sqrt[n]{c_{n}^{G}(M)} exists and is equal to inf_{n⩾1} \sqrt[n]{c_{n}^{G}(M)}.

We begin with some obvious properties of the invariant γ_{G}(M).

Lemma 2.6. For any kG-module we have:

(i) 0 ⩽ γ_{G}(M) ⩽ dim M, (ii) γ_{G}(M) = 0 if and only if M is projective, (iii) If M is not projective then γ_{G}(M) ⩾ 1.

Proof. Part (i) is because

c_{n}^{G}(M) = dim core_{G}(M^{⊗n}) ⩽ dim M^{⊗n} = (dim M)^{n}.

If M is projective then clearly γ_{G}(M) = 0. Conversely, if M is not projective then, by Lemma 2.3, no c_{n}^{G}(M) is 0. Since this number is an integer we have c_{n}^{G}(M) ⩾ 1, proving the other two parts. □

Lemma 2.7. We have γ_{G}(M^{∗}) = γ_{G}(M).

Proof. We have core_{G}(M^{∗}) ≅ core_{G}(M)^{∗} and so c_{n}^{G}(M^{∗}) = c_{n}^{G}(M). □

Recall that we have the syzygy operator Ω, where ΩM is defined to be the kernel of a projective cover P → M. Similarly, Ω^{−1}M is defined to be the cokernel of an injective hull M → I. Since projective kG-modules are the same as injective modules, we have Ω(Ω^{−1}M) ≅ core_{G}(M) ≅ Ω^{−1}(ΩM).

For n > 0, Ω^{n}M denotes Ω(Ω^{n−1}M), Ω^{−n}M denotes Ω^{−1}(Ω^{−n+1}M), and Ω^{0}M denotes core_{G}(M). For n ∈ ℤ we have core_{G}(Ω^{n}k ⊗ M) ≅ Ω^{n}M.

Lemma 2.8. We have γ_{G}(Ωk) = γ_{G}(Ω^{−1}k) = 1, provided that p divides |G|.

Proof. We have core_{G}((Ωk)^{⊗n}) ≅ Ω^{n}k, and dim Ω^{n}k grows polynomially in n (see for example [3] §5.3). Therefore γ_{G}(Ωk) = 1. Since (Ωk)^{∗} ≅ Ω^{−1}k, Lemma 2.7 shows that γ_{G}(Ω^{−1}k) = 1. □

Lemma 2.9. If core_{G}(M) ≅ core_{G}(N) then γ_{G}(M) = γ_{G}(N).

Proof. If core_{G}(M) ≅ core_{G}(N) then, since (proj) ⊗ (anything) = (proj), we have

core_{G}(M^{⊗n}) ≅ core_{G}(core_{G}(M)^{⊗n}) ≅ core_{G}(core_{G}(N)^{⊗n}) ≅ core_{G}(N^{⊗n}).

Thus c_{n}^{G}(M) = c_{n}^{G}(N), and so γ_{G}(M) = γ_{G}(N). □

Lemma 2.10. If H is a subgroup of G and M is a kG-module then γ_{H}(M) ⩽ γ_{G}(M).

Proof. We have c_{n}^{H}(M) ⩽ c_{n}^{G}(M) for all n. □

Lemma 2.11. If K is an extension field of k then γ_{G}(K ⊗_{k} M) = γ_{G}(M).

Proof. This will follow immediately if we can show that for any kG-module N, we have K ⊗_{k} core_{G}(N) ≅ core_{G}(K ⊗_{k} N). For this, we need to show that if K ⊗_{k} N has a projective summand P then N also has a projective summand. Consider the restrictions of K ⊗_{k} N and P from KG to kG. The restriction P ↓_{kG} must be a sum of finite dimensional indecomposable projective kG-modules; let P′ be one of them. It is a summand of K ⊗_{k} M↓_{kG}, which is a sum of copies of N. Because it is finite dimensional, P′ is a summand of a finite sum of copies of N, hence is a summand of N, by the Krull–Schmidt Theorem. □

Example 2.12. Let M be the 3 dimensional faithful uniserial module for G = ℤ/3 × ℤ/3 = ⟨g, h⟩ over 𝔽_{3} given by

g ↦ (\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) h ↦ (\begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

Then Ω^{2}M ≅ M, M is algebraic (Craven [11], Section 3.3.2), and ΩM has dimension 6. The modules M′ = M ⊗ M^{∗} and ΩM′ ≅ ΩM ⊗ M^{∗} ≅ M ⊗ Ω(M^{∗}) are indecomposable, of dimensions 9 and 18 respectively. The indecomposable summands of tensor powers of M are determined by the equations

M ⊗ M ≅ M^{∗} ⊕ Ω(M^{∗}),  M ⊗ M′ ≅ 2M ⊕ 2ΩM ⊕ P,
M ⊗ ΩM ≅ M^{∗} ⊕ Ω(M^{∗}) ⊕ P,  M ⊗ ΩM′ ≅ 2M ⊕ 2ΩM ⊕ 4P,

where P is the 9 dimensional projective module. These equations imply that

M^{⊗5} ≅ 8M^{⊗2} ⊕ 19P.

It follows that for n ⩾ 5 we have c_{n}(M) = 8c_{n−3}(M), and so γ_{G}(M) = 2. Similarly we have γ_{G}(M′) = 4.

3. Short Exact Sequences and Direct Sums

Lemma 3.1. Let a_{n}, b_{n} and c_{n} be sequences of non-negative real numbers, satisfying

c_{n} ⩽ ∑_{i=0}^{n} \binom{n}{i}a_{i}b_{n−i}.

Then

lim sup_{n→∞} \sqrt[n]{c_{n}} ⩽ lim sup_{n→∞} \sqrt[n]{a_{n}} + lim sup_{n→∞} \sqrt[n]{b_{n}}.

Proof. The statement that lim sup_{n→∞} \sqrt[n]{a_{n}} = α implies that for all ε > 0, there exists m such

that for all n ⩾ m we have a_{n} ⩽ (α + ε)^{n}. Introducing a positive constant A, we can assume that a_{n} ⩽ A(α + ε)^{n} for all n ⩾ 0. Similarly, if lim sup_{n→∞} \sqrt[n]{b_{n}} = β then for all ε > 0 there

exists a positive constant B such that for all n ⩾ 0 we have b_{n} ⩽ B(β + ε)^{n}. Thus for all ε > 0 there is a positive constant C = AB such that for all n ⩾ 0 we have

c_{n} ⩽ ∑_{i=0}^{n} \binom{n}{i}A(α + ε)^{i}B(β + ε)^{n−i} = C(α + β + 2ε)^{n},

and so lim sup_{n→∞} \sqrt[n]{c}_{n} ⩽ α + β. □

Theorem 3.2. If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of kG-modules then

γ_{G}(M_{2}) ⩽ γ_{G}(M_{1}) + γ_{G}(M_{3}).

If the sequence splits then we also have

max{γ_{G}(M_{1}), γ_{G}(M_{3})} ⩽ γ_{G}(M_{2}).

Proof. The module M_{2}^{⊗n} has a filtration of length 2^{n} where the filtered quotients are \binom{n}{i} copies of M_{1}^{⊗i} ⊗ M_{2}^{⊗(n−i)} (0 ⩽ i ⩽ n). Projective summands of a filtered quotient split off the entire module, since they are also injective. So

c_{n}^{G}(M_{2}) = dim core_{G}(M_{2}^{⊗n}) ⩽ ∑_{i=0}^{n} \binom{n}{i} dim core_{G}(M_{1}^{⊗i} ⊗ M_{3}^{⊗(n−i)})

⩽ ∑_{i=0}^{n} \binom{n}{i}c_{i}^{G}(M_{1})c_{n−i}^{G}(M_{3})

Applying Lemma 3.1, we deduce that

γ_{G}(M_{2}) ⩽ γ_{G}(M_{1}) + γ_{G}(M_{3}).

If the sequence splits, then each c_{n}^{G}(M_{2}) is at least as big as c_{n}^{G}(M_{1}) and also at least as big as c_{n}^{G}(M_{3}). □

Example 3.3. Let M be the two dimensional natural module for SL(2, 𝔽_{4}) and let N be its Frobenius twist. Then M ⊗ N is the four dimensional Steinberg module St, which is projective. Furthermore, M^{⊗3} ≅ M ⊕ M ⊕ St and N^{⊗3} ≅ N ⊕ N ⊕ St. Thus we have

core((M ⊕ N)^{⊗n}) ≅ core(M^{⊗n}) ⊕ core(N^{⊗n}),

and

γ_{G}(M) = γ_{G}(N) = γ_{G}(M ⊕ N) = \sqrt{2}.

This shows that the first inequality in the theorem is not always an equality, even for direct sums.

We shall make further use of this example in Remark 5.9.

On the other hand, for sums of isomorphic modules, we have the following.

Theorem 3.4. If N is isomorphic to a direct sum of m copies of a kG-module M then γ_{G}(N) = m · γ_{G}(M).

Proof. The module N^{⊗n} is isomorphic to a direct sum of m^{n} copies of M, so we have c_{n}^{G}(N) = m^{n} · c_{n}^{G}(M) and \sqrt[n]{c_{n}^{G}(M)} = m · (\sqrt[n]{c_{n}^{G}(M)}). Now take lim sup_{n→∞}. □

Here is another useful bound.

Theorem 3.5. If M_{1} ⊗ ⋯ ⊗ M_{m} is not projective then γ_{G}(M_{1} ⊕ ⋯ ⊕ M_{m}) ⩾ m.

Proof. By Lemma 2.3, no power of M_{1} ⊗ ⋯ ⊗ M_{m} is projective, so neither is any module of the form M_{1}^{i_{1}} ⊗⋯⊗M_{m}^{i_{m}}. The module (M_{1}⊕⋯⊕M_{m})^{⊗n} thus has at least m^{n} non-projective summands, so we have

c_{n}^{G}(M_{1} ⊕ ⋯ ⊕ M_{m}) ⩾ m^{n}

and

\sqrt[n]{c_{n}^{G}(M_{1} ⊕ ⋯ ⊕ M_{m})} ⩾ m.

Now take lim sup_{n→∞}. □

Corollary 3.6. If M is not projective then γ_{G}(k ⊕ M) ⩾ 2.

Proof. This follows by taking m = 2, M_{1} = k and M_{2} = M in Theorem 3.5. □

Remark 3.7. We shall prove in the next section, using the theory of submultiplicative sequences, that γ_{G}(k ⊕ M) is always equal to 1 + γ_{G}(M).

4. Submultiplicative Sequences

In this section, we investigate the submultiplicative properties of γ_{G}, and deduce Theorem 4.6. We shall revisit this from the point of view of Banach algebras and Gelfand’s spectral radius theorem later on, but for the moment we shall try to stay elementary.

Definition 4.1. We say that a sequence c_{0}, c_{1}, c_{2}, … of non-negative real numbers is submultiplicative if c_{0} = 1, and for all m, n ⩾ 0 we have c_{m+n} ⩽ c_{m}.c_{n}.

Lemma 4.2. If M is a kG-module then c_{n}(M) is a submultiplicative sequence, provided that p divides |G|.

Proof. This follows from the fact that

core_{G}(M^{⊗m} ⊗ M^{⊗n}) ≅ core_{G}(core_{G}(M^{⊗m}) ⊗ core_{G}(M^{⊗n})).  □

Lemma 4.3 (Fekete [15]). If c_{n} is a submultiplicative sequence then

lim sup_{n→∞} \sqrt[n]{c_{n}} = lim_{n→∞} \sqrt[n]{c_{n}} = inf_{n⩾1} \sqrt[n]{c_{n}}.

Proof. It suffices to show that lim sup_{n→∞} \sqrt[n]{c_{n}} ≤ inf_{n⩾1} \sqrt[n]{c_{n}}. If some c_{n} is equal to zero, then so are all subsequent ones. So we assume that all c_{n} > 0. Suppose that L is a number such that

inf_{n→∞} \sqrt[n]{c_{n}} < L.

Then there is an m ⩾ 1 with \sqrt[m]{c_{m}} < L. For n > m we use division with remainder to write n = mq_{m} + r_{m} with 0 ⩽ r_{m} < m. By the definition of submultiplicativity, we have

c_{n} = c_{mq_{m}+r_{m}} ⩽ c_{mq_{m}}c_{r_{m}} ⩽ (c_{m})^{q_{m}}c_{r_{m}}.

Now q_{m} ⩽ n/m, so q_{m}/n ⩽ 1/m. So we have

\sqrt[n]{c_{n}} ⩽ \sqrt[m]{c_{m}} \sqrt[n]{c_{r_{m}}} < L.\sqrt[n]{c_{r_{m}}}.

As n tends to infinity, the numbers \sqrt[n]{c_{0}}, …, \sqrt[n]{c_{m−1}} all tend to one, and so

lim sup_{n→∞} \sqrt[n]{c_{n}} < L.  □

Theorem 4.4. If M is a kG-module then γ_{G}(M) = lim_{n→∞} \sqrt[n]{c_{n}^{G}(M)} = inf_{n⩾1} \sqrt[n]{c_{n}^{G}(M)}.

Proof. This follows from Lemmas 4.2 and 4.3. □

Proposition 4.5. Suppose that a_{n} and b_{n} are submultiplicative sequences. Define a sequence c_{n} by

c_{n} = ∑_{i=0}^{n} \binom{n}{i}a_{i}b_{n−i}.

Then c_{n} is also a submultiplicative sequence, and we have

lim_{n→∞} \sqrt[n]{c_{n}} = lim_{n→∞} \sqrt[n]{a_{n}} + lim_{n→∞} \sqrt[n]{b_{n}}.

Proof. Using the fact that

\binom{m + n}{ℓ} = ∑_{i+j=ℓ} \binom{m}{i}\binom{n}{j}

and the submultiplicativity of the sequences a_{n} and b_{n}, we have

∑_{ℓ=0}^{m+n} \binom{m + n}{ℓ}a_{ℓ}b_{m+n−ℓ} ⩽ (∑_{i=0}^{m} \binom{m}{i}a_{i}b_{m−i}) . (∑_{j=0}^{n} \binom{n}{j}a_{j}b_{n−j})

and so the sequence c_{n} is submultiplicative.

By Lemma 3.1 we have

lim_{n→∞} \sqrt[n]{c_{n}} ⩽ lim_{n→∞} \sqrt[n]{a_{n}} + lim_{n→∞} \sqrt[n]{b_{n}}.

The reverse inequality is proved similarly. If lim_{n→∞} \sqrt[n]{a_{n}} = α and lim_{n→∞} \sqrt[n]{b_{n}} = β then given ε > 0 there exist positive constants A and B such that for all n ⩾ 0 we have a_{n} ⩾ A(α − ε)^{n} and b_{n} ⩾ B(β − ε)^{n}. So for all ε > 0 there is a positive constant C = AB such that for all n ⩾ 0 we have

c_{n} ⩾ ∑_{i=0}^{n} \binom{n}{i}A(α − ε)^{i}B(β − ε)^{n−i} = C(α + β − 2ε)^{n},

and so

lim_{n→∞} \sqrt[n]{c_{n}} ⩾ α + β − 2ε = lim_{n→∞} \sqrt[n]{a_{n}} + lim_{n→∞} \sqrt[n]{b_{n}} − 2ε.  □

Theorem 4.6. If M is a kG-module then we have γ_{G}(k ⊕ M) = 1 + γ_{G}(M).

Proof. We have

c_{n}^{G}(k ⊕ M) = ∑_{i=0}^{n} \binom{n}{i}c_{i}^{G}(M).

So we can apply Proposition 4.5 with a_{n} = c_{n}^{G}(M), b_{n} = 1, and c_{n} = c_{n}^{G}(k ⊕ M). □

Remark 4.7. Another proof of Theorem 4.6 can be given using Pringsheim’s Theorem 8.3. See Remark 9.10 for details.

Corollary 4.8. If M is a kG-module, and N is isomorphic to a direct sum of a copies of k and b copies of M then we have γ_{G}(N) = a + b.γ_{G}(M).

Proof. This follows inductively from Theorems 3.4 and 4.6. □

5. Tensor Products

Theorem 5.1. We have γ_{G}(M ⊗ N) ⩽ γ_{G}(M)γ_{G}(N). Proof. We have

core_{G}(M ⊗ N) = core_{G}(core_{G}(M) ⊗ core_{G}(N)).

Therefore

c_{n}^{G}(M ⊗ N) ⩽ c_{n}^{G}(M)c_{n}^{G}(N)

and

\sqrt[n]{c_{n}^{G}(M ⊗ N)} ⩽ \sqrt[n]{c_{n}^{G}(M)}\sqrt[n]{c_{n}^{G}(N)}.

Now apply lim sup_{n→∞} to both sides. □

This inequality may be strict. For example, it is possible for M ⊗ N to be projective with neither M nor N projective. However for tensor powers of a single module, we have the following. Theorem 5.2. We have γ_{G}(M^{⊗m}) = γ_{G}(M)^{m}. Proof. By Theorem 5.1 we have γ_{G}(M^{⊗m}) ⩽ γ_{G}(M)^{m}. Conversely, if n = ms + i with 0 ⩽ i < m then

M^{⊗n} = M^{⊗i} ⊗ (M^{⊗m})^{⊗s}

and so

c_{n}^{G}(M) ⩽ (dim M)^{m}c_{s}^{G}(M^{⊗m}).

Thus

\sqrt[n]{c_{n}^{G}(M)} ⩽ \sqrt[ms]{c_{n}^{G}(M)} ⩽ \sqrt[s]{dim M} \sqrt[m]{\sqrt[s]{c_{s}^{G}(M^{⊗m})}}.

Applying lim sup_{n→∞}, the factor \sqrt[s]{dim M} tends to 1. It follows that

γ_{G}(M) ⩽ \sqrt[m]{γ_{G}(M^{⊗m})}.  □

Conjecture 5.3. We have γ_{G}(M ⊗ M^{∗}) = γ_{G}(M)^{2}. Theorem 5.4. We have γ_{G}(ΩM) = γ_{G}(M). Proof. We have core_{G}(Ωk ⊗ M) ≅ core(ΩM) = ΩM. So by Lemma 2.9, Theorem 5.1 and Lemma 2.8 we have

γ_{G}(ΩM) = γ_{G}(Ωk ⊗ M) ⩽ γ_{G}(Ωk)γ_{G}(M) = γ_{G}(M).

The reverse inequality follows in the same way from the fact that

core_{G}(Ω^{−1}k ⊗ ΩM) ≅ core_{G}(M).  □

Example 5.5. Let M be the three dimensional module Soc^{2}(kG) for G = ℤ/3×ℤ/3 = ⟨g, h⟩ over 𝔽_{3}, given by the following matrices and diagram:

g ↦ (\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})  h ↦ (\begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})  [figure: •; •; •]

Then M is non-periodic and non-algebraic, and (Craven [11], Section 3.3.2) we have

M ⊗ M ≅ M^{∗} ⊕ Ω(M^{∗}).

Here, M^{∗} ≅ kG/Rad^{2}(kG) of dimension three, and Ω(M^{∗}) ≅ Soc^{3}(kG) of dimension six. Using this, and the fact that M′ = M ⊗ M^{∗} is a non-projective indecomposable module, it is easy to compute that core_{G}(M^{⊗n}) has 2^{n−2} non-projective summands if n is divisible by three, and 2^{n−1} non-projective summands otherwise. So using Theorem 3.5 we have c_{n}^{G}(M) ⩾ 2^{n−2} and γ_{G}(M) ⩾ 2. On the other hand, using Theorem 5.2, Theorem 3.2, Lemma 2.7 and Theorem 5.4 we have

γ_{G}(M)^{2} = γ_{G}(M ⊗ M) = γ_{G}(M^{∗} ⊕ Ω(M^{∗})) ⩽ γ_{G}(M^{∗}) + γ_{G}(Ω(M^{∗})) = 2γ_{G}(M)

and so γ_{G}(M) ⩽ 2. Combining these, we have γ_{G}(M) = 2. Similarly, we have γ_{G}(M′) = 4.

Corollary 5.6. Let 0 → M_{1} → M_{2} → M_{3} → 0 be a short exact sequence of kG-modules. Then each γ_{G}(M_{i}) is at most the sum of the other two.

Proof. This follows from Theorems 3.2 and 5.4, together with the observation that there are short exact sequences

0 → M_{2} → M_{3} ⊕ (proj) → Ω^{−1}(M_{1}) ⊕ (proj) → 0
0 → Ω(M_{3}) ⊕ (proj) → M_{1} ⊕ (proj) → M_{2} → 0.  □

Definition 5.7. Recall that a kG-module M is endotrivial if M ⊗ M^{∗} ≅ k ⊕ (proj).

Theorem 5.8. If M is neither projective nor endotrivial then γ_{G}(M) ⩾ \sqrt{2}.

Proof. Suppose that M is neither projective nor endotrivial. We divide into two cases according to whether the dimension of M is divisible by p.

If the dimension of M is divisible by p, then by Proposition 2.1, (M ⊗ M^{∗})^{⊗2} has a direct summand isomorphic to a direct sum of two copies of M ⊗ M^{∗}. Thus, γ_{G}(M ⊗ M^{∗})^{2} = γ_{G}((M ⊗M^{∗})^{⊗2}) ⩾ 2γ_{G}(M ⊗M^{∗}). As M is not projective, neither is M ⊗M^{∗} by Lemma 2.2 and we have γ_{G}(M ⊗ M^{∗}) ≠ 0, so γ_{G}(M ⊗ M^{∗}) ⩾ 2.

On the other hand, if the dimension of M is not divisible by p and M is not endotrivial then M ⊗ M^{∗} ≅ k ⊕ X with X non-projective. Then γ_{G}(X) ⩾ 1, so γ_{G}(M ⊗ M^{∗}) ⩾ 2, by Corollary 3.6.

In both cases, using Theorem 5.1 and Lemma 2.7, we have

2 ⩽ γ_{G}(M ⊗ M^{∗}) ⩽ γ_{G}(M)γ_{G}(M^{∗}) = γ_{G}(M)^{2}

and so γ_{G}(M) ⩾ \sqrt{2}. □

Remark 5.9. Example 3.3 shows that equality can occur in Theorem 5.8. See Section 12 for more about what happens when γ_{G}(M) ⩾ \sqrt{2}.

We can deduce a theorem of Carlson on finite dimensional idempotent kG-modules (see Theorem 3.5 of [9]) as a corollary.

Corollary 5.10. If M ⊗ M ≅ M ⊕ (proj) and M is not projective then M ≅ k ⊕ (proj).

Proof. The hypothesis implies that γ_{G}(M) = 1, so by Theorem 5.8, M is endotrivial. The endotrivial modules (modulo projective summands) form a group under tensor product, so the only idempotent element is the identity. □

6. Faithful Modules

Definition 6.1. We say that a kG-module M is p-faithful if it is not 0 and no element of order p in G acts trivially on M. So faithful implies p-faithful, and p-faithful is equivalent to being faithful on restriction to a Sylow p-subgroup.

Lemma 6.2. Let M be a kG-module. Then some tensor power M^{⊗n} with n ⩾ 1 has a non-zero projective summand if and only if M is p-faithful.

Proof. The lemma is clearly true if M = 0, so assume that M ≠ 0. If M is not p-faithful, then there is an element g ∈ G of order p acting trivially on M. It therefore acts trivially on M^{⊗n}, so this module has no projective summands.

On the other hand, if M is p-faithful then the kernel of the action on M is a p′-subgroup H ⩽ G. Projective kG/H-modules are projective kG-modules, so we may assume that H = 1. This case is handled in Bryant and Kovács [7]. □

Theorem 6.3. We have γ_{G}(M) < dim M if and only if M is p-faithful.

Proof. Again, we may assume that M ≠ 0. We use Lemma 6.2. If M is not p-faithful then for all n we have core_{G}(M^{⊗n}) = M^{⊗n} and so γ_{G}(M) = dim M. Conversely, if M is p-faithful, then some tensor power has a projective summand, say M^{⊗m} = P ⊕ N with P a non-zero projective module. Thus using Theorem 5.2 we have

γ_{G}(M)^{m} = γ_{G}(M^{⊗m}) ⩽ dim N < (dim M)^{m}

and so γ_{G}(M) < dim M. □

7. Restriction to Elementary Abelian Subgroups

Theorem 7.1. There exists a constant B, which depends only on p and G, such that if M is a kG-module then

dim core_{G}(M) ⩽ B · max_{E⩽G} dim core_{E}(M)

where the maximum is taken over the set of elementary abelian p-subgroups E of G.

Proof. See Theorem 3.7 of Carlson [8]. □

Theorem 7.2. Let M be a kG-module. Then γ_{G}(M) = max_{E⩽G} γ_{E}(M).

Proof. By Theorem 7.1 and Lemma 2.10 we have

max_{E⩽G} \sqrt[n]{c_{n}^{E}(M)} ⩽ \sqrt[n]{c_{n}^{G}(M)} ⩽ \sqrt[n]{B}. max_{E⩽G} \sqrt[n]{c_{n}^{E}(M)}.

Taking lim sup_{n→∞}, the factor of \sqrt[n]{B} tends to 1. □

Example 7.3. Let G be a generalised quaternion group and let k be a field of characteristic two. Then G has only one elementary abelian 2-subgroup E = ⟨z⟩, where z is the central element of order two. Let X = 1 + z, an element of kG satisfying X^{2} = 0. If M is a kG-module then the restriction to kE is a direct sum of dim(Ker(X, M)/Im(X, M)) copies of the trivial module plus a free module. It follows that

γ_{G}(M) = dim(Ker(X, M)/Im(X, M)).

In particular, this is an integer.

Proposition 7.4 (Dade [13, 14]). If E is an elementary abelian p-group, then the only endotrivial kE-modules are the syzygies Ω^{n}(k) (n ∈ ℤ) of the trivial module. □

Theorem 7.5. A kG-module M is endotrivial if and only if γ_{G}(M) = 1.

Proof. If M is projective then γ_{G}(M) = 0. If M is neither projective nor endotrivial then by Theorem 5.8 we have γ_{G}(M) ⩾ \sqrt{2}.

Conversely if M is endotrivial then its restriction to every elementary abelian p-subgroup of G is endotrivial. So by Theorem 7.2, we may assume that G = E is an elementary abelian p-group. By Proposition 7.4, M is a syzygy of the trivial module, so by Theorem 5.4 we have γ_{E}(M) = 1. □

Warning 7.6. If E is an elementary abelian group and M is a kE-module then γ_{E}(M) does depend on the Hopf algebra structure of kE. If we regard E as a restricted Lie algebra with trivial bracket, kE as its universal enveloping algebra and use the corresponding comultiplication, then γ_{E}(M) may change. For example, restrict the module of Example 3.3 to a Sylow 2-subgroup, which is elementary abelian of order four. Then by Theorem 7.2, we have γ_{E}(M) = \sqrt{2}. But if we use the Lie comultiplication then M ⊗ M ≅ M ⊕ M, and so γ_{E}(M) = 2.

8. Radius of Convergence

Another way of studying the invariant γ_{G}(M) is to consider power series; we begin with a well known lemma from analysis.

Lemma 8.1 (Cauchy, Hadamard). Let ϕ: ℤ_{⩾0} → ℂ. Then the radius of convergence r of the power series

f(t) = ∑_{n=0}^{∞} ϕ(n)t^{n}

is given by

1/r = lim sup_{n→∞} \sqrt[n]{|ϕ(n)|}.

Strictly inside the radius, the convergence is uniform and absolute.

Proof. See for example Conway [10], Theorem III.1.3. □

Corollary 8.2. Let M be a kG-module. Consider the power series

f_{M}(t) = ∑_{n=0}^{∞} c_{n}^{G}(M)t^{n},

and let r be the radius of convergence of f_{M}(t). Then

1/r = γ_{G}(M).

Next we have a slightly less well known theorem, and for the convenience of the reader we provide a proof. See also Statement (7.21) in Chapter VII of Titchmarsh [20].

Theorem 8.3 (Pringsheim). Suppose that ϕ: ℤ_{⩾0} → ℝ_{⩾0}, and that the power series

f(t) = ∑_{n=0}^{∞} ϕ(n)t^{n}

has radius of convergence r. Then t = r is a singular point of f(t).

Proof. Expand f(t) as a Taylor series about t = \frac{r}{2}:

f(t) = ∑_{n=0}^{∞} \frac{f^{(n)}(\frac{r}{2})}{n!}(t − \frac{r}{2})^{n}

where f^{(n)}(t)/n! = ∑_{m⩾n} ϕ(m)\binom{m}{n}t^{m−n}. Thus

f(t) = ∑_{n=0}^{∞}(t − \frac{r}{2})^{n} ∑_{m=n}^{∞} ϕ(m)\binom{m}{n}(\frac{r}{2})^{m−n}.

If t = r is not a singular point of f(t) then for ε small enough this converges at t = r+ε. The terms are all non-negative reals, so the sum is absolutely convergent, and we may rearrange the terms to get

f(r + ε) = ∑_{n=0}^{∞}(\frac{r}{2} + ε)^{n} ∑_{m=n}^{∞} ϕ(m)\binom{m}{n}(\frac{r}{2})^{m−n}

= ∑_{m=0}^{∞} ϕ(m) ∑_{n=0}^{m} \binom{m}{n}(\frac{r}{2} + ε)^{n}(\frac{r}{2})^{m−n}

= ∑_{m=0}^{∞} ϕ(m)(r + ε)^{m}.

The convergence of this sum implies that the radius of convergence of f is larger than r, contradicting the hypotheses of the theorem. □

9. Banach Algebras

We recall the basics of the theory of norms and spectral radius, referring to Chapters 17–18 of Lax [18] for proofs. We always work over the field of complex numbers.

Definition 9.1. A normed space is a vector space B over ℂ, together with a norm B → ℝ, x ↦ ∥x∥, satisfying

∥x + y∥ ⩽ ∥x∥ + ∥y∥, ∥cx∥ = |c|∥x∥,

for x, y ∈ B, c ∈ ℂ, such that ∥x∥ ⩾ 0 and ∥x∥ = 0 if and only if x = 0. A Banach space is a normed space that is complete with respect to the norm.

A (unital) normed algebra is an associative algebra A over ℂ with identity 𝟙 that is also a normed space, with the norm satisfying the additional conditions

∥𝟙∥ = 1, ∥xy∥ ⩽ ∥x∥∥y∥

for x, y ∈ A. A Banach algebra is a normed algebra that is also a Banach space. Note that we are assuming that all our Banach algebras are unital.

If a is an element of a Banach algebra A, we write σ(a) for the spectrum of a, namely the set of λ ∈ ℂ such that λ𝟙 − a is not invertible in A. It is a non-empty closed bounded subset of ℂ. The spectral radius of a ∈ A, denoted ρ(a), is defined to be sup_{λ∈σ(a)} |λ|.

Notice that if A is finite dimensional and a ∈ A, then the spectral radius ρ(a) is just the largest absolute value of an eigenvalue of the linear map induced by multiplying by a.

Let A(G) = ℂ ⊗_{ℤ} a(G), where a(G) is the Green ring or representation ring of finitely generated kG-modules. Following [5], we write elements of A(G) in the form ∑_{i} a_{i}[M_{i}] where a_{i} ∈ ℂ and M_{i} are indecomposable kG-modules. If M = ⨁_{i} n_{i}M_{i} is a kG-module, we write [M] for ∑_{i} n_{i}[M_{i}] ∈ A(G). Multiplication is extended bilinearly from [M][N] = [M ⊗ N].

We write A(G, 1) for the ideal of A(G) spanned by the elements [P] with P projective, and A_{0}(G, 1) for the linear span of the elements of the form [M_{2}] − [M_{1}] − [M_{3}] where 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence. Thus we have

A(G) = A(G, 1) ⊕ A_{0}(G, 1).

We put a norm on A(G)/A(G, 1) ≅ A_{0}(G, 1) by setting

∥∑_{i} a_{i}[M_{i}]∥ = ∑_{i} |a_{i}| dim core_{G}(M_{i}) = ∑_{M_{i} non-projective} |a_{i}| dim M_{i}.

The reason for choosing this particular norm is that it has two good properties:

(i) If M is a kG-module then ∥[M]∥ = dim core_{G}(M); thus ∥ · ∥ is additive on direct sums of genuine (as opposed to virtual) modules.

(ii) If H is a subgroup of G then restriction can only reduce the norm: ∥M↓_{H}^{G}∥ ⩽ ∥M∥. This makes A(G)/A(G, 1) into a normed algebra, which we may complete with respect to the norm to obtain a commutative Banach algebra which we shall denote Â_{1}(G). Thus A(G)/A(G, 1) is a dense subalgebra of Â_{1}(G).

Warning 9.2. If p does not divide |G| then Â_{1}(G) = 0, which is not a Banach algebra because it does not satisfy the condition ∥𝟙∥ = 1. In this paper we always implicitly assume that the characteristic of the field divides the order of the group.

The role of the invariant γ_{G}(M) in this context is that by Theorem 4.4 we have

(9.3)  γ_{G}(M) = lim_{n→∞} \sqrt[n]{∥[M]^{n}∥}.

Proposition 9.4 (Spectral radius formula, Gelfand [16]). If A is a Banach algebra and a ∈ A then the spectral radius of a is related to the norm by the formula

(9.5)  ρ(a) = lim_{n→∞} \sqrt[n]{∥a^{n}∥}.

Proof. See for example §17.1 of Lax [18]. □

Remark 9.6. In fact, slightly more is true. By Lemma 8.1, the right hand side of (9.5) is 1/r, where r is the radius of convergence of f_{a}(t) = ∑_{n=0}^{∞} ∥a^{n}∥t^{n}. On the boundary of the circle of convergence, if |λ| = 1/r then λ is a singularity of f_{a}(t) if and only if λ ∈ σ(a).

Theorem 9.7. If M is a kG-module then γ_{G}(M) = ρ([M]), where [M] is the corresponding element of Â_{1}(G).

Proof. This follows from (9.3) and Proposition 9.4. □

Lemma 9.8. Let a be an element with ρ(a) = r, in a Banach algebra A. Then r (as a real number) is an element of σ(a) ⊆ ℂ if and only if ρ(𝟙 + a) = 1 + r.

Proof. It is clear that σ(𝟙 + a) is the set of 1 + λ with λ ∈ σ(a). So σ(𝟙 + a) is contained in a disc of radius r centred at 1 ∈ ℂ. The only point in this disc at distance 1+r from the origin is the real number 1 + r. Now using the fact that σ(a) is closed, we see that the spectral radius of 𝟙 + a is 1 + r if and only if 1 + r ∈ σ(𝟙 + a), namely if and only if r ∈ σ(a). □

Theorem 9.9. Let M be a kG-module. Then the real number γ_{G}(M) is an element of σ([M]) ⊆ ℂ.

We give two proofs of Theorem 9.9.

First proof. By Theorem 4.6, we have γ_{G}(k ⊕ M) = 1 + γ_{G}(M). By Theorem 9.7, it follows that on Â_{1}(G) we have ρ(𝟙 + [M]) = ρ([M]). By Lemma 9.8 this implies that γ_{G}(M) ∈ σ([M]). □

Second proof. The power series f_{M}(t) = ∑_{n=0}^{∞} c_{n}^{G}(M)t^{n} has non-negative integer coefficients.

So by Pringsheim’s Theorem 8.3, γ_{G}(M) is a singular point on boundary of the circle of convergence of f_{M}(t). By Remark 9.6, this implies that γ_{G}(M) ∈ σ([M]). □

Remark 9.10. This second proof can be reversed to show that Pringsheim’s theorem gives another proof of Theorem 4.6.

The way to connect spectral radius with the species of the Green ring in the sense of Benson and Parker [5] is the following.

Theorem 9.11. An element a of a commutative Banach algebra A is invertible if and only if ϕ(a) ≠ 0 for all algebra homomorphisms ϕ: A → ℂ.

Proof. See Theorem 3 in Chapter 18 of Lax [18]. □

Remark 9.12. Note that if A is a commutative Banach algebra and ϕ: A → ℂ is an algebra homomorphism then for all a ∈ A we have |ϕ(a)| ⩽ ∥a∥. It follows that ϕ is automatically continuous with respect to the norm. See Theorem 1 in Chapter 18 of Lax [18].

Corollary 9.13. If a is an element of a commutative Banach algebra then σ(a) is the set of values of ϕ(a) as ϕ runs over the algebra homomorphisms A → ℂ. The spectral radius ρ(a) is equal to sup_{ϕ: A→ℂ} |ϕ(a)|.

Proof. It follows from Theorem 9.11 that λ𝟙 − a is not invertible if and only if there exists an algebra homomorphism ϕ: A → ℂ such that ϕ(a) = λ. □

Definition 9.14. Recall from [5] that a species of a(G) is a ring homomorphism s: a(G) → ℂ. A species of a(G) extends to give an algebra homomorphism s: A(G) → ℂ, and all algebra homomorphisms have this form.

We say that a species s of a(G) is core-bounded if for all kG-modules M we have

|s([M])| ⩽ dim core_{G}(M).

In particular, the extension of a core-bounded species to A(G) vanishes on A(G, 1), and so defines an algebra homomorphism A(G)/A(G, 1) → ℂ.

Lemma 9.15. If s is any core-bounded species then |s([Ω^{i}(k)])| = 1 for any i ∈ ℤ.

Proof. We have |s([Ωk])|^{i} = |s([Ωk]^{i})| = |s([Ω^{i}k])| ⩽ dim core_{G}(Ω^{i}k). But dim core_{G}(Ω^{i}k) grows polynomially in i (see for example [3] §5.3), so |s([Ωk])| ⩽ 1. The same holds for |s([Ω^{−1}k])|; but |s([Ωk])|·|s([Ω^{−1}k])| = |s([k])| = 1, so we must have |s([Ωk])| = |s([Ω^{−1}k])| = 1. The general case follows from the first formula in this proof. □

Examining the species for ℤ/2 × ℤ/2 described in Appendix 1 of [5], we see that not every species that vanishes on A(G, 1) is core-bounded. In this example, of the species parametrised by non-zero z ∈ ℂ, only the ones with z on the unit circle are core-bounded.

The following proposition shows that there is a natural correspondence between corebounded species of a(G) and algebra homomorphisms Â_{1}(G) → ℂ.

Proposition 9.16. For a species s: A(G) → ℂ, the following are equivalent:

(i) s is core-bounded.

(ii) For all x ∈ A(G) we have |s(x)| ⩽ ∥x∥.

(iii) s is continuous with respect to the norm.

(iv) s extends to an algebra homomorphism Â_{1}(G) → ℂ.

Proof. The implications (ii) ⇒ (iii) ⇒ (iv) are clear, and the implication (iv) ⇒ (i) follows from Remark 9.12. So it remains to prove that (i) ⇒ (ii). Suppose that s is core-bounded, and write x = ∑_{i} a_{i}[M_{i}] where the M_{i} are indecomposable. Then s(x) = ∑_{i} a_{i}s([M_{i}]) and so

|s(x)| ⩽ ∑_{i} |a_{i}| · |s([M_{i}])| ⩽ ∑_{i} |a_{i}| dim core(M_{i}) = ∥x∥.  □

Theorem 9.17. If M is a kG-module then

γ_{G}(M) = sup_{s: a(G)→ℂ} |s([M])|

where the supremum runs over the core-bounded species of a(G). Furthermore, there exists a core-bounded species s of a(G) such that s([M]) = γ_{G}(M).

Proof. This equality follows from Theorem 9.7, Corollary 9.13 and Proposition 9.16. The final statement follows from Theorem 9.9. □

Corollary 9.18. For any kG-module M, the restriction of γ_{G} to the sub-semiring of A(G) consisting of elements of the form f([M]) = ∑_{i=0}^{n} a_{i}[M]^{i} with the a_{i} real and non-negative is additive and multiplicative.

Proof. By Theorem 9.17, we may choose a species s that maximises |s([M])|, and such that s([M]) = γ_{G}(M). Such an s also maximises |s(f([M]))|. □

Question 9.19. What can be said about the quasi-nilpotent elements of Â_{1}(G), namely the elements a satisfying lim_{n→∞} \sqrt[n]{∥a^{n}∥} = 0? These are the elements on which all core-bounded species vanish.

Remark 9.20. Many of the properties of γ_{G} that we have described correspond to wellknown properties of the spectral radius in a Banach algebra, although we have chosen an exposition that is self-contained except for Theorem 9.4. This applies to Theorem 1.2 (i), (ii), (iv), (viii), the second inequality of (x), (xi), (xiii) and (xiv). Others require a Banach lattice: these can record the special role played by the linear combinations of modules with real non-negative coefficients, which roughly approximate the image of genuine modules as opposed to virtual ones; see, for example, the book by Schaefer [19]. The first inequality of Theorem 1.2 (x) and also (xii) correspond to facts about Banach lattices, as does Theorem 9.9 ([19, Prop. V 4.1]).

10. Cyclic groups

The computations in this section are based on Green [17]. For the purpose of this section only, let G = ℤ/p be the cyclic group of order p and let M_{j} be the indecomposable kG-module of dimension j for 1 ⩽ j ⩽ p. Then we have

M_{2} ⊗ M_{j} ≅ \begin{cases}M_{2} & j = 1 \\ M_{j+1} ⊕ M_{j−1} & 2 ⩽ j ⩽ p − 1 \\ M_{p} ⊕ M_{p} & j = p. \end{cases}

Let U_{j}(x) be the Chebyshev polynomial of the second kind, defined by the recurrence relation U_{0}(x) = 1, U_{1}(x) = 2x, U_{j}(x) = 2xU_{j−1}(x) − U_{j−2}(x) (j ⩾ 2). These polynomials are designed so that

U_{j}(cos θ) = \frac{sin(j + 1)θ}{sin θ}.

The roots of U_{j}(x) are real and distinct, symmetric around x = 0, and given by

x = cos(kπ/(j + 1)), 1 ⩽ k ⩽ j.

We define f_{j}(x) = U_{j−1}(x/2). So f_{1}(x) = 1, f_{2}(x) = x, and xf_{j}(x) = f_{j+1}(x) + f_{j−1}(x) (j ⩾ 2). Then we have

f_{j} (\frac{sin 2θ}{sin θ}) = \frac{sin jθ}{sin θ}.

Note that f_{p}(x) is an irreducible polynomial in x^{2}. For example, we have f_{3}(x) = x^{2} − 1 and f_{5}(x) = x^{4} − 3x^{2} + 1. The roots of f_{p}(x) are given by x = 2 cos(kπ/p) (1 ⩽ k ⩽ p − 1)

In the ring a(G)/a(G, 1), we have [M_{2}][M_{j}] = [M_{j+1}] + [M_{j−1}] and [M_{p}] = 0. It follows that f_{j}([M_{2}]) = [M_{j}] and f_{p}([M_{2}]) = 0. We therefore have

a(G)/a(G, 1) ≅ ℤ[X]/(f_{p}(X))

where X corresponds to [M_{2}]. The core-bounded species of a(G) are just the non-Brauer species of a(G), and are given by

s_{k}: [M_{2}] ↦ 2 cos(kπ/p), (1 ⩽ k ⩽ p − 1).

Theorem 10.1. In the case of the indecomposable module M_{j} (1 ⩽ j ⩽ p − 1) the cyclic group G of order p in characteristic p, we have

γ_{G}(M_{j}) = \frac{sin(jπ/p)}{sin(π/p)} = f_{j}(γ_{G}(M_{2}))

where

γ_{G}(M_{2}) = \frac{sin(2π/p)}{sin(π/p)} = 2 cos(π/p).

In fact, γ_{G} is additive and multiplicative on modules, so this determines γ_{G} on any module.

Proof. It follows from Theorem 9.17 that for fixed j we need to maximise |s_{k}([M_{j}])|. By the discussion above,

s_{k}([M_{j}]) = s_{k}(f_{j}([M_{2}]) = f_{j}(s_{k}([M_{2}])) = f_{j}(2 cos(kπ/p)) = U_{j−1}(cos kπ/p) = \frac{sin(jkπ/p)}{sin(kπ/p)}.

Express the sines in terms of e^{ikπ/p}, expand as a geometric series and pair conjugate terms. If j is odd, say j = 2r + 1, the result is 1 + ∑_{s=1}^{r} cos(skπ/p); the case when j is even is similar and is left to the reader.

Thus we want to maximise the sum of the elements of a certain class of r-element subsets of {cos(tπ/p) : 1 ⩽ t ⩽ p − 1}. Clearly, the maximum over all r-element subsets is obtained by choosing the r largest elements, i.e., t = 1, …, r. In our case, this is achieved by choosing k = 1.

Note that k = 1 yields the maximum for all the M_{j}. Since each s_{1}([M_{j}]) is a positive number, s_{1} also yields the maximum on all sums of the M_{j}, i.e., on all modules. We have γ_{G}(M) = s_{1}([M]) for all modules and the last part of the theorem follows. □

11. Methods of Calculation

We recall some basic facts from Banach theory that we will use.

A linear operator (i.e., linear map) T from a Banach space B to itself is said to be bounded if ∥T ∥_{op} := sup{∥T x∥ : ∥x∥ = 1} is finite. The space of all such bounded operators forms a Banach space B(B) with norm ∥ · ∥_{op}. If B is finite dimensional then σ(T) is just the finite set of eigenvalues, so the set of roots of the characteristic polynomial of T, and ρ(T) is the largest of the absolute values of these.

If we start with a Banach algebra A, then any a ∈ A yields a bounded linear operator T_{a} ∈ B(A) by T_{a}x = ax for x ∈ A. It is easy to check that ∥T_{a}∥_{op} = ∥a∥; we will usually omit the subscript op and often identify T_{a} with a. The spectra might differ, but it follows from Proposition 9.4 that the spectral radii agree: ρ(T_{a}) = ρ(a).

We saw in Lemma 9.15 that for a core-bounded species s we have s([Ωk]) = λ for some complex number λ with |λ| = 1. It follows that s vanishes on the ideal of Â_{1}(G) generated by [Ωk] − λ[k] and so factors through the quotient Banach algebra by the closure of this ideal, which we denote by Â_{1}(G)/(Ω − λ). Thus we can find γ_{G}(M) = ρ(T_{M}) by calculating it on each of these quotients and taking the maximum value.

Because of the form of the definition of γ_{G}(M), we can calculate it on any subalgebra of Â_{1}(G) that contains all the tensor powers of M and similarly for Â_{1}(G)/(Ω−λ). If we consider the operator T_{M}, we can even restrict to any Banach subspace that contains some tensor power of M^{⊗n} and that is closed under tensor product with M^{⊗n}, by Theorem 1.2 (xiv).

Our strategy will be to use these observations to reduce to the finite dimensional case. Of course, an operator A on a finite dimensional vector space with given basis can be rep-

resented by a matrix (A_{i,j}). There are various possible norms that we can use on matrices. One is the operator norm induced from a norm on the vector space. Another, which we will use later, is ∥A∥_{max} = max_{i,j}{|A_{i,j}|}; this is a Banach space norm, but it is not submultiplicative. However, any two vector space norms on a finite dimensional vector space, in this case the vector space of n × n matrices, are commensurate, i.e., there is a positive real number c such that c^{−1}∥x∥_{1} ⩽ ∥x∥_{2} ⩽ ∥x∥_{1} for all x. It follows that lim_{n→∞} \sqrt[n]{∥x∥} is independent of the norm, so yields ρ(x), regardless of whether the norm is submultiplicative or not.

Note that if the entries A_{i,j} in the matrix for A are integers then ρ(A) must be an algebraic integer.

The following standard lemma will be useful later, when we look more closely at the quotient Â_{1}(G)/(Ω − λ). For a matrix B, we write |B| for the matrix of absolute values of the entries of B. If A_{1} and A_{2} are matrices with real entries, we write A_{1} ⩽ A_{2} to indicate that each entry of A_{1} is less than or equal to the corresponding entry of A_{2}.

Lemma 11.1. Let A be a matrix with non-negative real entries, and let B be a complex matrix satisfying |B| ⩽ A. Then ρ(B) ⩽ ρ(A).

Proof. We have in general |XY | ⩽ |X| · |Y |, hence |B^{n}| ⩽ A^{n}. Thus ∥B^{n}∥_{max} ⩽ ∥A^{n}∥_{max}. Taking nth roots and then the limit as n tends to infinity yields the result. □

Finally, we formulate a result that depends heavily on the fact that our norm is additive on modules; it could be generalised to a Banach lattice with a norm that is additive on the positive cone.

Proposition 11.2. Suppose that we have a bounded operator T on Â_{1}(G) that takes modules to modules and for some m ∈ ℕ we have modules S_{1}, …, S_{m} and Y_{1}, …, Y_{m} with none of the S_{i} projective. Suppose that there are non-negative integers A_{i,j} such that

T [S_{i}] = ∑_{j} A_{i,j}[S_{j}] + [Y_{i}], i = 1, …, m

and consider the matrix A = (A_{i,j}). Then ρ(T) ⩾ ρ(A).

Proof. By induction, for any n ⩾ 1 there are modules Z_{i,n} such that

T^{n}[S_{i}] = ∑_{j}(A^{n})_{i,j}[S_{j}] + [Z_{i,n}], i = 1, …, m.

Note that this does not require the [S_{i}] to be linearly independent. Because the norm is additive on sums of modules with non-negative coefficients, we obtain

∥T^{n}∥_{op}∥[S_{i}]∥ ⩾ ∥T^{n}[S_{i}]∥ = ∥ ∑_{j}(A^{n})_{i,j}[S_{j}] + [Z_{i,n}]∥ ⩾ max_{j}{(A^{n})_{i,j}}∥[S_{j}]∥ ⩾ max_{j}{(A^{n})_{i,j}},

since S_{j} is not projective, so ∥[S_{j}]∥ ⩾ 1. It follows that for some i we have

∥T^{n}∥_{op}∥[S_{i}]∥ ⩾ max_{i,j}{(A^{n})_{i,j}} = ∥A^{n}∥_{max}.

Taking nth roots and then the limit as n tends to infinity yields ρ(T) ⩾ ρ(A). □

12. Modules M with γ_{G}(M) = \sqrt{2}

In Theorem 5.8, we showed that if M is neither projective nor endotrivial then we have γ_{G}(M) ⩾ \sqrt{2}. In this section, we investigate the case of equality.

Lemma 12.1. For any kG-module we have:

(i) If M = E ⊕ N with E endotrivial and γ_{G}(M ⊗ M^{∗}) < 4 then N is projective.

(ii) If M ⊗ M^{∗} is endotrivial then so is M.

Proof. We have M ⊗ M^{∗} ≅ k ⊕ (E ⊗ N^{∗}) ⊕ (E^{∗} ⊗ N) ⊕ (N ⊗ N^{∗}). Theorem 3.5 shows that γ_{G}(M ⊗ M^{∗}) ⩾ 4 unless the tensor product of the modules in the sum is projective. Lemmas 2.2 and 2.3 show that if the latter occurs then N is projective.

For the second part, notice that if M⊗M^{∗} is endotrivial then p cannot divide the dimension of M. Thus k is a summand of M ⊗M^{∗}, and since M ⊗M^{∗} is endotrivial, the complementary summand is projective. □

Proposition 12.2. If 1 < γ_{G}(M ⊗ M^{∗}) < 1 + \sqrt{2}, then p divides the dimension of M.

Proof. Suppose that the dimension of M is not divisible by p. By Theorem 7.2, without loss of generality we may assume that G is elementary abelian. By Lemma 2.11, we may also assume that k is algebraically closed.

We first show that we may suppose that M is indecomposable. Otherwise, choose an indecomposable summand M_{1} of M with dimension not divisible by p. Then γ_{G}(M_{1} ⊗M_{1}^{∗}) < 1 + \sqrt{2}, and we claim that 1 < γ_{G}(M_{1} ⊗ M_{1}^{∗}) so that we may replace M by M_{1}. If not, then M_{1} ⊗ M_{1}^{∗} is endotrivial by Theorem 7.5, so M_{1} is endotrivial by Lemma 12.1 (ii). Now apply Lemma 12.1 (i) to M to see that M = M_{1} ⊕ (proj), so γ_{G}(M_{1} ⊗ M_{1}^{∗}) = γ_{G}(M ⊗ M^{∗}) > 1, a contradiction. Thus we may assume that M is indecomposable.

We have M ⊗ M^{∗} ≅ k ⊕ X with X non-projective. So using Theorem 4.6 we have

1 + \sqrt{2} > γ_{G}(M ⊗ M^{∗}) = γ_{G}(k ⊕ X) = 1 + γ_{G}(X).

By Theorem 5.8, the only possibility is γ_{G}(X) = 1, and by Theorem 7.5, X is endotrivial. By Proposition 7.4 we have X ≅ Ω^{r}k ⊕ (proj). We know that X is self dual, hence if G is not cyclic we have r = 0 and so M ⊗M^{∗} ≅ k⊕k⊕(proj). This contradicts Theorem 2.1 of Benson and Carlson [4] (this is where we need to use the statements that M is indecomposable of dimension not divisible by p, and k is algebraically closed). If G is cyclic then Ω has period two and the only other possibility is r = 1. Thus M ⊗M^{∗} ≅ k⊕Ωk⊕(proj). This contradicts our assumption that p does not divide the dimension of M, so the lemma is proved. □

The next proposition involves a number α ≈ 2.839286755 …, which is the real root of the polynomial x^{3} − 4x^{2} + 4x − 2.

Proposition 12.3. If M has dimension divisible by p and M ⊗ M^{∗} ⊗ M ≅ M ⊕ M ⊕ X with X not projective, then γ_{G}(M ⊗ M^{∗}) ⩾ α.

Proof. The dimension of X is divisible by p, so by Proposition 2.1 (ii), X ⊗ X^{∗} ⊗ X ≅ X ⊕ X ⊕ Y, but we have no information on whether Y is projective. We have

(M ⊗ M^{∗}) ⊗ (X ⊗ M^{∗}) ≅ 2(X ⊗ M^{∗}) ⊕ (X ⊗ X^{∗})
(M ⊗ M^{∗}) ⊗ (X ⊗ X^{∗}) ≅ (M ⊗ M^{∗} ⊗ X ⊗ X^{∗})
(M ⊗ M^{∗}) ⊗ (M ⊗ M^{∗} ⊗ X ⊗ X^{∗}) ≅ 2(X ⊗ M^{∗}) ⊕ 2(M ⊗ M^{∗} ⊗ X ⊗ X^{∗}) ⊕ (Y ⊗ M^{∗}).

Set T = M ⊗ M^{∗}, S_{1} = X ⊗ M^{∗}, S_{2} = X ⊗ X^{∗} and S_{3} = M ⊗ M^{∗} ⊗ X ⊗ X^{∗}. None of the S_{i} are projective. For S_{2} this is just Lemma 2.2. Since S_{2} is isomorphic to a summand of S_{1} ⊗ M ⊗ M^{∗}, it follows that S_{1} is not projective. Finally, S_{3} ≅ S_{1} ⊗ S_{1}^{∗}, so S_{3} is not projective by Lemma 2.2 again.

We regard tensoring with T as an operator and, ignoring Y ⊗ M^{∗} for the moment, the information above can be recorded by the matrix

A = (\begin{matrix}2 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & 0 & 2 \end{matrix}) .

The characteristic polynomial of this matrix is the polynomial used to define α, so ρ(A) = α. Applying Proposition 11.2, we obtain

γ_{G}(M ⊗ M^{∗}) = ρ(T) ⩾ ρ(A) = α.  □

Remark 12.4. The appeal to Proposition 11.2 at the end of the proof of Proposition 12.3 can be expressed more naïvely as follows. We have T ⊗ T ≅ 2T ⊕ S_{1}, T ⊗ S_{1} ≅ 2S_{1} ⊕ S_{2}, T ⊗ S_{2} ≅ S_{3}, T ⊗ S_{3} ≅ 2S_{1} ⊕ 2S_{2} ⊕ (Y ⊗ M^{∗}). So ignoring some summands, we see that T^{⊗(n+2)} has

( 1 0 0 ) A^{n} ( \begin{matrix}S_{1} \\ S_{2} \\ S_{3} \end{matrix} )

as a direct summand. It follows that the number of non-projective direct summands of T^{⊗(n+2)} is at least the sum of the entries in ( 1 0 0 ) A^{n}. By Frobenius–Perron theory, this number is bounded below by a constant multiple of α^{n}, since α is the largest real root of this matrix. So we have γ_{G}(T) ⩾ α.

Theorem 12.5. For any non-projective kG-module M, the following conditions are equivalent.

(i) 1 < γ_{G}(M ⊗ M^{∗}) < 1 + \sqrt{2}.

(ii) γ_{G}(M ⊗ M^{∗}) = 2.

(iii) M ⊗ M^{∗} ⊗ M ≅ M ⊕ M ⊕ (proj).

(iv) [M ⊗ M^{∗}]^{2} ≅ 2[M ⊗ M^{∗}] in the Green ring modulo projectives.

Proof. We will show that (i) implies (iii). The other implications are easy.

If the dimension of M is divisible by p then by Proposition 2.1 we have

M ⊗ M^{∗} ⊗ M ≅ M ⊕ M ⊕ X.

If X is not projective then by Proposition 12.3 we have γ_{G}(M ⊗ M^{∗}) ⩾ α > 1 + \sqrt{2}, contradicting (i).

On the other hand, if the dimension of M is not divisible by p then Proposition 12.2 shows that γ_{G}(M ⊗ M^{∗}) cannot lie between 1 and 1 + \sqrt{2}, also contradicting (i). □

Corollary 12.6. If a kG-module M satisfies 1 < γ_{G}(M) < \sqrt{1 + \sqrt{2}} ≈ 1.553773974 … then it satisfies the conditions of Theorem 12.5.

Proof. The inequalities in the corollary imply those in condition (i) of the theorem by Theorem 5.1 and Lemma 12.1 (ii). □

Corollary 12.7. If γ_{G}(M) < \sqrt{1 + \sqrt{2}} then the only possibilities for γ_{G}(M) are 0, 1 and \sqrt{2}. □

Remarks 12.8. (i) If Conjecture 5.3 holds, then the converse of Corollary 12.6 also

holds, provided M is not projective. (ii) If M satisfies M ⊗ M^{∗} ⊗ M ≅ M ⊕ M ⊕ (proj) then the restriction of M to any cyclic subgroup of G is projective. This can be seen by noting that on restriction the module becomes self-dual, and the relation [M]^{3} = 2[M] in the Green ring modulo projectives yields that γ_{G}(M) is equal to 0 or \sqrt{2}; the latter can be seen to be impossible from the calculations in Section 10. It is not clear whether there are any such modules except in the case that p = 2 and a Sylow 2-subgroup of G is isomorphic to ℤ/2 × ℤ/2.

We end this section with a related conjecture.

Conjecture 12.9. If M is a kG-module with γ_{G}(M) < 2 then γ_{G}(M) = 2 cos(π/m) for some integer m ⩾ 2.

Remark 12.10. The case m = 2 gives γ_{G}(M) = 0, m = 3 gives γ_{G}(M) = 1, and the case m ⩾ 4 gives γ_{G}(M) ⩾ \sqrt{2}. This fits well with Theorems 5.8, 7.5 and 12.6. The two dimensional representation of ℤ/p in characteristic p is an example with γ_{G}(M) = 2 cos(π/p), see Theorem 10.1. It follows from the computations in Alperin [1] and Section 3 of Craven [12] that if M is the two dimensional natural module for SL(2, q) with q a prime power then M is algebraic and γ_{G}(M) = 2 cos(π/q).

Question 12.11. What are the general properties of modules M with γ_{G}(M) < 2?

13. Eventually Recursive Functions

Definition 13.1. We say that a function ϕ: ℤ_{⩾0} → ℤ_{⩾0} is eventually recursive of degree d if there exists a homogeneous linear recurrence relation with constant coefficients, in other words a recurrence relation of the form

ϕ(n) + a_{1}ϕ(n − 1) + ⋯ + a_{d}ϕ(n − d) = 0

with a_{i} ∈ ℤ, which is satisfied for all large enough integers n. The recurrence relation of minimal degree eventually satisfied by ϕ(n) is uniquely determined, and the corresponding polynomial

z^{d} + a_{1}z^{d−1} + ⋯ + a_{d}

has a_{d} ≠ 0 and is called the characteristic polynomial of the recurrence relation.

The following is a standard theorem from the theory of recurrence relations.

Theorem 13.2. If ϕ is eventually recursive and r is the radius of convergence of the generating function f(t) = ∑_{n=0}^{∞} ϕ(n)t^{n} then 1/r is an algebraic integer. It is the largest positive real root of the characteristic polynomial.

Proof. It is easy to check that (1 + a_{1}t + ⋯ + a_{d}t^{d})f(t) is a polynomial in t, and therefore f(t) is a rational function with denominator equal to 1 + a_{1}t + ⋯ + a_{d}t^{d}. The poles of this rational function are the 1/λ_{i} where the roots of the characteristic polynomial are λ_{i}. The radius of convergence r is therefore the smallest of the 1/|λ_{i}|. Now apply Pringsheim’s Theorem 8.3 to f(t) to see that r is a root of the characteristic polynomial. □

Conjecture 13.3. If M is a kG-module then the function ϕ(n) = c_{n}^{G}(M) is eventually recursive.

Remarks 13.4. (i) If M is an algebraic module then Conjecture 13.3 holds for M.

(ii) If Conjecture 13.3 holds for a module M, then by Theorem 13.2, γ_{G}(M) is an

algebraic integer. (iii) In Example 2.12, the minimal equation for M modulo projectives is x^{2}(x − 2)^{3} = 0.

The largest solution to this has |x| = 2, and so 1/r = γ_{G}(M) = 2. The associated recurrence relation is c_{n}^{G}(M) − 8c_{n−3}^{G}(M) = 0 for n ⩾ 5.

(iv) In Example 5.5, the module M is not algebraic, but nonetheless, the relation

M ⊗ M ≅ M^{∗} ⊕ Ω(M^{∗})

can be used to produce a recurrence relation for c_{n}^{G}(M) for large enough n.

The consequence of Conjecture 13.3 that γ_{G}(M) is an algebraic integer is at least consistent with the last result of this section.

Proposition 13.5. The invariant γ_{G}(M) can take only countably many values (for all primes, fields, groups and modules).

Proof. We know that γ_{G} does not vary with field extension, by Lemma 2.11, and any representation over any field can be realised over a finitely generated field – the subfield generated over 𝔽_{p} by the entries of the matrices in some matrix form of the representation – so we only need to consider finitely generated fields. A finitely generated field is countable and there are only countably many such fields. To see this, add the generators one by one. If a generator is algebraic over the field produced at the previous stage then there are only countably many possibilities for its minimal polynomial and hence for the new field. If it is transcendental, the new field is already determined. Thus, for a given group, field and dimension, there are only countably many representations (consider matrices again) and so only countably many possible values of γ_{G}. There are also only countably many possibilities for group, field and dimension. □

This proof also works for any other invariant on kG-modules that is preserved under field extension.

14. Omega-Algebraic Modules

Definition 14.1. A kG-module M is called Omega-algebraic if the non-projective indecomposable direct summands of the modules M^{⊗n} fall into finitely many orbits of the syzygy functor Ω.

An example of an Omega-algebraic module may be found in Remark 13.4 (iv). We’ll see more examples in the next section, as well as evidence that not all modules are Omegaalgebraic.

Conjecture 14.2. A weaker form of Conjecture 13.3 is that it holds for Omega-algebraic modules.

We can calculate γ_{G}(M) for an Omega-algebraic module M as follows, using the methods of Section 11. First we restrict to the subspace of Â_{1}(G) generated by all the indecomposable summands of all the tensor powers of M, together with all their syzygies. Choose representatives of the Ω-orbits of these indecomposable summands, say M_{1}, …, M_{d}; by hypothesis there are only finitely many. Each M ⊗ M_{i} decomposes as a direct sum of modules of the form Ω^{m}(M_{j}). This gives us a d × d matrix X(Ω) for the effect of tensoring with M, whose entries are Laurent polynomials in the operator Ω which have non-negative coefficients.

Now consider the quotients Â_{1}(G)/(Ω−λ) for |λ| = 1. These are all finite dimensional with basis the images of the [M_{i}]; the matrix corresponding to tensoring with M is X(λ), meaning that we substitute λ for Ω. Since the Laurent polynomials in Ω have non-negative coefficients we have |X(λ)| ⩽ X(1), so we can apply Lemma 11.1 to see that ρ(X(λ)) ⩽ ρ(X(1)). It follows that γ_{G}(M) is just the largest eigenvector of the matrix X(1). In particular, it is an algebraic integer.

15. Some examples

Example 5.5 is an example of an Omega-algebraic module which is not algebraic. Here are some more complicated examples. The computations use the methods outlined in the previous section and in Section 11.

Example 15.1. Let G = ⟨g, h⟩ ≅ ℤ/3 × ℤ/3 and k = 𝔽_{3}. Let M_{6} be the six-dimensional module given by the following matrices and diagram:

g ↦ (\begin{matrix}1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})  h ↦ (\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})  [figure: •; •; •; •; •; •]

Then M_{6} is Omega-algebraic. Letting M_{3} = k_{⟨g⟩}↑^{G} and P be the projective indecomposable module, we have

M_{6} ⊗ M_{6} ≅ Ω(M_{6}) ⊕ Ω^{−1}(M_{6}^{∗}) ⊕ M_{3} ⊕ P

M_{6} ⊗ M_{6}^{∗} ≅ Ω^{−1}(M_{6}) ⊕ Ω(M_{6}^{∗}) ⊕ M_{3} ⊕ P
M_{6} ⊗ M_{3} ≅ Ω(M_{3}) ⊕ Ω(M_{3}) ⊕ Ω(M_{3}).

The rows of the table in Figure 15.1 give the effect of tensoring with M_{6}.

M_{6} M_{6}^{∗} M_{3} P
M_{6} Ω Ω^{−1} 1 1
M_{6}^{∗} Ω^{−1} Ω 1 1
M_{3} 0  0 3Ω 0
P  0  0 0 6

Figure 15.1. Table for Example 15.1

We delete the projectives in this table and replace Ω by 1, to obtain the matrix

(\begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 3 \end{matrix})

Then we find the eigenvalues of this matrix, which are 3, 2 and 0. The largest of these is 3, so γ_{G}(M_{6}) = 3.

Example 15.2. Let G = ⟨g, h⟩ ≅ ℤ/3 × ℤ/3 and k = 𝔽_{3}. This time, let M_{6} be the self-dual six dimensional module given by the following matrices and diagram:

g ↦ (\begin{matrix}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})  h ↦ (\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})  [figure: •; •; •; •; •; •]

Then M_{6} is Omega-algebraic. There are modules M_{15}, M_{21}, M_{27}, M_{27}^{∗}, M_{39}, M_{39}^{∗} and the projective indecomposable module P, such that the rows of the table in Figure 15.2 give the effect of tensoring with M_{6}.

If we throw out the projectives, and replace Ω by 1, the largest real eigenvalue of the remaining matrix will give the value of γ_{G}(M). In this example, the eigenvalues are 4, 3, 1, 0, 0, 0, −2, −2, so γ_{G}(M) = 4.

Example 15.3. Let G = ℤ/5 × ℤ/5, and k = 𝔽_{5}. Let M_{3} be the three dimensional module kG/J^{2}(kG). Then M_{3} is Omega-algebraic. More precisely, there are modules M_{6}, M_{8}, M_{10}, M_{15}, M_{30}, M_{35}, M_{45} and M_{65} such that M_{8}, M_{35} and M_{65} are self-dual, and the rows of the table in Figure 15.3 give the effect of tensoring with M_{3}.

Replacing Ω by 1, the characteristic equation is

(x^{2} − 3x + 1)(x^{6} − 4x^{3} − 1)(x^{8} + 3x^{7} + 8x^{6} + 6x^{5} + 5x^{4} − x^{3} + 12x^{2} + 7x + 9),

so the eigenvalues are 1 + τ and 2 − τ; τ and 1 − τ times cube roots of unity; and the roots of a degree eight irreducible with no real roots. The largest real eigenvalue is 1 + τ ≈ 2.618,

M_{6} M_{15} M_{21} M_{27} M_{27}^{∗} M_{39} M_{39}^{∗} M_{66} P
M_{6}  1  1
M_{15}  2  1  1
M_{21}  1  1  1  1
M_{27}  2  2  6
M_{27}^{∗}  2  2  6
M_{39}  Ω  1  1  1  2Ω  6
M_{39}^{∗} Ω^{−1}  1  1  1  2Ω^{−1}  6
M_{66}  2  2  2  Ω^{−1}  Ω  18
P  6

Figure 15.2. Table for Example 15.2

k M_{3} M_{3}^{∗} M_{6} M_{6}^{∗} M_{8} M_{10} M_{10}^{∗} M_{15} M_{15}^{∗} M_{30} M_{30}^{∗} M_{35} M_{45} M_{45}^{∗} M_{65}
k  1
M_{3} 1 1
M_{3}^{∗} 1  1
M_{6}  1  1
M_{6}^{∗}  1  1
M_{8}  1  1  1
M_{10}  Ω^{−1}  1
M_{10}^{∗}  1
M_{15}  1  1
M_{15}^{∗}  1  1
M_{30}  2  Ω^{−1}  1
M_{30}^{∗}  Ω  2  1
M_{35}  2  1  1
M_{45}  Ω^{−1}  1  1
M_{45}^{∗}  Ω^{−1}  1  1
M_{65}  Ω^{−1}  Ω  1

Figure 15.3. Table for Example 15.3

and so γ_{G}(M) = 1 + τ. This happens to be the same as γ_{H}(M) for each cyclic subgroup H of G in this case.

The next example shows how we can make use of examples already computed.

Example 15.4. Let G = ℤ/3 × ℤ/3 = ⟨g, h⟩ and k = 𝔽_{3}. Let M_{5} be the five dimensional kG-module given by the following matrices and diagram:

g ↦ (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})  h ↦ (\begin{matrix}1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})  [figure: •; •; •; •; •]

Consider the restriction of M_{5} to the cyclic subgroup generated by g. This is a sum of the trivial module and two copies of the indecomposable module of dimension 2. By Theorem 10.1 we have γ_{⟨g⟩}(M_{5}) = 1 + 4 cos(π/3) = 3 and by Theorem 2.10 γ_{G}(M_{5}) ⩾ 3.

There is a short exact sequence

0 → M_{5} → Ω^{−1}(M_{3}) → k → 0,

where M_{3} is the three dimensional module of Example 5.5. Since γ_{G}(M_{3}) = 2, by Theorem 5.4 we have γ_{G}(Ω^{−1}(M_{3})) = 2. So using Corollary 5.6 we have γ_{G}(M_{5}) ⩽ 3. Combining these bounds, we have γ_{G}(M_{5}) = 3. This example is quite difficult to compute directly.

Not all modules are Omega-algebraic. For example, we could take the module of Example 5.5 and inflate it to (ℤ/3)^{3}, with the last factor acting trivially. It is less clear whether faithful modules can fail to be Omega-algebraic, but some evidence is provided by the following example.

Example 15.5. Calculations using Magma give the following. Let M be the self-dual four dimensional module for G = ⟨g, h⟩ ≅ ℤ/3 × ℤ/3 over 𝔽_{3} given by the following matrices and diagram:

g ↦ (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})  h ↦ (\begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})  [figure: •; •; •; •]

The beginning of the pattern is as follows:

M ⊗ M ≅ k ⊕ V_{2} ⊕ W
M ⊗ V_{2} ≅ M ⊕ V_{3}
M ⊗ W ≅ M ⊕ X ⊕ X^{∗}.

Here, V_{2} and W are self-dual of dimensions 5 and 10, and X has dimension 18 and its dual satisfies X^{∗} ≅ Ω(X) ≅ Ω^{−1}X.

Thereafter, at least as far as we have calculated, there are self-dual indecomposable modules denoted V_{i} for all i ⩾ 1, where V_{1} = M and V_{2} and V_{3} are the same modules as above. The dimension of V_{i} is 3i + a(i), where a(i) depends only on the residue class of i modulo 6.

\begin{matrix}i (mod 6) & 0 & 1 & 2 & 3 & 4 & 5 \\ a(i) & 11 & 1 & −1 & 7 & 2 & 7 \end{matrix}

The tensor product with M modulo projectives appears to be given by the rows of the table in Figure 15.4. In this figure, the shift operator s replaces the subscript i from the row by i + 1 for the column, and s^{−1} replaces it by i − 1. The subscript on the V_{i} indexing the rows must be at least 2.

X X^{∗} V_{3j} V_{3j+1} V_{3j+2}
X  2 1
X^{∗} 1 2
V_{3j}  1 1  s  s^{−1}
V_{3j+1}  s^{−1}  s
V_{3j+2}  s s^{−1}

Figure 15.4. Table for Example 15.5

Notice that X ⊕ X^{∗} is an eigenvector with eigenvalue 3; this does not depend on having correctly spotted the pattern. Thus γ_{G}(M) ⩾ 3.

Now consider the diagram that describes M. If we remove the right hand vertex we obtain the diagram for the three dimensional module of Example 5.5, which we will denote by M_{3}. We know that γ_{G}(M_{3}) = 2 and there is a short exact sequence

0 → k → M → M_{3} → 0.

By Corollary 5.6 we have γ_{G}(M) ⩽ γ_{G}(M_{3}) + γ_{G}(k) = 3. Thus γ_{G}(M) ⩽ 3. Combining this with the previous bound gives γ_{G}(M) = 3.

It appears that this example is not Omega-algebraic, but it still seems to satisfy Conjecture 13.3, although we have not been able to write down a proof of this.

Conjecture 15.6. If M is an absolutely indecomposable module for ℤ/p × ℤ/p and the dimension of M is divisible by p then M is Omega-algebraic.

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Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom

School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom