SOME CONJECTURES AND THEIR CONSEQUENCES FOR TENSOR PRODUCTS OF MODULES OVER A FINITE p-GROUP

DAVE BENSON

Abstract. We make some conjectures about tensor products of modular representations for a finite p-group in characteristic p, for p = 2 and p = 3. The easiest to state is that if M is a module of odd dimension for a 2-group over an algebraically closed field of characteristic two, then k is the only direct summand of M ⊗ M^{∗} of odd dimension, and the remaining indecomposable summands have dimension divisible by four. As a consequence, the odd dimensional modules form an abelian group, where the composition law is to take the unique indecomposable summand of odd dimension of the tensor product, and where the inverse of an odd dimensional module is its dual. In characteristic three the conjectures are more complicated, and in larger characteristics it is not at all clear what the corresponding conjectures should be.

1. Introduction

Let G be a finite group and k be an algebraically closed field of characteristic p. The kG-modules studied in this paper are finite dimensional. We shall work with p′-modules, namely modules whose indecomposable summands have dimension not divisible by p.

We shall say that an indecomposable p′-module M is p′-invertible if M ⊗M^{∗} is isomorphic to the direct sum of trivial module k and indecomposable modules of dimension divisible by p. The isomorphism classes of p′-invertible modules form an abelian group 𝔄(G). If M and N are p′-invertible, then M ⊗ N has a unique indecomposable p′-summand. This gives the composition law in 𝔄(G). The inverse of a p′-invertible module M as an element of 𝔄(G) is the dual, M^{∗}.

The purpose of this paper is to discuss the following conjectures and their consequences.

Conjecture 1.1. Let p = 2 and G be a finite 2-group. If M is an odd dimensional indecomposable kG-module then M ⊗ M^{∗} is a direct sum of k and indecomposable modules of dimension divisible by four. In particular, M is 2′-invertible.

We discuss Conjecture 1.1 in Section 3.

Conjecture 1.2. Let p = 3 and G be a finite 3-group. If M is an indecomposable 3′-module for G then M ⊗ M has a direct summand which is 3′-invertible.

Conjecture 1.3. Let p = 3 and let G be a finite 3-group of rank two. If M is an indecomposable 3′-module for G and M is not 3′-invertible then the generic Jordan type of M has two non-projective Jordan blocks.

2010 Mathematics Subject Classification. 20C20.

Key words and phrases. Modular representations, Tensor product, finite p-groups.

The following is our main theorem, and shows how Conjectures 1.2 and 1.3 are related. This theorem is proved at the end of Section 5.

Theorem 1.4. Let k have characteristic three, and let M be an indecomposable 3′-module for a finite group G. If the generic Jordan type of M has two non-projective summands, then M ⊗ M has a direct summand which is 3′-invertible. In this case, the McKay graph Γ_{M} is a quotient of a graph of the form ℤT by an admissible group of automorphisms, where T is a directed labelled tree. The associated undirected labelled tree T̄ is uniquely determined by Γ_{M} and is either the Dynkin diagram A_{3m−1} for some positive integer m, a Euclidean diagram D̃_{n}, Ẽ_{6}, Ẽ_{7} or Ẽ_{8}, or one of the infinite diagrams A_{∞}, D_{∞} or A_{∞}^{∞}.

In Section 6, we describe some examples of this phenomenon for the group ℤ/3 × ℤ/3, thereby illustrating Conjectures 1.2 and 1.3. In particular, these examples show that the diagrams D̃_{4} and D_{∞} occur in this case. In Section 7, we give examples of the Euclidean diagrams B̃_{5} and B̃_{2} over a field which is not algebraically closed. It would be interesting to decide which of the remaining cases can occur.

Acknowledgement. This material is based on work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester.

The author thanks Jon Carlson, Karin Erdmann, Pavel Etingof and Peter Symonds for conversations relating to this material.

2. Trivial summands of tensor products

Our basic tool is the following theorem and corollary from Benson [2, §2.18, Exercise 3], Benson and Carlson [5, Theorem 2.1 and Proposition 2.2]. Note that the assumption that k is algebraically closed is necessary for this theorem and corollary.

Theorem 2.1. If M and N are indecomposable kG-modules, then k is a direct summand of M ⊗ N if and only if N ≅ M^{∗} and the dimension of M is not divisible by p. □

Corollary 2.2. Let M be an indecomposable module of dimension divisible by p, and let N be any kG-module. Then every direct summand of M ⊗ N has dimension divisible by p. □

This means that we can work modulo the indecomposable modules of dimension divisible by p, as they form an ideal. We write M ≡ N if the sum of the p′-summands of M is isomorphic to the sum of the p′-summands of N. This is obviously an equivalence relation, and the corollary shows that tensor product is well defined on equivalence classes.

Proposition 2.3. Let M_{1}, M_{2} and M_{3} be indecomposable p′-modules. Then the following are equivalent:

(i) k is isomorphic to a direct summand of M_{1} ⊗ M_{2} ⊗ M_{3} (ii) M_{1}^{∗} is isomorphic to a direct summand of M_{2} ⊗ M_{3} (iii) k is isomorphic to a direct summand of M_{1}^{∗} ⊗ M_{2}^{∗} ⊗ M_{3}^{∗}. (iv) M_{1} is isomorphic to a direct summand of M_{2}^{∗} ⊗ M_{3}^{∗}

Proof. This follows immediately from Theorem 2.1. □

Proposition 2.4. Suppose that M is an indecomposable p′-module and N is a p′-invertible summand of M ⊗ M. Then

(i) N ⊗ M^{∗} ≡ M.

(ii) If X and Y are indecomposable p′-modules then Y is isomorphic to a summand of M ⊗ X if and only if N ⊗ X is isomorphic to a summand of M ⊗ Y.

Proof. (i) Since N^{∗} is isomorphic to a direct summand of M^{∗} ⊗ M^{∗}, by Proposition 2.3, M is isomorphic to a direct summand of N ⊗ M^{∗}. Similarly, M^{∗} is isomorphic to a direct summand of N^{∗}⊗M, and so N ⊗M^{∗} is isomorphic to a direct summand of N ⊗N^{∗}⊗M ≡ M.

(ii) If Y is isomorphic to a direct summand of M ⊗ X then by Proposition 2.3, X is isomorphic to a direct summand of M^{∗} ⊗Y and so N ⊗X is isomorphic to a direct summand of N ⊗ M^{∗} ⊗ Y. By part (i), the latter is equivalent to M ⊗ Y. □

3. The conjecture for p = 2

In this section we give a brief discussion of Conjecture 1.1. I have run hundreds of examples with the computer algebra package Magma [6], and not found a counterexample to the conjecture. The interesting part is the weaker conjecture that the summands of M ⊗ M^{∗} other than k have even dimension. I shall call this weaker statement Conjecture 1.1′. That they have dimension divisible by four seems more mysterious, and I really only conjecture this because of the evidence from the examples.

If Conjecture 1.1′ is true, then under these hypotheses, every odd dimensional indecomposable module for a finite 2-group is 2′-invertible. It follows that 𝔄(G) is the entire set of isomorphism classes of odd dimensional indecomposable modules.

Here are some cases where the conjecture is known to be true.

Example 3.1. If G = ℤ/2×ℤ/2 then the only odd dimensional indecomposable modules are the Ω^{n}(k) with n ∈ ℤ. We have Ω^{m}(k) ⊗ Ω^{n}(k) ≅ Ω^{m+n}(k) ⊕ (proj), and the projectives have dimension divisible by four. Thus Conjecture 1.1 holds in this case, and we have 𝔄(G) ≅ ℤ.

Example 3.2. In the case where G is a dihedral group D_{2^{n}} with n ⩾ 3, Archer [1] proved that Conjecture 1.1′ holds, and that 𝔄(G) is an infinitely generated torsion free abelian group. The proof of the conjecture in this case consists of observing that the odd dimensional indecomposables are string modules, and the restriction to a cyclic subgroup of G of order two has just one trivial direct summand, the rest being projective. It follows that the restriction to a subgroup isomorphic to ℤ/2 × ℤ/2 has exactly one odd dimensional indecomposable summand, and so using Example 3.1, it follows that the divisibility by four Conjecture 1.1 holds in this case.

Example 3.3. In the case where G is a semidihedral group of order 2^{n}

SD_{2^{n}} = ⟨g, h | g^{2^{n−1}} = 1, h^{2} = 1, hgh = g^{2^{n−2}−1}⟩ (n ⩾ 4)

the same argument works. The indecomposables are described in Crawley-Boevey [7] (see also Geiß [10]). The odd dimensional indecomposables are asymmetric or symmetric string modules, and again the restriction to a cyclic subgroup of G of order two has just one trivial direct summand, the rest being projective (Erdmann [8, Lemma II.10.5]). So Conjecture 1.1′ holds in this case. Then the same argument as in Example 3.2 using restriction to a subgroup isomorphic to ℤ/2 × ℤ/2 shows that Conjecture 1.1 holds in this case. In contrast with the dihedral case, there is an endotrivial module of dimension 2^{n−2} + 1, giving an element of 𝔄(G) which squares to the identity, so there is torsion.

Example 3.4. In the case of the generalised quaternion groups Q_{2^{n}} (n ⩾ 3), the endotrivial group is ℤ/4 × ℤ/2, so again there is torsion in 𝔄(G). Even though these groups have tame representation type, we do not currently know enough about the modules to prove Conjecture 1.1, or even Conjecture 1.1′, in this case.

4. Generic Jordan type

If G = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} is an elementary abelian p-group, we set X_{i} = g_{i} −1 ∈ J(kG). If α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k), we form the element

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ J(kG).

This satisfies X_{α}^{p} = 0. If M is a kG-module, we can ask for the Jordan canonical form of X_{α} on M. By a theorem of Friedlander, Pevtsova and Suslin [9], see also [4, Theorem 4.5.3], there is a dense open set U_{max}(M) ⊆ 𝔸^{r}(k) (depending on M) such that the Jordan canonical form of X_{α} on M is constant for α ∈ U_{max}(M). It is the same as the Jordan canonical form obtained by extending scalars to a large transcendental extension of k and using independent transcendentals for λ_{1}, …, λ_{r}. This is called the generic Jordan type of M. If M and N are kG-modules then by [4, Theorem 4.6.1], U_{max}(M ⊗ N) ⊇ U_{max}(M) ∩ U_{max}(N) and the generic Jordan type of M ⊗ N is the tensor product of those for M and for N.

Definition 4.1. We say that M has generic Jordan type [1]^{a_{1}} … [p]^{a_{p}} and stable generic Jordan type [1]^{a_{1}} … [p − 1]^{a_{p−1}} if there are a_{i} Jordan blocks of length i in the generic Jordan type of M for 1 ⩽ i ⩽ p.

If the generic Jordan type of M is just [p]^{a_{p}} then M is generically projective. In this case, every indecomposable summand of M has dimension divisible by p.

Lemma 4.2. If M is a kG-module with stable generic Jordan type [1] or [p − 1] then M is p′-invertible.

Proof. The stable generic Jordan type of M ⊗ M^{∗} is [1]. Thus M ⊗ M^{∗} ≅ k ⊕ N where N is generically projective. □

Definition 4.3. If M has generic Jordan type [1] or [p − 1] then we say that M is strongly p′-invertible. By Lemma 4.2, strongly p′-invertible modules are p′-invertible.

It also follows from [4, Corollary 4.6.2] that in odd characteristic the generic Jordan type of the symmetric and exterior squares are the symmetric and exterior squares of the generic Jordan type. Thus we have the following.

Proposition 4.4. Suppose that p = 3 and M is an indecomposable 3′-module whose generic Jordan type has exactly two non-projective summands. Then M ⊗M has a summand N which is strongly 3′-invertible. Furthermore, for m, n ⩾ 0, every direct summand of M^{⊗m} ⊗ M^{∗ ⊗n} has stable generic Jordan type either [1]^{s} or [2]^{s} for some s ⩾ 0.

Proof. The hypotheses imply that the stable generic Jordan type of M is [1]^{2} or [2]^{2}. We have [2] ⊗ [2] = [1] ⊕ [3], so by induction on the number of tensor factors, M^{⊗m} ⊗ M^{∗ ⊗n} has stable generic Jordan type either [1]^{2^{m+n}} or [2]^{2^{m+n}}. So every direct summand has stable Jordan type [1]^{s} or [2]^{s} for some s ⩾ 0.

For stable generic Jordan types, we have S^{2}[1] = [1], Λ^{2}[1] = 0, S^{2}[2] = 0, a d Λ^{2}[2] = [1]. We also have M ⊗ M = S^{2}(M) ⊕ Λ^{2}(M). So set N = Λ^{2}(M) if M has type [1^{2}] and N = S^{2}(M) if M has type [2^{2}]. Then N has stable generic Jordan type [1], and so N is strongly 3′-invertible. □

5. The McKay graph

Definition 5.1. Suppose that M be an absolutely indecomposable module of dimension not divisible by p, and that M is not isomorphic to k. The McKay graph of M is a directed graph Γ_{M} whose vertices are labelled by the indecomposable direct summands of tensor products of copies of M and M^{∗}, of dimension not divisible by p. We put a directed edge going from X to Y if Y is isomorphic to a direct summand of M ⊗ X. If Y occurs as a summand of M ⊗ X with multiplicity greater than one, then there is still only one directed edge X → Y, but it is labelled with this multiplicity. Note that by Theorem 2.1, there is a directed edge X → Y if and only if there is a directed edge Y^{∗} → X^{∗}.

Example 5.2. If M ⊗ M ≡ k then Γ_{M} takes the form

\xymatrix{k \ar[r] & M \ar[l]} .

Example 5.3. If M is p′-invertible of infinite order then the McKay graph Γ_{M} is as follows:

\xymatrix{⋯ \ar[r] & M_{2}^{∗} \ar[r] & M_{1}^{∗} \ar[r] & k \ar[r] & M_{1} \ar[r] & M_{2} \ar[r] & M_{3} \ar[r] & ⋯}

Here, M_{i} denotes the unique summand of M^{⊗i} whose dimension is not divisible by p. In particular, M_{1} = M.

Now suppose that M is an indecomposable p′-module, and that M ⊗ M has a p′-invertible summand N. Then there is an automorphism τ: Γ_{M} → Γ_{M}, sending X to the unique p′- module X′ such that X is isomorphic to a summand of N ⊗ X′. The inverse τ^{−1} sends X to the unique indecomposable p′-summand of N ⊗ X.

Lemma 5.4. For modules X and Y labelling vertices of Γ_{M}, there is a directed edge from Y to X if and only if there is a directed edge from τ(X) to Y.

Proof. This follows from Proposition 2.4. □

At this stage, we can apply the Riedtmann Structure Theorem [12] (see also [3, §4.15]).

Definition 5.5. A (stable) translation quiver is a directed graph Γ containing no loops or multiple edges, together with an automorphism τ: Γ → Γ with the property that for vertices x and y of Γ, there is a directed edge from y to x if and only if there is a directed edge from τ(x) to y.

Examples 5.6. (i) If T is a directed tree then ℤT is the translation quiver whose vertices are the pairs (n, x) with n ∈ ℤ and x a vertex of T, and with directed edges (n, x) → (n, y) and (n, y) → (n − 1, x) for each directed edge x → y in T. The translation is defined by τ(n, x) = (n + 1, x). If an edge x → y in T is given a pair of labels (ℓ, ℓ′) then we can associate the label ℓ to each directed edge (n, x) → (n, y) and ℓ′ to each directed edge (n, y) → (n − 1, x).

(ii) By Lemma 5.4, if M is a p′-module and M ⊗ M has a p′-invertible summand N then Γ_{M} is a translation quiver with τ defined as above.

Theorem 5.7 (Riedtmann’s Structure Theorem). Given a connected translation quiver Γ, there is a directed tree T and an admissible group of automorphisms Π ⩽ Aut(ℤT) such that Γ ≅ ℤT/Π. The undirected graph T̄ associated to T is determined by Γ uniquely up to canonical isomorphism, and Π is uniquely defined up to conjugation in Aut(ℤT).

Proof. See [12, Struktursatz] or [3, Theorem 4.15.6]. □

Note that in this theorem, if Γ is a labelled tree, then each edge x—y of the tree T̄ comes with a pair of labels, one associated with each direction, x → y and y → x.

Definition 5.8. An subadditive function on a translation quiver Γ is a function f from the vertices of Γ to the positive integers with the property that f(x) = f(τ(x)), and the sum over x → y of f(y), counted with multiplicity, is less than or equal to 2f(x). If equality holds, then f is additive.

Theorem 5.9. Suppose that p = 3, and M is a 3′-module whose generic Jordan type has exactly two non-projective summands. Then Γ_{M} is a translation quiver.

Let f be the function sending each vertex X of Γ_{M} to the number of non-projective Jordan blocks in the generic Jordan type of X. Then f is a subadditive function on Γ_{M}, which takes value 1 at the vertex corresponding to k and value 2 at the vertex corresponding to M. At vertices where f is not additive, 2f(x) is a multiple of three more than the sum over x → y of f(y).

Proof. By Proposition 4.4, M ⊗ M has a direct summand N which is strongly p′-invertible, so we have f(X) = f(τ(X)). The number of non-projective summands in the Jordan type of M ⊗ X is twice that of X, and so the sum over X → Y of f(Y), counted with multiplicity, is less than or equal to 2f(x).

Strict inequality will hold when there are summands of dimension divisible by 3 which are not generically projective. Using Proposition 4.4, the generic Jordan type of such a summand has to have a multiple of three Jordan blocks of length 1 or 2. □

Theorem 5.10. Suppose that a connected labelled undirected graph T̄ admits a subadditive function. Then T̄ is either a Dynkin diagram, a Euclidean diagram, or one of the infinite diagrams one of the infinite trees A_{∞}, B_{∞}, C_{∞}, D_{∞} or A_{∞}^{∞}.

If the subadditive function is additive, then the diagram is Euclidean or one of the five infinite trees. If it is not additive, then the diagram is Dynkin or A_{∞}. If it is unbounded then the diagram is A_{∞}.

Proof. See Happel, Preiser and Ringel [11] or [3, Theorem 4.5.8]. Pictures of these diagrams can be found in [3, Section 4.5]. □

Proposition 5.11. Let f be a subadditive function on a Dynkin diagram. Suppose that f takes value 1 at some vertex, and that if x is a vertex at which f is not additive, 2f(x) is a multiple of three more than the sum over x → y of f(y). Then the Dynkin diagram is not of type D_{n}, E_{6}, E_{7} or E_{8}. If it is of type A_{n} then n has the form 3m − 1 for some positive integer m.

Proof. Suppose that the type is one of D_{n}, E_{6}, E_{7} or E_{8}. Then there is a vertex u with three edges, u—v_{i}, 1 ⩽ i ⩽ 3, and possibly arms going further out from at most two of the v_{i}. Then we have

2f(u) ⩾ f(v_{1}) + f(v_{2}) + f(v_{3}), 2f(v_{1}) ⩾ f(u), 2f(v_{2}) ⩾ f(u), 2f(v_{3}) ⩾ f(u).

Doubling the first equation and adding two of the other three, we see that f(u) ⩾ f(v_{i}) for 1 ⩽ i ⩽ 3. Since one of the arms, say the one with i = 1, ends in a vertex w with f(w) = 1, it follows that f(v_{1}) ⩾ f(u) − 1. Since f(v_{2}) + f(v_{3}) ⩾ f(u), this implies that f(v_{1}) + f(v_{2}) + f(v_{3}) ⩾ 2f(u) − 1. From the hypotheses, it now follows that u is a vertex at which f is additive. Furthermore, f must be additive on the arm of v_{1}, since again a vertex at which it is strictly subadditive will violate the hypotheses. So f(u) = f(v_{1}) + 1 is the length of the arm containing v_{1}, and after switching v_{2} and v_{3} if necessary we have f(u) = 2f(v_{2}) and f(v_{3}) = f(v_{2})+1. In particular, f(u) is even. It is now easy to check case by case that the diagram cannot have type D_{n}, E_{6}, E_{7}, or E_{8}. Finally, if the Dynkin type is A_{n}, then looking at f modulo three, we see that n is congruent to −1 modulo three. □

Proof of Theorem 1.4. It follows from Proposition 4.4 that M ⊗ M has a direct summand which is 3′-invertible. By Theorem 5.9, the McKay graph Γ_{M} is a quotient of a graph of the form ℤT by an admissible group of automorphisms, where T is a directed labelled tree. By Theorem 5.7, the tree T̄ is uniquely determined by Γ_{M}. Using Theorem 5.10 on the subadditive function f of Theorem 5.9, we deduce that T̄ is a Dynkin diagram, a Euclidean diagram or one of the infinite trees A_{∞}, B_{∞}, C_{∞}, D_{∞} or A_{∞}^{∞}.

Since we are working with trees, we can ignore the Euclidean diagram Ã_{n}. Since we are working over an algebraically closed field, the multiplicities on the edges of T̄ are symmetric. This rules out some of the trees allowed in the theorem, leaving the simply laced Dynkin diagrams of type A, D and E, the Euclidean diagrams of type Ã_{12}, D̃, Ẽ, and the infinite trees A_{∞}, D_{∞} and A_{∞}^{∞}. Type Ã_{12} cannot occur, as the subadditive function f would have to take the same value on both vertices, whereas k and M have different stable generic Jordan types. Finally, Proposition 5.11 rules out Dynkin types D_{n}, E_{6}, E_{7} and E_{8}, and shows that if the Dynkin type is A_{n} then n has the form 3m − 1 for some positive integer m. □

Remark 5.12. We shall see that some of the other trees in Theorem 5.10 occur over non algebraically closed fields; examples where the trees B̃_{5} and B̃_{2} occur are discussed in Examples 7.1 and 7.2.

6. Examples

Example 6.1. Let M be a seven dimensional indecomposable module for

G = ⟨g, h⟩ ≅ ℤ/3 × ℤ/3

\xymatrix{ & Ω^{−2}(4_{1}) \ar[ddr] &  & Ω^{−1}(2_{1}) \ar[ddr] &  & 4_{1} \ar[ddr] &  & Ω(2_{1}) \ar[ddr] &  & Ω^{2}(4_{1}) \\  & Ω^{−2}(4_{2}) \ar@{-->}[dr] &  & Ω^{−1}(2_{2}) \ar@{-->}[dr] &  & 4_{2} \ar[dr] &  & Ω(2_{2}) \ar@{-->}[dr] &  & Ω^{2}(4_{2}) \\ ⋯ &  & Ω^{−2}(7) \ar[uur] \ar@{-->}[ur] \ar@{-->}[dr] \ar[ddr] &  & 7^{∗} \ar[uur] \ar[ur] \ar[dr] \ar[ddr] &  & 7 \ar[uur] \ar[ur] \ar[dr] \ar[ddr] &  & Ω^{2}(7^{∗}) \ar[uur] \ar@{-->}[ur] \ar@{-->}[dr] \ar[ddr] &  & ⋯ \\  & Ω^{−2}(4_{3}) \ar@{-->}[ur] &  & Ω^{−1}(2_{3}) \ar@{-->}[ur] &  & 4_{3} \ar[ur] &  & Ω(2_{3}) \ar@{-->}[ur] &  & Ω^{2}(4_{3}) \\  & Ω^{−2}(k) \ar[uur] &  & Ω^{−1}(8) \ar[uur] &  & k \ar[uur] &  & Ω(8) \ar[uur] &  & Ω^{2}(k)}

Figure 6.1. Figure for Example 6.1 of type D̃_{4}

given by the matrices

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

Then the McKay graph Γ_{M} is shown in Figure 6.1.

In this diagram, modules are indicated by their dimensions, with modules of the same dimension distinguished by subscripts. The diagram continues infinitely to the left and to the right, to form a ℤD̃_{4} translation quiver. Translating to the right is tensoring with the ten dimensional summand N = Ω(8) of M ⊗ M, and duality is reflection about a vertical axis going through the trivial module.

Example 6.2. Let M be the very similar looking seven dimensional indecomposable module (just one matrix entry has changed) for

G = ⟨g, h⟩ ≅ ℤ/3 × ℤ/3

given by the matrices

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

Then the McKay graph Γ_{M} is displayed in Figure 6.2. Again, modules are indicated by their dimensions, with modules of the same dimension distinguished by subscripts. The diagram continues infinitely to the left, to the right, and upwards to form a ℤD_{∞} translation quiver.

\xymatrix{Ω^{−2}(17) \ar[dr] & ⋮ & Ω^{−1}(22) \ar[dr] & ⋮ & 17 \ar[dr] & ⋮ & Ω(22) \ar[dr] & ⋮ & Ω^{2}(17) \ar[dr] \\ ⋯ & Ω^{−2}(19_{2}) \ar[dr] \ar[ur] &  & 19_{2}^{∗} \ar[dr] \ar[ur] &  & 19_{2} \ar[dr] \ar[ur] &  & Ω^{2}(19_{2}^{∗}) \ar[dr] \ar[ur] &  & Ω^{2}(19_{2}) \\ Ω^{−2}(14) \ar[dr] \ar@{-->}[ur] &  & Ω^{−1}(10_{3}) \ar@{-->}[dr] \ar@{-->}[ur] &  & 14 \ar[dr] \ar[ur] &  & Ω(10_{3}) \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(14) \ar[dr] \ar@{-->}[ur] & ⋯ \\ ⋯ & Ω^{−2}(16_{2}) \ar[dr] \ar@{-->}[ur] &  & 16_{2}^{∗} \ar[dr] \ar[ur] &  & 16_{2} \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(16_{2}^{∗}) \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(16_{2}) \\ Ω^{−2}(23) \ar[dr] \ar@{-->}[ur] &  & Ω^{−1}(16_{1}) \ar[dr] \ar@{-->}[ur] &  & 23 \ar[dr] \ar[ur] &  & Ω(16_{1}) \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(23) \ar[dr] \ar@{-->}[ur] & ⋯ \\ ⋯ & Ω^{−2}(7_{2}) \ar[dr] \ar@{-->}[ur] &  & 7_{2}^{∗} \ar[dr] \ar[ur] &  & 7_{2} \ar[dr] \ar[ur] &  & Ω^{2}(7_{2}^{∗}) \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(7_{2}) \\ Ω^{−2}(5) \ar[dr] \ar@{-->}[ur] &  & Ω^{−1}(10_{2}) \ar@{-->}[dr] \ar[ur] &  & 5 \ar[dr] \ar[ur] &  & Ω(10_{2}) \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(5) \ar[dr] \ar@{-->}[ur] & ⋯ \\ ⋯ & Ω^{−2}(19_{1}) \ar[dr] \ar@{-->}[ur] &  & 19_{1}^{∗} \ar[dr] \ar[ur] &  & 19_{1} \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(19_{1}^{∗}) \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(19_{1}) \\ Ω^{−2}(8) \ar[dr] \ar@{-->}[ur] &  & Ω^{−1}(4) \ar[dr] \ar@{-->}[ur] &  & 8 \ar[dr] \ar[ur] &  & Ω(4) \ar[dr] \ar@{-->}[ur] &  & Ω^{2}(8) \ar[dr] \ar@{-->}[ur] & ⋯ \\ Ω^{−2}(10_{1}) \ar[r] & Ω^{−2}(7_{1}) \ar[dr] \ar[r] \ar@{-->}[ur] & Ω^{−1}(20) \ar[r] & 7_{1}^{∗} \ar[dr] \ar[r] \ar[ur] & 10_{1} \ar[r] & 7_{1} \ar[dr] \ar[r] \ar[ur] & Ω(20) \ar[r] & Ω^{2}(7_{1}^{∗}) \ar[dr] \ar[r] \ar@{-->}[ur] & Ω^{2}(10_{1}) \ar[r] & Ω^{2}(7_{1}) \\ Ω^{−2}(1) \ar@{-->}[ur] &  & Ω^{−1}(2) \ar[ur] &  & 1 \ar[ur] &  & Ω(2) \ar@{-->}[ur] &  & Ω^{2}(1) \ar@{-->}[ur] & ⋯}

Figure 6.2. Figure for Example 6.2 of type D_{∞}

Translating to the right is tensoring with the summand N = Ω(2) of M ⊗ M, and duality is reflection about a vertical axis going through the trivial module.

Remark 6.3. To illustrate how different things are in higher characteristic, a random example for ℤ/7 × ℤ/7 is shown in Figure 6.3.

7. Non algebraically closed fields

We were deliberately a bit vague about the multiplicities occurring in Section 5. Over an algebraically closed field, the multiplicities are symmetric, but not over a field which is not algebraically closed.

Example 7.1. Let G = ℤ/3 × ℤ/3 and let M be the four dimensional representation over 𝔽_{9} = 𝔽_{3}[i]/(i^{2} + 1) given by the matrices

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

\xymatrix{ &  &  & 50 \ar[dl] \\  &  & 60^{∗} \ar[dl] \ar[dr] & 64 \ar[l] \ar[ddll] \ar[ddr] & 60 \ar[l] \ar[ul] \\  & 45^{∗} \ar[dl] \ar[dr] &  & 20_{b} \ar@{-->}[ur] \ar[dl] &  & 45 \ar[ul] \ar[d] \\ 20_{a}^{∗} \ar@{-->}[dr] & 36^{∗} \ar[ddrr] \ar[u] & 20^{∗} \ar[dr] \ar[uur] \ar[dl] &  & 20 \ar[ul] \ar[ddl] \ar[ur] & 36 \ar[uull] \ar[d] & 20_{a} \ar@{-->}[ul] \\  & 10^{∗} \ar[dr] \ar[u] &  & 6 \ar[ur] \ar[dl] &  & 10 \ar[ur] \ar[ul] \\  &  & 4^{∗} \ar[dr] \ar[r] & 15 \ar[uurr] \ar[r] \ar[uul] & 4 \ar[ur] \ar[ul] \\  &  &  & k \ar[ur]}

Figure 6.3. Figure for Remark 6.3 in characteristic seven

\xymatrix{Ω^{2}(1) \ar@{-->}[dr] &  & Ω^{2}(7) \ar@{-->}[dr] &  & 4_{1} \ar[dr] &  & Ω^{−2}(7^{∗}) \ar@{-->}[dr] &  & Ω^{−2}(1) \ar@{-->}[dr] & ⋯ \\ ⋯ & Ω^{2}(8) \ar@{-->}[dr] \ar@{=>}[ur] &  & Ω(16) \ar@{-->}[dr] \ar@{=>}[ur] &  & Ω^{−1}(16^{∗}) \ar@{-->}[dr] \ar@{=>}[ur] &  & Ω^{−2}(8^{∗}) \ar@{-->}[dr] \ar@{=>}[ur] &  & Ω^{−3}(10) \\ Ω^{2}(14_{1}) \ar[dr] \ar@{-->}[ur] &  & Ω(4_{2}) \ar@{-->}[dr] \ar@{-->}[ur] &  & 14_{1} \ar[dr] \ar@{-->}[ur] &  & Ω^{−1}(4_{2}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−2}(14_{1}) \ar@{-->}[dr] \ar@{-->}[ur] & ⋯ \\ ⋯ & Ω(16^{∗}) \ar@{=>}[dr] \ar@{-->}[ur] &  & 8^{∗} \ar@{=>}[dr] \ar[ur] &  & 8 \ar@{=>}[dr] \ar[ur] &  & Ω^{−1}(16) \ar@{=>}[dr] \ar@{-->}[ur] &  & Ω^{−3}(16^{∗}) \\ Ω^{2}(4_{1}) \ar@{-->}[ur] &  & 7^{∗} \ar[ur] &  & 1 \ar[ur] &  & 7 \ar[ur] &  & Ω^{−2}(4_{1}) \ar@{-->}[ur] & ⋯}

Figure 7.1. First Figure for Example 7.1 of type B̃_{5}

Then M is 3′-invertible, but not isomorphic to its Frobenius twist M^{F}. When we restrict scalars to get an 8 dimensional representation over 𝔽_{3}, it is no longer invertible. But the generic Jordan type has two summands of size one plus two projective summands, so we can apply the theory in this case. Figure 7.1 shows the McKay graph, which this time has asymmetric multiplicities. We have Γ_{M} ≅ ℤB̃_{5}. Duality acts by reflection about a vertical axis going through the trivial module. The functor Ω^{2} acts as a glide reflection.

This can be viewed as being obtained from the ℤA_{∞}^{∞}/Π diagram Γ_{M⊕M^{F}} for the decomposable module M ⊕ M^{F} by folding over, using the Frobenius map. Figure 7.2 shows Γ_{M⊕M^{F}}, where M is labelled 4. The group Π ≅ ℤ identifies the top and bottom rows to form an infinite cylinder. The Frobenius F acts by reflection through a plane cutting through the

\xymatrix{Ω^{2}(7_{1}) \ar@{-->}[dr] &  & Ω(2) \ar@{-->}[dr] &  & 7_{1}^{F} \ar@{-->}[dr] &  & Ω^{−1}(2^{F}) \ar@{-->}[dr] &  & Ω^{−2}(7_{1}) \ar@{-->}[dr] \\ ⋯ & Ω^{2}(4) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω(8_{1}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−1}(8_{1}^{F ∗}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−2}(4^{F ∗}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−2}(4) \\ Ω^{2}(1) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{2}(7) \ar@{-->}[dr] \ar@{-->}[ur] &  & 4_{1} \ar[dr] \ar[ur] &  & Ω^{−2}(7^{∗}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−2}(1) \ar@{-->}[dr] \ar@{-->}[ur] & ⋯ \\ ⋯ & Ω^{2}(4^{F}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω(8_{1}^{F}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−1}(8_{1}^{∗}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−2}(4^{∗}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−2}(4^{F}) \\ Ω^{2}(7_{1}^{F}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω(2^{F}) \ar@{-->}[dr] \ar@{-->}[ur] &  & 7_{1} \ar[dr] \ar[ur] &  & Ω^{−1}(2) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−2}(7_{1}^{F}) \ar@{-->}[dr] \ar@{-->}[ur] & ⋯ \\ ⋯ & Ω(8_{1}^{F ∗}) \ar@{-->}[dr] \ar@{-->}[ur] &  & 4^{F ∗} \ar[dr] \ar@{-->}[ur] &  & 4 \ar[dr] \ar[ur] &  & Ω^{−1}(8_{1}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−3}(8_{1}^{F ∗}) \\ Ω^{2}(4_{1}) \ar@{-->}[dr] \ar@{-->}[ur] &  & 7^{∗} \ar[dr] \ar[ur] &  & 1 \ar[dr] \ar[ur] &  & 7 \ar[dr] \ar[ur] &  & Ω^{−2}(4_{1}) \ar@{-->}[dr] \ar@{-->}[ur] & ⋯ \\ ⋯ & Ω(8_{1}^{∗}) \ar@{-->}[dr] \ar@{-->}[ur] &  & 4^{∗} \ar[dr] \ar[ur] &  & 4^{F} \ar@{-->}[dr] \ar[ur] &  & Ω^{−1}(8_{1}^{F}) \ar@{-->}[dr] \ar@{-->}[ur] &  & Ω^{−3}(8_{1}^{∗}) \\ Ω^{2}(7_{1}) \ar@{-->}[ur] &  & Ω(2) \ar@{-->}[ur] &  & 7_{1}^{F} \ar@{-->}[ur] &  & Ω^{−1}(2^{F}) \ar@{-->}[ur] &  & Ω^{−2}(7_{1}) \ar@{-->}[ur]}

Figure 7.2. Second Figure for Example 7.1

\xymatrix{ &  & Ω^{2}(4) \ar[dr] &  & 4 \ar[dr] &  & Ω^{−2}(4) \ar@{-->}[dr] \\ ⋯ & Ω^{2}(10) \ar@{=>}[ur] \ar@{=>}[dr] &  & 10^{∗} \ar@{=>}[ur] \ar@{=>}[dr] &  & 10 \ar@{=>}[ur] \ar@{=>}[dr] &  & Ω^{−2}(10^{∗}) & ⋯ \\  &  & Ω^{2}(1) \ar@{-->}[ur] &  & 1 \ar[ur] &  & Ω^{−2}(1) \ar@{-->}[ur]}

Figure 7.3. Figure for Example 7.2 of type B̃_{2}

cylinder at the trivial module and 4_{1}. Duality acts as an order two rotation centred at the trivial module, noting that 7_{1}^{F} ≅ 7_{1}^{∗} and 4_{1} ≅ 4_{1}^{F} ≅ 4_{1}^{∗} ≅ 4_{1}^{F ∗}. The functor Ω^{2} acts by translating the cylinder to the left and then rotating it by half a turn.

Example 7.2. As a variant of the previous example, let G = ℤ/3 × ℤ/3 and let M be the five dimensional representation over 𝔽_{9} = 𝔽_{3}[i]/(i^{2} + 1) given by the matrices

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

and restrict scalars to 𝔽_{3} to get a 10 dimensional representation. Then the McKay quiver is of type B̃_{2}, and is shown in Figure 7.3. The reader is invited to write down the unfolded diagram ℤA_{∞}^{∞}/Π, a cylinder of half the circumference of the one in the previous example.

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