The structure of the invariant ring of two matrices of degree 3 Kazunori NAKAMOTO

Department of Mathematics Faculty of Science Kyoto University nakamoto@kusm.kyoto-u.ac.jp

Let M_{3} × M_{3} be the scheme of two 3 × 3 matrices over ℤ and let ℤ[M_{3} × M_{3}] be its coordinate ring. The group scheme PGL_{3} acts on the scheme M_{3} × M_{3} by (A, B) ↦ (P^{−1}AP, P^{−1}BP) for (A, B) ∈ M_{3} × M_{3} and P ∈ PGL_{3}. In this paper we determine the structure of the invariant ring ℤ[M_{3} × M_{3}]^{PGL^{3}}. The spectrum of ℤ[M_{3} × M_{3}]^{PGL^{3}} is isomorphic to a hypersurface of 𝔸_{ℤ}^{11}.

The invariant ring ℤ[M_{3} × M_{3}]^{PGL^{3}} is closely related to various geometric objects. Through the structure of ℤ[M_{3}×M_{3}]^{PGL^{3}} we can obtain explicit descriptions of certain quiver varieties and semiuniversal deformation spaces of simple singularities ([7]). For another example, we can investigate the singularity of the moduli space of rank 3 semistable vector bundles over a nonsingular projective curve of genus 2 by studying the structure of ℤ[M_{3} × M_{3}]^{PGL^{3}}.

The author specially thanks Doctor Ken-ichi Tokunaga, who made several programs for computing relations in the invariant ring. The author also thanks Doctor Yasuhiro Omoda for his useful advice on quiver varieties.

1. Generators of ℤ[M_{3} × M_{3}]^{PGL^{3}}

1.1. Definition of Saito character ring. In this section we introduce the Saito character ring. The Saito character ring of degree 3 for the free group F_{2} of rank 2 is an approximation of the invariant ring ℤ[M_{3} × M_{3}]^{PGL^{3}}. We can deduce several relations in the invariant ring from the Saito character ring.

Definition 1.1 ([9],[6]). Let Γ be a group. We define the representation ring of degree n for Γ by

A_{n}(Γ) := ℤ[a_{ij}(γ) | γ ∈ Γ, 1 ≤ i, j ≤ n]/I(Γ),

where I(Γ) is the ideal of the polynomial ring ℤ[a_{ij}(γ) | γ ∈ Γ, 1 ≤ i, j ≤ n] generated by all entries of the n × n matrices σ(e) − I_{n} and σ(γ)σ(δ) − σ(γδ) for each γ, δ ∈ Γ, where σ(γ) := (a_{ij}(γ))_{1≤i,j≤n}. Here we denote by e the identity element of Γ. We regard each a_{ij}(γ) as an independent variable in the polynomial ring ℤ[a_{ij}(γ)].

Remark 1.2. In [9] and [5], the affine scheme Spec A_{n}(Γ) is called the representation variety of degree n for Γ. The scheme Spec A_{n}(Γ) represents the following contravariant functor from the category of schemes to the category of sets:

Rep_{n}(Γ) : (Sch) → (Sets)
X ↦ {ρ : Γ → GL_{n}(Γ(X, O_{X})) representation }.

We denote the universal representation of degree n for Γ by σ_{Γ} : Γ → GL_{n}(A_{n}(Γ)).

Notation 1.3 ([9],[6]). Let ℤ[M_{n}^{⊕m}] be the coordinate ring of the scheme of m matrices of size n × n over ℤ. We denote by A_{1}, A_{2}, ..., A_{m} the universal matrices on Specℤ[M_{n}^{⊕m}]. We write ℤ[M_{3} × M_{3}] instead of ℤ[M_{3}^{⊕2}].

If R is a commutative ring, then we define

R[M_{n}^{⊕m}] := ℤ[M_{n}^{⊕m}] ⊗_{ℤ} R.

The symbol det A denotes Π_{k=1}^{n} det A_{k}. The symbol R[M_{n}^{⊕m}]_{det A} denotes the localization of R[M_{n}^{⊕m}] by det A.

Remark 1.4 ([9],[6]). Let F_{m} be the free group of rank m. Then the representation ring A_{n}(F_{m}) is isomorphic to ℤ[M_{n}^{⊕m}]_{det A}.

Definition 1.5 ([9],[6]). Let Γ be a group. We define the action of the group scheme PGL_{n} over ℤ on the representation variety Rep_{n}(Γ) by

Rep_{n}(Γ) × PGL_{n} → Rep_{n}(Γ)
(ρ, P)  ↦ P^{−1}ρP.

Notation 1.6. Let R be a commutative ring. For A ∈ M_{n}(R), we define c_{i}(A) for 1 ≤ i ≤ n by

det(xI_{n} − A) = x^{n} − c_{1}(A)x^{n−1} + c_{2}(A)x^{n−2} − ⋯ + (−1)^{n}c_{n}(A).

In particular we denote the trace c_{1}(A) and the determinant c_{n}(A) by s(A) and d(A), respectively. Let Γ be a group. If σ_{Γ} : Γ → GL_{n}(A_{n}(Γ)) is the universal representation of degree n for Γ, then we denote c_{i}(σ_{Γ}(γ)) by c_{i}(γ) for γ ∈ Γ.

The group scheme PGL_{n} acts on the scheme of m matrices of size n × n by (A_{1}, …, A_{m}) ↦ (P^{−1}A_{1}P, …, P^{−1}A_{m}P). We quote an important theorem on the invariant subring ℤ[M_{n}^{⊕m}]^{PGL^{n}} of the coordinate ring ℤ[M_{n}^{⊕m}] by PGL_{n}.

Theorem 1.7 (S.Donkin [1]). The invariant ring ℤ[M_{n}^{⊕m}]^{PGL^{n}} is generated by the coefficients of the characteristic polynomials of all matrices A_{i_{1}}A_{i_{2}} ⋯ A_{i_{r}} for i_{1}, i_{2}, …, i_{r} ∈ {1, 2, ⋯, m} and r ∈ ℕ over ℤ.

By the above theorem and Remark 1.4 we have the following corollary.

Corollary 1.8. For the free group F_{m} of rank m, the invariant subring A_{n}(F_{m})^{PGL^{n}} of the representation ring A_{n}(F_{m}) by PGL_{n} is generated by {c_{i}(γ) | γ ∈ F_{m}, 1 ≤ i ≤ n} over ℤ.

In [9] Kyoji Saito defined the Saito character ring of degree 2, which was called by him “character ring” simply. This ring was constructed as an approximation of the invariant subring of the representation ring. In the present paper, we define the Saito character ring of degree 3 and higher degree. Before we define the Saito character ring of degree 3, we introduce an easy lemma.

Lemma 1.9. Let R be a commutative ring. For A ∈ GL_{n}(R), we have

c_{n−1}(A) = d(A)s(A^{−1}).

Proof. It suffices to prove that the statement is true when R is the representation ring A_{n}(F_{1}) and A is the universal matrix because of Remark 1.2. The ring A_{n}(F_{1}) is isomorphic to ℤ[M_{n}]_{det(A)}, and it can be embedded into the complex number field ℂ. Therefore we can assume that A ∈ GL_{n}(ℂ). Let α_{1}, ⋯, α_{n} be eigenvalues of A. We see that c_{n−1}(A) = Σ_{i=1}^{n}α_{1} ⋯ α̂_{i} ⋯ α_{n} and that d(A)s(A^{−1}) = α_{1} ⋯ α_{n}(α_{1}^{−1} + ⋯ + α_{n}^{−1}). Thus we have proved the equality.

Notation 1.10. In the degree 3 case, we simply denote c_{2}(A) by c(A). For A ∈ M_{3}(R), we have

det(xI_{3} − A) = x^{3} − s(A)x^{2} + c(A)x − d(A).

From the above lemma we have c(A) = d(A)s(A^{−1}) for A ∈ GL_{3}(R).

Here we define the Saito character ring of degree 3.

Definition 1.11. Let Γ be a group and let e be the identity element of Γ. We denote by Γ^{ab} the group Γ/[Γ, Γ]. For γ ∈ Γ, we denote by d(γ) the equivalence class containing γ in Γ^{ab}. Then we define the Saito character ring of degree 3 for Γ by

R_{3}(Γ) := ℤΓ^{ab}[s(γ) | γ ∈ Γ]/I(Γ),

where I(Γ) is the ideal generated by the following elements:

s(e) − 3, (1) s(γδ) − s(δγ), (2) s(γ^{2}δ) − s(γ)s(γδ) + d(γ)s(γ^{−1})s(δ) − d(γ)s(γ^{−1}δ), (3)

for all γ, δ ∈ Γ. Here by ℤΓ^{ab}[s(γ) | γ ∈ Γ] we understand the polynomial ring over ℤΓ^{ab}. We regard s(γ) as an independent variable for each γ ∈ Γ. In the Saito character ring R_{3}(Γ), we define c(γ) by

c(γ) := d(γ)s(γ^{−1})

for each γ ∈ Γ.

Here let us explain the relations (1)–(3) in the Saito character ring. Let R be a commutative ring and Γ a group. If ρ : Γ → GL_{3}(R) is a representation of degree 3 for Γ, then we have the following equalities:

s(e) = 3, (4) s(γδ) = s(δγ), (5) s(γ^{2}δ) − s(γ)s(γδ) + d(γ)s(γ^{−1})s(δ) − d(γ)s(γ^{−1}δ) = 0, (6)

for each γ, δ ∈ Γ. Here we simply denote s(ρ(γ)) and d(ρ(γ)) by s(γ) and d(γ), respectively. It is obvious that the equalities (4) and (5) hold. Let us prove the equality (6). Put A := ρ(γ) and B := ρ(δ). By the Cayley-Hamilton theorem and Lemma 1.9 we have

A^{3} − s(A)A^{2} + d(A)s(A^{−1})A − d(A)I_{3} = 0.

Multiplying A^{−1}B from the right, we obtain

A^{2}B − s(A)AB + d(A)s(A^{−1})B − d(A)A^{−1}B = 0.

By taking the traces we get the equality (6). The relations (1)–(3) in the Saito character ring come from the equalities (4)–(6).

We define the canonical map from the Saito character ring to the representation ring by

Φ : R_{3}(Γ) → A_{3}(Γ)
d(γ) ↦ det(σ_{Γ}(γ))
s(γ) ↦ tr(σ_{Γ}(γ)),

where σ_{Γ} is the universal representation of degree 3 for Γ. From Donkin’s theorem (Theorem 1.7) we see that ImΦ = A_{3}(Γ)^{PGL^{3}} for free groups Γ. However the author does not know whether Φ is injective. It is still open whether all relations in the invariant ring can be deduced from the relations (1), (2), and (3).

We give an easy example of the Saito character ring of degree 3.

Example 1.12 (The case Γ = F_{1}). Let F_{1} be the free group of rank one, and let α be a generator of F_{1}. Then the elements s(α), c(α), and d(α) are algebraically independent in R_{3}(F_{1}). The Saito character ring R_{3}(F_{1}) is isomorphic to

ℤ[s(α), c(α), d(α), d(α)^{−1}].

The ring R_{3}(F_{1}) can be regarded as a subring of A_{3}(F_{1}) through Φ.

We introduce the following notation for Definition 1.14.

Notation 1.13. We denote by s_{i} the i-th symmetric polynomial in ℤ[x_{1}, x_{2}, ⋯, x_{n}], that is, s_{i} := ∑_{1≤j_{1}<···<j_{i}≤n} x_{j_{1}} ⋯ x_{j_{i}}. By f_{m} we understand the unique polynomial such that x_{1}^{m} + x_{2}^{m} + ⋯ + x_{n}^{m} = f_{m}(s_{1}, s_{2}, …, s_{n}) for each positive integer m.

We now define the Saito character ring of every degree.

Definition 1.14. Let Γ be a group and let n be a positive integer. We define the Saito character ring of degree n for Γ by

R_{n}(Γ) := ℤ[s(γ), c_{2}(γ), ⋯, c_{n−1}(γ), d(γ) | γ ∈ Γ]/I(Γ),

where I(Γ) is the ideal of the polynomial ring ℤ[s(γ), c_{2}(γ), ⋯, c_{n−1}(γ), d(γ) | γ ∈ Γ] over ℤ generated by the next elements:

s(e) − n, d(e) − 1, c_{i}(e) − \binom{n}{i}, (7) d(αβ) − d(α)d(β), (8) s(αβ) − s(βα), c_{i}(αβ) − c_{i}(βα), (9) s(α^{n}β) − s(α)s(α^{n−1}β) + ∑_{2≤i≤n−1}(−1)^{i}c_{i}(α)s(α^{n−i}β) (10)

+(−1)^{n}d(α)s(β),

s(α^{m}) − f_{m}(s(α), c_{2}(α), ⋯, d(α)), (11)

for α, β ∈ Γ, m ∈ ℕ_{+}, and 2 ≤ i ≤ n − 1.

The following proposition implies that the Saito character ring for an arbitrary group is a quotient of the one for a free group.

Proposition 1.15. Let Γ be a group. Suppose that Γ has a presentation Γ = F_{I}/R, where F_{I} is a free group with a set of free generators I and R is a set of fundamental relations. Then we have

R_{n}(Γ) ≅ R_{n}(F_{I})/(s(xδ) − s(x), c_{i}(xδ) − c_{i}(x), d(δ) − 1 | x ∈ F_{I}, δ ∈ R).

Proof. Put A := R_{n}(F_{I})/(s(xδ) − s(x), c_{i}(xδ) − c_{i}(x), d(δ) − 1 | x ∈ F_{I}, δ ∈ R). Let π : F_{I} → Γ be the canonical projection. The map π induces the ring homomorphism π_{∗} : R_{n}(F_{I}) → R_{n}(Γ). Then π_{∗} factors through A and we obtain the ring homomorphism \overline{π_{∗}} : A → R_{n}(Γ).

Now we take a section of π : F_{I} → Γ as a morphism of sets. Call it ρ. Then we define the ring homomorphism ρ_{∗} by

ρ_{∗} : ℤ[s(γ), c_{2}(γ), …, d(γ) | γ ∈ Γ] →  A
s(γ) ↦ s(ρ(γ))
c_{i}(γ) ↦ c_{i}(ρ(γ))
d(γ) ↦ d(ρ(γ)).

We show that ρ_{∗} does not depend on choices of sections ρ : Γ → F_{I}. For γ ∈ Γ we pick α, α′ ∈ F_{I} such that π(α) = π(α′) = γ. Then we can write α = α′ ∏_{i=1}^{t} g_{i}^{−1}δ_{i}g_{i}, where δ_{i} ∈ R and g_{i} ∈ F_{I}. In A we have

s(α) = s(α′ ∏_{i=1}^{t} g_{i}^{−1}δ_{i}g_{i})

= s(g_{t}(α′ ∏_{i=1}^{t−1} g_{i}^{−1}δ_{i}g_{i})g_{t}^{−1}δ_{t})

= s(g_{t}(α′ ∏_{i=1}^{t−1} g_{i}^{−1}δ_{i}g_{i})g_{t}^{−1})

= s(α′ ∏_{i=1}^{t−1} g_{i}^{−1}δ_{i}g_{i}).

By induction on t we obtain s(α) = s(α′). We can also prove that c_{i}(α) = c_{i}(α′) and that d(α) = d(α′) in a similar way. Hence we see that ρ_{∗} does not depend on choices of ρ.

It is easy to see that Kerρ_{∗} contains all elements in (7)–(11). Therefore we obtain \overline{ρ_{∗}} : R_{n}(Γ) → A. Since \overline{π_{∗}} ∘ \overline{ρ_{∗}} = 1 and \overline{ρ_{∗}} ∘ \overline{π_{∗}} = 1, we have proved the statement.

In the sequel we will deal with the Saito character ring of degree 3 mainly. Before we enter the next section, we show that Definition 1.11 is compatible with Definition 1.14 in the degree 3 case.

Proposition 1.16. The Saito character ring of degree 3 defined in Definition 1.11 is isomorphic to the one defined in Definition 1.14.

Proof. Throughout this proof, we put tildes on the symbols of the Saito character ring of degree 3 defined in Definition 1.14. We define the ring homomorphism ϕ : R_{3}(Γ) → R̃_{3}(Γ) by s(γ) ↦ s̃(γ) and d(γ) ↦ d̃(γ). We show that ϕ is an isomorphism.

To prove that ϕ is surjective, it suffices to show that c̃(γ) ∈ Imϕ for each γ ∈ Γ. From the relation (11) with m = 2, we have

s̃(γ^{2}) = s̃(γ)^{2} − 2c̃(γ).

By the relations (10) and (7), we obtain

s̃(γ^{2}) − s̃(γ)^{2} + 3c̃(γ) − d̃(γ)s̃(γ^{−1}) = 0.

Then from these equalities, we have c̃(γ) = d̃(γ)s̃(γ^{−1}). Therefore c̃(γ) ∈ Imϕ.

To prove that ϕ is injective, it suffices to show that the relation (11) holds in R_{3}(Γ). First we consider the case Γ = F_{1}. Let α be a generator of F_{1}. The relation (11) holds in A_{3}(F_{1}), and hence it also holds in the subring R_{3}(F_{1}) (cf. Example 1.12). In general, for each γ ∈ Γ, let us consider the group homomorphism ψ : F_{1} → Γ sending α to γ. The map ψ induces the ring homomorphism from R_{3}(F_{1}) to R_{3}(Γ). Since the relation (11) holds in the case Γ = F_{1}, so does it in the arbitrary group case. Thus we have proved the proposition.

By the above proposition, we identify the Saito character ring of degree 3 defined in Definition 1.11 with the one defined in Definition 1.14.

1.2. Generators of the Saito character ring of degree 3. In this section, we obtain a set of generators of the Saito character ring of degree 3 over ℤ. We choose finitely many generators of the Saito character ring of degree 3 for F_{2} over ℤ.

Proposition 1.17. Let Γ be a group with a subset G generating Γ. Then the Saito character ring R_{3}(Γ) is generated by

∪_{n∈ℕ}{s(α_{1}^{ε_{1}} ⋯ α_{n}^{ε_{n}}) | α_{i} ∈ G, α_{i} ≠ α_{i+1}, ε_{i} ∈ {−1, 1}}

over ℤΓ^{ab}.

Proof. For γ ∈ Γ we write γ = α_{1}^{l_{1}} ⋯ α_{n}^{l_{n}}, where α_{i} ∈ G, α_{i} ≠ α_{i+1}, and l_{i} ∈ ℤ. We claim that s(γ) is contained in the subring generated by the above set in the statement over ℤΓ^{ab}. We show the claim by induction on the number m := ♯{i | l_{i} ∉ {−1, 1}}.

For the case m = 0 the claim is obvious. Suppose that the claim is true for the case m − 1. We now consider the case m. By the equality (5), we may assume that s(γ) = s(α_{1}^{l_{1}}δ), where l_{1} ∉ {−1, 1}, δ = α_{2}^{l_{2}} ⋯ α_{n}^{l_{n}} ∈ Γ, and ♯{i | l_{i} ∉ {−1, 1}, i = 2, …, n} = m − 1. By the equality (6), the next equality holds for each integer l:

s(α_{1}^{l}δ) = s(α_{1})s(α_{1}^{l−1}δ) − d(α_{1})s(α_{1}^{−1})s(α_{1}^{l−2}δ) + d(α_{1})s(α_{1}^{l−3}δ).

From the assumption for the case m − 1, we see that s(α_{1}^{−1}δ), s(δ), and s(α_{1}δ) are contained in the subring generated by the above set. By induction on l_{1} we also see that s(α_{1}^{l_{1}}δ) is contained in the subring. Thus we have shown the claim.

Corollary 1.18. Let Γ be a group with a subset G generating Γ. Then the Saito character ring R_{3}(Γ) is generated by {d(α)^{±1} | α ∈ G} and ∪_{n∈ℕ}{s(α_{1}^{ε_{1}} ⋯ α_{n}^{ε_{n}}) | α_{i} ∈ G, α_{i} ≠ α_{i+1}, ε_{i} ∈ {−1, 1}} over ℤ.

Proof. The ring ℤΓ^{ab} is generated by {d(α)^{±1} | α ∈ G} over ℤ. The statement follows from the previous proposition.

In a similar way we obtain the next proposition.

Proposition 1.19. Let Γ be a group with a subset G generating Γ. Then the Saito character ring R_{3}(Γ) is generated by

∪_{n∈ℕ}{s(α_{1}^{ε_{1}} ⋯ α_{n}^{ε_{n}}) | α_{i} ∈ G, α_{i} ≠ α_{i+1}, ε_{i} ∈ {1, 2}} ∪ {c(α) | α ∈ G}

over ℤΓ^{ab}.

We now introduce some terminologies.

Definition 1.20. Let Γ be a group with a subset G generating Γ, and let γ ∈ Γ. We say that γ has length n if γ can be expressed as α_{1}^{l_{1}} ⋯ α_{n}^{l_{n}}, where α_{i} ∈ G, α_{i} ≠ α_{i+1}, and l_{i} ∈ ℤ.

Let J be {−1, 1} or {1, 2}. We say that γ has J-length n if γ can be expressed as α_{1}^{ε_{1}} ⋯ α_{n}^{ε_{n}}, where α_{i} ∈ G, α_{i} ≠ α_{i+1}, and ε_{i} ∈ J.

We say that γ has s-length n if s(γ) is contained in the subring of R_{3}(Γ) generated by {s(δ) | δ ∈ Γ, J-length (δ) ≤ n} ∪ {c(α) | α ∈ G} over ℤΓ^{ab}.

Remark 1.21. Suppose that ♯G ≥ 2. From the definition, if γ has length n − 1 (resp. s-length n − 1), then γ has length n (resp. s-length n). By the discussion in the proof of Proposition 1.17, we see that if γ has length n, then γ has s-length n.

From now on we restrict ourselves to the case Γ = F_{2}. We determine finitely many generators of R_{3}(F_{2}) over ℤ.

Definition 1.22. Let F_{2} be the free group of rank 2 with a subset of free generators G := {α, β}. Let J be {−1, 1} or {1, 2}. Suppose that γ ∈ F_{2} has J-length n. Then γ ( or s(γ) ) is called reducible if γ has s-length n − 1.

Remark 1.23. If γ is α^{ε_{1}}β^{ε_{2}} ⋯ α^{ε_{2m−1}}β^{ε_{2m}}α^{ε_{2m+1}} or β^{ε_{1}}α^{ε_{2}} ⋯ β^{ε_{2m−1}}× α^{ε_{2m}}β^{ε_{2m+1}}, then s(γ) = s(α^{ε_{1}+ε_{2m+1}}β^{ε_{2}} ⋯ α^{ε_{2m−1}}β^{ε_{2m}}) or s(β^{ε_{1}+ε_{2m+1}} × α^{ε_{2}} ⋯ β^{ε_{2m−1}}α^{ε_{2m}}), respectively. Therefore if γ has J-length 2m + 1, then γ is reducible.

In the next lemma we investigate the case that γ has J-length 2m with J = {−1, 1}.

Lemma 1.24. Let J = {−1, 1}. Let γ = α^{ε^{1}}β^{ε^{2}} ⋯ α^{ε^{2m−1}}β^{ε^{2m}} ∈ F_{2} with ε_{i} ∈ J.

(i) If there exists i such that ε_{i} = ε_{i+2}, then s(γ) is reducible. (ii) If γ has J-length 2m with m ≥ 4, then either there exists i such

that ε_{i} = ε_{i+2} and hence s(γ) is reducible, or γ contains a subsequence αεβ^{ε′}α^{−ε}β^{−ε′}αεβ^{ε′}α^{−ε}β^{−ε′} with ε, ε′ ∈ J.

(iii) The Saito character ring R_{3}(F_{2}) is finitely generated by {s(γ) | γ ∈ F_{2}, J-length(γ) ≤ 6} over ℤF_{2}^{ab}.

Proof. (i) Suppose that γ = α^{ε^{1}}β^{ε^{2}} ⋯ α^{ε^{2m−1}}β^{ε^{2m}} with ε_{i} = ε_{i+2}. By using the equality (5), we may assume that ε_{1} = ε_{3} without loss of generality (if necessary, change α and β). Then we have γ = (α^{ε^{1}}β^{ε^{2}})^{2}β^{−ε_{2}+ε_{4}}α^{ε^{5}}β^{ε^{6}} ⋯ α^{ε_{2m−1}}β^{ε_{2m}}. Put δ = α^{ε^{5}}β^{ε^{6}} ⋯ α^{ε_{2m−1}}β^{ε_{2m}}. Since

s(γ) = s((α^{ε^{1}}β^{ε^{2}})^{2}β^{−ε_{2}+ε_{4}}δ)
= s(α^{ε^{1}}β^{ε^{2}})s(α^{ε^{1}}β^{ε^{2}}β^{−ε_{2}+ε_{4}}δ) − d(α^{ε^{1}}β^{ε^{2}})s((α^{ε^{1}}β^{ε^{2}})^{−1})×
s(β^{−ε_{2}+ε_{4}}δ) + d(α^{ε^{1}}β^{ε^{2}})s(α^{−ε_{1}}β^{−ε_{2}+ε_{4}}α^{ε^{5}} ⋯ β^{ε_{2m}−ε_{2}}),

we see that s(γ) is reducible.

(ii) For γ = α^{ε^{1}}β^{ε^{2}}α^{ε^{3}}β^{ε^{4}}α^{ε^{5}}β^{ε^{6}}α^{ε^{7}}β^{ε^{8}} ⋯ α^{ε^{2m−1}}β^{ε^{2m}} with m ≥ 4, assume that ε_{i} ≠ ε_{i+2} for any i. Then we have ε_{1} = −ε_{3} = ε_{5} = −ε_{7} and ε_{2} = −ε_{4} = ε_{6} = −ε_{8}. Therefore γ contains a subsequence α^{ε^{1}}β^{ε^{2}}α^{−ε_{1}}β^{−ε_{2}}α^{ε^{1}}β^{ε^{2}}α^{−ε_{1}}β^{−ε_{2}}.

(iii) By Proposition 1.17 and Remark 1.23 it suffices to show that s(γ) is reducible for any γ ∈ F_{2} with J-length 2m ≥ 8. We may assume that the top of the expression of γ is α because of symmetry. Moreover from (ii) we may assume that γ contains a subsequence α^{ε^{1}}β^{ε^{2}}α^{−ε_{1}}β^{−ε_{2}}α^{ε^{1}}β^{ε^{2}}α^{−ε_{1}}β^{−ε_{2}}. Then s(γ) = s((α^{ε^{1}}β^{ε^{2}}α^{−ε_{1}}β^{−ε_{2}})^{2}δ), where δ = α^{ε^{9}} ⋯ α^{ε^{2m−1}}β^{ε^{2m}}. Therefore we can prove that s(γ) is reducible in the same way as (i). Thus we have proved the lemma.

From the above lemma, we obtain a set of generators of R_{3}(F_{2}) as an algebra over ℤ.

Proposition 1.25. The next elements generate R_{3}(F_{2}) as an algebra over ℤ:

d(α)^{±1}, d(β)^{±1}, s(α^{±1}), s(β^{±1}), s(αβ), s(αβ^{−1}), s(α^{−1}β), s(α^{−1}β^{−1}), s(αβα^{−1}β^{−1}), s(αβ^{−1}α^{−1}β).

Proof. From Lemma 1.24, we have only to consider γ ∈ F_{2} with J-length ≤ 6. For γ ∈ F_{2} with J-length(γ) = 6, we may assume that γ = α^{ε^{1}}β^{ε^{2}}α^{ε^{3}}β^{ε^{4}}α^{ε^{5}}β^{ε^{6}} without loss of generality. Then at least one of ε_{1} = ε_{3}, ε_{3} = ε_{5}, and ε_{5} = ε_{1} holds. Hence we see that s(γ) is reducible by Lemma 1.24 (i) and the equality (5). Therefore any element with J-length 6 is reducible. In a similar way, any element with J-length 4 except s(αβα^{−1}β^{−1}) and s(αβ^{−1}α^{−1}β) is reducible. Thus we have proved the statement.

We also obtain another set of generators of R_{3}(F_{2}).

Corollary 1.26. The next elements generate R_{3}(F_{2}) as an algebra over ℤ:

d(α)^{±1}, d(β)^{±1}, s(α), s(β), c(α), c(β), s(αβ), s(αβ^{2}), s(α^{2}β), s(α^{2}β^{2}), s(αβα^{2}β^{2}), s(αβ^{2}α^{2}β).

Proof. We recall that c(γ) = d(γ)s(γ^{−1}). By the equalities s(α^{2}β) = s(α)s(αβ) − c(α)s(β) + d(α)s(α^{−1}β) and so on, it is easy to prove the statement from the above proposition.

We have seen that R_{3}(F_{2}) is a finitely generated algebra over ℤ. However the author does not know whether R_{3}(Γ) is a finitely generated algebra over ℤ for any finitely generated group Γ.

1.3. Generators of ℤ[M_{3} × M_{3}]^{PGL^{3}}. In the previous section, we have obtained sets of generators of R_{3}(F_{2}) as an algebra over ℤ. Now we consider generators of ℤ[M_{3} × M_{3}]^{PGL^{3}}. In R_{3}(F_{2}) the element d(γ) is invertible, however it is not invertible in ℤ[M_{3} × M_{3}]^{PGL^{3}}. For the difference between R_{3}(F_{2}) and ℤ[M_{3} × M_{3}]^{PGL^{3}} we need to introduce the following definition.

Definition 1.27. Let F_{m} be the free group of rank m. For the Saito character ring R_{n}(F_{m}) and the representation ring A_{n}(F_{m}), we define the canonical map Φ : R_{n}(F_{m}) → A_{n}(F_{m}) by s(γ) ↦ tr(σ_{F_{m}}(γ)), d(γ) ↦ det(σ_{F_{m}}(γ)), and c_{i}(γ) ↦ c_{i}(σ_{F_{m}}(γ)) for γ ∈ F_{m}, 2 ≤ i ≤ n−1. Here we denote by σ_{F_{m}} the universal representation of degree n for F_{m}.

Recall that A_{n}(F_{m}) is isomorphic to ℤ[M_{n}^{⊕m}]_{det A} (cf. Remark 1.4). For x ∈ R_{n}(F_{m}), we say that x is integral if Φ(x) is contained in the subring ℤ[M_{n}^{⊕m}]^{PGL^{n}} of ℤ[M_{n}^{⊕m}]_{det A} ≅ A_{n}(F_{m}).

Under the above definition, we get the next proposition.

Proposition 1.28. Let {α_{1}, α_{2}, …, α_{m}} be a system of free generators of F_{m}. Let x = d(α_{1}^{i_{1}}α_{2}^{i_{2}} ⋯ α_{m}^{i_{m}})s(α_{k_{1}}^{l_{1}}α_{k_{2}}^{l_{2}} ⋯ α_{k_{r}}^{l_{r}}) ∈ R_{n}(F_{m}). For 1 ≤ p ≤ m, we define j_{p} := ∑_{{q|k_{q}=p,l_{q}<0}} l_{q}. If i_{p} + j_{p} ≥ 0 for 1 ≤ p ≤ m, then x is integral.

Proof. The proof will be done by induction on the number t := −(∑_{p=1}^{m} j_{p}). In the case t = 0, we see that x is integral, since l_{q} ≥ 0 for 1 ≤ q ≤ r and i_{p} ≥ 0 for 1 ≤ p ≤ m. Suppose that x is integral until the case t − 1. We now consider the case t > 0. From t = −(∑_{p=1}^{m} j_{p}) > 0, we may assume that j_{1} < 0, k_{1} = 1 and that l_{1} < 0 without loss of generality (if necessary, use the relation (9) in Definition 1.14). Since i_{1} + j_{1} ≥ 0, we have i_{1} > 0. Then x = d(γ)d(α_{1})s(α_{1}^{−1}δ), where γ = α_{1}^{i_{1}−1}α_{2}^{i_{2}} ⋯ α_{m}^{i_{m}} and δ = α_{1}^{l_{1}+1}α_{k_{2}}^{l_{2}} ⋯ α_{k_{r}}^{l_{r}}. From the relation (10) in Definition 1.14, we have

d(γ)d(α_{1})s(α_{1}^{−1}δ) = (−1)^{n−1}d(γ){s(α_{1}^{n−1}δ) − s(α_{1})s(α_{1}^{n−2}δ)
+c_{2}(α_{1})s(α_{1}^{n−3}δ) − ⋯
⋯ + (−1)^{n−1}c_{n−1}(α_{1})s(δ)}.

By induction we see that the elements d(γ)s(α_{1}^{n−1}δ), d(γ)s(α_{1}^{n−2}δ), d(γ)s(α_{1}^{n−3}δ), …, and d(γ)s(δ) are integral. Therefore x is integral, and hence we have proved the proposition.

Corollary 1.29. With the same hypotheses as Proposition 1.28, the element x can be expressed as a polynomial in s(α_{q_{1}}α_{q_{2}} ⋯ α_{q_{r}}), c_{2}(α_{p}), ⋯, c_{n−1}(α_{p}), d(α_{p}) for q_{1}, …, q_{r}, p ∈ {1, 2, …, m} and r ∈ ℕ.

Proof. From the proof of Proposition 1.28, the statement can be easily proved.

Here we introduce several examples of integral elements of R_{n}(F_{m}). It is useful for understanding what Proposition 1.28 means.

Example 1.30. We have many integral elements of R_{n}(F_{m}) by using Proposition 1.28: d(α_{1})s(α_{1}^{−1}), d(α_{1}^{2}α_{2})s(α_{1}^{−1}α_{2}^{−1}α_{3}), d(α_{1}^{4}α_{2}^{5}α_{3}^{2}) s(α_{1}α_{2}^{−3}α_{1}^{−1}α_{3}^{−1}α_{2}^{−1}), and so on. For example, we obtain

d(αβ)s(α^{−1}β^{−1}) =

s(α^{2}β^{2}) − s(α)s(αβ^{2}) − s(β)s(α^{2}β) − c(α)c(β) + s(α)s(β)s(αβ).

Now we investigate the invariant ring ℤ[M_{3}×M_{3}]^{PGL^{3}}. The following proposition gives us a set of generators of ℤ[M_{3}×M_{3}]^{PGL^{3}} as an algebra over ℤ. Through the canonical map Φ : R_{3}(F_{2}) → A_{3}(F_{2}), we identify x ∈ R_{3}(F_{2}) with Φ(x) ∈ A_{3}(F_{2}).

Proposition 1.31. The next elements generate ℤ[M_{3} ×M_{3}]^{PGL^{3}} as an algebra over ℤ:

d(α), d(β), s(α), s(β), c(α), c(β), s(αβ), s(αβ^{2}), s(α^{2}β), s(α^{2}β^{2}),

s(αβα^{2}β^{2}), s(αβ^{2}α^{2}β).

Proof. Let A be the subalgebra of ℤ[M_{3} × M_{3}]^{PGL^{3}} generated by the 12 elements above d(α), d(β), s(α), …, s(αβ^{2}α^{2}β) over ℤ. It suffices to prove that any coefficients of characteristic polynomials of matrices are contained in A (cf. Theorem 1.7). It is clear that any determinants of matrices are contained in A.

First we prove that any traces of matrices s(α^{l_{1}}β^{l_{2}} ⋯ α^{l_{2r−1}}β^{l_{2r}}) with l_{i} ≥ 0 are contained in A. As in the proofs of Propositions 1.17 and 1.19, we have only to show that s(α^{ε^{1}}β^{ε^{2}} ⋯ α^{ε^{2m−1}}β^{ε^{2m}}) ∈ A for ε_{i} ∈ {1, 2}. For proving this, we need the next lemma:

Lemma 1.32. Let J = {1, 2}. Let γ = α^{ε^{1}}β^{ε^{2}} ⋯ α^{ε^{2m−1}}β^{ε^{2m}} ∈ F_{2} with ε_{i} ∈ J.

(i) Suppose that there exists i such that ε_{i} = ε_{i+2}. Then s(γ) is reducible. Moreover if m ≥ 3, then s(γ) can be expressed as a polynomial in integral elements s(δ) with J-length(δ) ≤ (2m − 2) and {c(α), c(β), d(α), d(β)} in ℤ[M_{3} × M_{3}]^{PGL^{3}} or R_{3}(F_{2}).

(ii) If m ≥ 4, then there exists i such that ε_{i} = ε_{i+2}, or γ contains a subsequence α^{ε^{1}}β^{ε^{2}}α^{ε′^{1}}β^{ε′^{2}}α^{ε^{1}}β^{ε^{2}}α^{ε′^{1}}β^{ε′^{2}} such that ε_{1}, ε_{2}, ε′_{1}, ε′_{2} ∈ J and ε_{1} ≠ ε′_{1}, ε_{2} ≠ ε′_{2}.

(iii) In particular, if δ ∈ F_{2} has J-length 2m ≥ 8, then s(δ) ∈ A.

Proof of Lemma 1.32. We can prove this lemma in the same way as Lemma 1.24.

(i) In the case m = 2, we can prove that s(γ) is reducible directly. For example, if γ = αβαβ, then we have

s(γ) = s(αβ)^{2} − 2d(αβ)s(α^{−1}β^{−1})
= s(αβ)^{2} − 2s(α^{2}β^{2}) + 2s(α)s(αβ^{2})
+2s(β)s(α^{2}β) − 2s(α)s(β)s(αβ) + 2c(α)c(β).

Assume that m ≥ 3. Let C denote the subalgebra of ℤ[M_{3} × M_{3}]^{PGL^{3}} ( or R_{3}(F_{2}) ) generated by {s(δ) | J-length(δ) ≤ 2m−2} and {c(α), c(β), d(α), d(β)}. If ε_{1} = ε_{3}, then γ = (α^{ε^{1}}β^{ε^{2}})^{2}β^{−ε_{2}+ε_{4}}α^{ε^{5}}β^{ε^{6}} ⋯ α^{ε_{2m−1}}β^{ε_{2m}}.

Put δ = α^{ε^{5}}β^{ε^{6}} ⋯ α^{ε_{2m−1}}β^{ε_{2m}}. Then we obtain

s(γ) = s((α^{ε^{1}}β^{ε^{2}})^{2}β^{−ε_{2}+ε_{4}}δ)
= s(α^{ε^{1}}β^{ε^{2}})s(α^{ε^{1}}β^{ε^{4}}δ) − d(α^{ε^{1}}β^{ε^{2}})s((α^{ε^{1}}β^{ε^{2}})^{−1})s(δβ^{−ε_{2}+ε_{4}})
+d(α^{ε^{1}}β^{ε^{2}})s(α^{−ε_{1}}β^{−ε_{2}+ε_{4}}α^{ε^{5}} ⋯ α^{ε^{2m−1}}β^{ε_{2m}−ε_{2}}).

The first term s(α^{ε^{1}}β^{ε^{2}})s(α^{ε^{1}}β^{ε^{4}}δ) of the right hand side is contained in C (cf. Remark 1.21). For the second term −d(α^{ε^{1}}β^{ε^{2}})s((α^{ε^{1}}β^{ε^{2}})^{−1}) s(δβ^{−ε_{2}+ε_{4}}), we can prove that d(α^{ε^{1}}β^{ε^{2}})s((α^{ε^{1}}β^{ε^{2}})^{−1}) is contained in the subalgebras A∩C as in the proof of Proposition 1.28. Since m ≥ 3, we have δβ^{−ε_{2}+ε_{4}} = α^{ε^{5}} ⋯ α^{ε_{2m−1}}β^{ε_{2m}−ε_{2}+ε_{4}} and ε_{2m} − ε_{2} + ε_{4} ≥ 0, and hence the second term is also contained in C.

In the third term d(α^{ε^{1}}β^{ε^{2}})s(α^{−ε_{1}}β^{−ε_{2}+ε_{4}}α^{ε^{5}} ⋯ α^{ε^{2m−1}}β^{ε_{2m}−ε_{2}}), we easily see that

the sum of exponents of β’s in d() + the sum of negative exponents of β’s in s() ≥ 0.

Therefore we see that the third term is integral, and that it is contained in C by using the discussion in the proof of Proposition 1.28. Thus we have proved (i).

(ii) It can be proved in the same way as Lemma 1.24 (ii).

(iii) It follows from (i) and (ii) and Proposition 1.28 (cf. Lemma 1.24).

We continue the proof of Proposition 1.31. From the above lemma, we can easily see that any s(α^{ε^{1}}β^{ε^{2}} ⋯ α^{ε^{2m−1}}β^{ε^{2m}}) are contained in A.

Next we prove that any c(α^{l_{1}}β^{l_{2}} ⋯ α^{l_{2r−1}}β^{l_{2r}}) with l_{i} ≥ 0 are contained in A. Since c(α^{l_{1}} ⋯ β^{l_{2r}}) = d(α^{l_{1}} ⋯ β^{l_{2r}})s((α^{l_{1}} ⋯ β^{l_{2r}})^{−1}), we see that c(α^{l_{1}}β^{l_{2}} ⋯ α^{l_{2r−1}}β^{l_{2r}}) ∈ A from Corollary 1.29. Therefore we have proved Proposition 1.31.

We also obtain another set of generators of ℤ[M_{3} × M_{3}]^{PGL^{3}}.

Corollary 1.33. The next elements generate ℤ[M_{3} × M_{3}]^{PGL^{3}} as an algebra over ℤ:

d(α), d(β), s(α), s(β), c(α), c(β), s(αβ), s(αβ^{2}), s(α^{2}β), c(αβ),
s(αβα^{2}β^{2}), s(αβ^{2}α^{2}β).

Moreover we may change s(αβ^{2}) to d(β)s(αβ^{−1}), or we may change s(α^{2}β) to d(α)s(α^{−1}β).

2. Relations among generators in ℤ[M_{3} × M_{3}]^{PGL^{3}}

2.1. The structure of ℤ[M_{3} × M_{3}]^{PGL^{3}}. In the previous section, we obtained sets of generators of ℤ[M_{3} × M_{3}]^{PGL^{3}}. In this section, we find relations among generators of ℤ[M_{3} × M_{3}]^{PGL^{3}}, and determine the structure of ℤ[M_{3} × M_{3}]^{PGL^{3}}.

We introduce the main theorem in this paper.

Theorem 2.1. The invariant ring ℤ[M_{3} × M_{3}]^{PGL^{3}} is isomorphic to

ℤ [ \begin{matrix}d(α), d(β), s(α), s(β), c(α), c(β), s(αβ), s(αβ^{2}), \\ s(α^{2}β), s(α^{2}β^{2}), s(αβα^{2}β^{2}) \end{matrix} ] /I,

where I is the principal ideal of the polynomial ring in 11 variables over ℤ generated by

s(αβα^{2}β^{2})^{2} − P (α, β)s(αβα^{2}β^{2}) + Q(α, β).

Here we denote by P (α, β) and Q(α, β) the polynomials in d(α), d(β), s(α), s(β), c(α), c(β), s(αβ), s(αβ^{2}), s(α^{2}β), and s(α^{2}β^{2}) as seen later. In particular, the invariant ring ℤ[M_{3} × M_{3}]^{PGL^{3}} is isomorphic to the coordinate ring of a hypersurface of 𝔸_{ℤ}^{11}.

Proof. We put B := {d(α), d(β), s(α), s(β), c(α), c(β), s(αβ), s(αβ^{2}), s(α^{2}β), s(α^{2}β^{2})}. In ℤ[M_{3} × M_{3}]^{PGL^{3}}, we have the following two relations:

s(αβα^{2}β^{2}) + s(αβ^{2}α^{2}β) = P (α, β), (12) s(αβα^{2}β^{2})s(αβ^{2}α^{2}β) = Q(α, β), (13)

where P (α, β), Q(α, β) are polynomials in the 10 elements of B as seen later. We will prove the relations (12) and (13) in the next section. From Proposition 1.31 and the equality (12), we see that ℤ[M_{3} ×M_{3}]^{PGL^{3}} is generated by B and s(αβα^{2}β^{2}) over ℤ. By using the equalities (12) and (13) we obtain the surjective morphism

ϕ : ℤ[B, s(αβα^{2}β^{2})]/I → ℤ[M_{3} × M_{3}]^{PGL^{3}}.

We remark that the 10 elements in B are algebraically independent in ℤ[M_{3}×M_{3}]^{PGL^{3}} since dim ℚ[M_{3}×M_{3}]^{PGL^{3}} = dim M_{3}×M_{3}−dim PGL_{3} = 10. The polynomial ring ℤ[B] can be regarded as a subring of ℤ[M_{3} × M_{3}]^{PGL^{3}}.

For proving the injectivity of ϕ, we have only to show that s(αβα^{2}β^{2}) is not contained in the subring ℤ[B] of ℤ[M_{3} × M_{3}]^{PGL^{3}}. Indeed, let cs(αβα^{2}β^{2}) + d ∈ Kerϕ with c, d ∈ ℤ[B]. If c ≠ 0, then s(αβα^{2}β^{2}) = −d/c is contained in the the subfield ℚ(B) of the quotient field of ℤ[M_{3}×M_{3}]^{PGL^{3}}. Because s(αβα^{2}β^{2}) is integral over the normal domain ℤ[B], we have s(αβα^{2}β^{2}) ∈ ℤ[B], which is a contradiction. Hence we get c = d = 0, and we can prove that Kerϕ = 0.

To prove that s(αβα^{2}β^{2}) is not contained in ℤ[B], we consider the following two representations of the free group F_{2} of rank 2:

ρ_{1} : F_{2} → GL_{3}(ℚ) ρ_{2} : F_{2} → GL_{3}(ℚ)
α ↦ A α ↦ A
β ↦ B_{1},  β ↦ B_{2},

where

A = ( \begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix} ) , B_{1} = ( \begin{matrix}1 & 1 & −1 \\ 1 & −1 & 1 \\ −1 & −1 & −1 \end{matrix} ) , B_{2} = ( \begin{matrix}1 & −1 & 1 \\ −1 & −1 & −1 \\ 1 & 1 & −1 \end{matrix} ) .

Then we have s(ρ_{1}(α)) = s(ρ_{2}(α)), s(ρ_{1}(β)) = s(ρ_{2}(β)), …, and s(ρ_{1}(α^{2}β^{2})) = s(ρ_{1}(α^{2}β^{2})). However s(ρ_{1}(αβα^{2}β^{2})) ≠ s(ρ_{2}(αβα^{2}β^{2})). Therefore s(αβα^{2}β^{2}) is not contained in ℤ[B]. This completes the proof.

Here we introduce the explicit forms of P (α, β) and Q(α, β), although they are very complicated.

P (α, β)

= −3d(α)d(β) + d(α)s(β)c(β) + d(β)s(α)c(α) + s(α)s(β)c(α)c(β)

−s(αβ)s(α)^{2}s(β)^{2} + s(α)s(β)^{2}s(α^{2}β) + s(α)^{2}s(β)s(αβ^{2}) +s(α^{2}β)s(αβ^{2}) + s(αβ)s(α^{2}β^{2}) − s(α)s(β)s(α^{2}β^{2}).

Q(α, β)

= 9d(α)^{2}d(β)^{2} − 6d(α)^{2}d(β)c(β)s(β) + d(α)^{2}d(β)s(β)^{3}

+d(α)^{2}c(β)^{3} − 6d(α)d(β)^{2}c(α)s(α) + d(α)d(β)^{2}s(α)^{3} −6d(α)d(β)s(α^{2}β^{2})s(αβ) + 3d(α)d(β)s(α^{2}β^{2})s(α)s(β) −6d(α)d(β)s(α^{2}β)s(αβ^{2}) − 3d(α)d(β)s(α^{2}β)c(β)s(α) +6d(α)d(β)s(α^{2}β)s(αβ)s(β) − 2d(α)d(β)s(α^{2}β)s(α)s(β)^{2} −3d(α)d(β)s(αβ^{2})c(α)s(β) + 6d(α)d(β)s(αβ^{2})s(αβ)s(α) −2d(α)d(β)s(αβ^{2})s(α)^{2}s(β) + 6d(α)d(β)c(α)c(β)s(αβ) −2d(α)d(β)c(α)c(β)s(α)s(β) + d(α)d(β)c(α)s(αβ)s(β)^{2} +d(α)d(β)c(β)s(αβ)s(α)^{2} + d(α)d(β)s(αβ)^{3} −5d(α)d(β)s(αβ)^{2}s(α)s(β) + 2d(α)d(β)s(αβ)s(α)^{2}s(β)^{2} −3d(α)s(α^{2}β^{2})s(αβ^{2})c(β) + d(α)s(α^{2}β^{2})s(αβ^{2})s(β)^{2} −2d(α)s(α^{2}β^{2})c(β)^{2}s(α) + 4d(α)s(α^{2}β^{2})c(β)s(αβ)s(β) −d(α)s(α^{2}β^{2})s(αβ)s(β)^{3} + 4d(α)s(α^{2}β)s(αβ^{2})c(β)s(β) −d(α)s(α^{2}β)s(αβ^{2})s(β)^{3} + d(α)s(α^{2}β)c(β)^{2}s(αβ) +2d(α)s(α^{2}β)c(β)^{2}s(α)s(β) − 4d(α)s(α^{2}β)c(β)s(αβ)s(β)^{2} +d(α)s(α^{2}β)s(αβ)s(β)^{4} + d(α)s(αβ^{2})^{3} + 4d(α)s(αβ^{2})^{2}c(β)s(α) −2d(α)s(αβ^{2})^{2}s(αβ)s(β) − d(α)s(αβ^{2})^{2}s(α)s(β)^{2} +4d(α)s(αβ^{2})c(α)c(β)^{2} − d(α)s(αβ^{2})c(α)c(β)s(β)^{2} +2d(α)s(αβ^{2})c(β)^{2}s(α)^{2} + d(α)s(αβ^{2})c(β)s(αβ)^{2} −8d(α)s(αβ^{2})c(β)s(αβ)s(α)s(β) + d(α)s(αβ^{2})s(αβ)^{2}s(β)^{2} +2d(α)s(αβ^{2})s(αβ)s(α)s(β)^{3} + 2d(α)c(α)c(β)^{3}s(α) −4d(α)c(α)c(β)^{2}s(αβ)s(β) + d(α)c(α)c(β)s(αβ)s(β)^{3} −2d(α)c(β)^{2}s(αβ)s(α)^{2}s(β) − d(α)c(β)s(αβ)^{3}s(β) +4d(α)c(β)s(αβ)^{2}s(α)s(β)^{2} − d(α)s(αβ)^{2}s(α)s(β)^{4} +d(β)^{2}c(α)^{3} − 3d(β)s(α^{2}β^{2})s(α^{2}β)c(α) +d(β)s(α^{2}β^{2})s(α^{2}β)s(α)^{2} − 2d(β)s(α^{2}β^{2})c(α)^{2}s(β) +4d(β)s(α^{2}β^{2})c(α)s(αβ)s(α) − d(β)s(α^{2}β^{2})s(αβ)s(α)^{3} +d(β)s(α^{2}β)^{3} + 4d(β)s(α^{2}β)^{2}c(α)s(β) −2d(β)s(α^{2}β)^{2}s(αβ)s(α) − d(β)s(α^{2}β)^{2}s(α)^{2}s(β) +4d(β)s(α^{2}β)s(αβ^{2})c(α)s(α) − d(β)s(α^{2}β)s(αβ^{2})s(α)^{3} +4d(β)s(α^{2}β)c(α)^{2}c(β) + 2d(β)s(α^{2}β)c(α)^{2}s(β)^{2} −d(β)s(α^{2}β)c(α)c(β)s(α)^{2} + d(β)s(α^{2}β)c(α)s(αβ)^{2} −8d(β)s(α^{2}β)c(α)s(αβ)s(α)s(β) + d(β)s(α^{2}β)s(αβ)^{2}s(α)^{2} +2d(β)s(α^{2}β)s(αβ)s(α)^{3}s(β) + d(β)s(αβ^{2})c(α)^{2}s(αβ) +2d(β)s(αβ^{2})c(α)^{2}s(α)s(β) − 4d(β)s(αβ^{2})c(α)s(αβ)s(α)^{2} +d(β)s(αβ^{2})s(αβ)s(α)^{4} + 2d(β)c(α)^{3}c(β)s(β) −4d(β)c(α)^{2}c(β)s(αβ)s(α) − 2d(β)c(α)^{2}s(αβ)s(α)s(β)^{2} +d(β)c(α)c(β)s(αβ)s(α)^{3} − d(β)c(α)s(αβ)^{3}s(α) +4d(β)c(α)s(αβ)^{2}s(α)^{2}s(β) − d(β)s(αβ)^{2}s(α)^{4}s(β) +s(α^{2}β^{2})^{3} − 2s(α^{2}β^{2})^{2}s(α^{2}β)s(β) − 2s(α^{2}β^{2})^{2}s(αβ^{2})s(α) −5s(α^{2}β^{2})^{2}c(α)c(β) + s(α^{2}β^{2})^{2}c(α)s(β)^{2} + s(α^{2}β^{2})^{2}c(β)s(α)^{2} +s(α^{2}β^{2})^{2}s(αβ)s(α)s(β) + s(α^{2}β^{2})s(α^{2}β)^{2}c(β) +s(α^{2}β^{2})s(α^{2}β)^{2}s(β)^{2} + s(α^{2}β^{2})s(α^{2}β)s(αβ^{2})s(αβ) +2s(α^{2}β^{2})s(α^{2}β)s(αβ^{2})s(α)s(β) + 9s(α^{2}β^{2})s(α^{2}β)c(α)c(β)s(β)

−2s(α^{2}β^{2})s(α^{2}β)c(α)s(β)^{3} − s(α^{2}β^{2})s(α^{2}β)c(β)s(αβ)s(α) −2s(α^{2}β^{2})s(α^{2}β)c(β)s(α)^{2}s(β) − s(α^{2}β^{2})s(α^{2}β)s(αβ)s(α)s(β)^{2} +s(α^{2}β^{2})s(αβ^{2})^{2}c(α) + s(α^{2}β^{2})s(αβ^{2})^{2}s(α)^{2} +9s(α^{2}β^{2})s(αβ^{2})c(α)c(β)s(α) − s(α^{2}β^{2})s(αβ^{2})c(α)s(αβ)s(β) −2s(α^{2}β^{2})s(αβ^{2})c(α)s(α)s(β)^{2} − 2s(α^{2}β^{2})s(αβ^{2})c(β)s(α)^{3} −s(α^{2}β^{2})s(αβ^{2})s(αβ)s(α)^{2}s(β) + 8s(α^{2}β^{2})c(α)^{2}c(β)^{2} −2s(α^{2}β^{2})c(α)^{2}c(β)s(β)^{2} − 2s(α^{2}β^{2})c(α)c(β)^{2}s(α)^{2} +s(α^{2}β^{2})c(α)c(β)s(αβ)^{2} − 8s(α^{2}β^{2})c(α)c(β)s(αβ)s(α)s(β) +2s(α^{2}β^{2})c(α)s(αβ)s(α)s(β)^{3} + 2s(α^{2}β^{2})c(β)s(αβ)s(α)^{3}s(β) −s(α^{2}β)^{3}c(β)s(β) − s(α^{2}β)^{2}s(αβ^{2})c(β)s(α) − s(α^{2}β)^{2}c(α)c(β)^{2} −4s(α^{2}β)^{2}c(α)c(β)s(β)^{2} + s(α^{2}β)^{2}c(α)s(β)^{4} +2s(α^{2}β)^{2}c(β)s(αβ)s(α)s(β) + s(α^{2}β)^{2}c(β)s(α)^{2}s(β)^{2} −s(α^{2}β)s(αβ^{2})^{2}c(α)s(β) − 8s(α^{2}β)s(αβ^{2})c(α)c(β)s(α)s(β) +s(α^{2}β)s(αβ^{2})c(α)s(αβ)s(β)^{2} + 2s(α^{2}β)s(αβ^{2})c(α)s(α)s(β)^{3} +s(α^{2}β)s(αβ^{2})c(β)s(αβ)s(α)^{2} + 2s(α^{2}β)s(αβ^{2})c(β)s(α)^{3}s(β) −8s(α^{2}β)c(α)^{2}c(β)^{2}s(β) + 2s(α^{2}β)c(α)^{2}c(β)s(β)^{3} +s(α^{2}β)c(α)c(β)^{2}s(αβ)s(α) + 2s(α^{2}β)c(α)c(β)^{2}s(α)^{2}s(β) −s(α^{2}β)c(α)c(β)s(αβ)^{2}s(β) + 8s(α^{2}β)c(α)c(β)s(αβ)s(α)s(β)^{2} −2s(α^{2}β)c(α)s(αβ)s(α)s(β)^{4} − s(α^{2}β)c(β)s(αβ)^{2}s(α)^{2}s(β) −2s(α^{2}β)c(β)s(αβ)s(α)^{3}s(β)^{2} − s(αβ^{2})^{3}c(α)s(α) −s(αβ^{2})^{2}c(α)^{2}c(β) − 4s(αβ^{2})^{2}c(α)c(β)s(α)^{2} +2s(αβ^{2})^{2}c(α)s(αβ)s(α)s(β) + s(αβ^{2})^{2}c(α)s(α)^{2}s(β)^{2} +s(αβ^{2})^{2}c(β)s(α)^{4} − 8s(αβ^{2})c(α)^{2}c(β)^{2}s(α) +s(αβ^{2})c(α)^{2}c(β)s(αβ)s(β) + 2s(αβ^{2})c(α)^{2}c(β)s(α)s(β)^{2} +2s(αβ^{2})c(α)c(β)^{2}s(α)^{3} − s(αβ^{2})c(α)c(β)s(αβ)^{2}s(α) +8s(αβ^{2})c(α)c(β)s(αβ)s(α)^{2}s(β) − s(αβ^{2})c(α)s(αβ)^{2}s(α)s(β)^{2} −2s(αβ^{2})c(α)s(αβ)s(α)^{2}s(β)^{3} − 2s(αβ^{2})c(β)s(αβ)s(α)^{4}s(β) −4c(α)^{3}c(β)^{3} + c(α)^{3}c(β)^{2}s(β)^{2} + c(α)^{2}c(β)^{3}s(α)^{2} −c(α)^{2}c(β)^{2}s(αβ)^{2} + 8c(α)^{2}c(β)^{2}s(αβ)s(α)s(β) −2c(α)^{2}c(β)s(αβ)s(α)s(β)^{3} − 2c(α)c(β)^{2}s(αβ)s(α)^{3}s(β) +c(α)c(β)s(αβ)^{3}s(α)s(β) − 4c(α)c(β)s(αβ)^{2}s(α)^{2}s(β)^{2} +c(α)s(αβ)^{2}s(α)^{2}s(β)^{4} + c(β)s(αβ)^{2}s(α)^{4}s(β)^{2}.

2.2. The art of finding relations. In this section, we deal with techniques for finding relations in ℤ[M_{3} ×M_{3}]^{PGL^{3}}. For finding and proving the equalities (12) and (13) we need to calculate by computer. Here we introduce not the algorithm but the art of obtaining relations.

Claim. The equalities (12) and (13) hold in ℤ[M_{3} × M_{3}]^{PGL^{3}}.

We prove this claim by the following steps.

Step 1. Finding relations.

The group scheme 𝔾_{m} × 𝔾_{m} acts on M_{3} × M_{3} by (A, B) ↦ (aA, bB) for (A, B) ∈ M_{3} × M_{3} and (a, b) ∈ 𝔾_{m} × 𝔾_{m}. We define a canonical grading on the ring ℤ[M_{3}^{⊕2}] as follows: f(A, B) ∈ ℤ[M_{3}^{⊕2}] has multidegree (α, β) if f(aA, bB) = a^{α}b^{β}f(A, B) for (a, b) ∈ 𝔾_{m} × 𝔾_{m}. For example, s(α) has multi-degree (1, 0), s(β) has multi-degree (0, 1), and s(αβ^{2}) has multi-degree (1, 2). We also define the canonical grading on ℤ[M_{3} × M_{3}]^{PGL^{3}}.

We put B := {d(α), d(β), s(α), s(β), c(α), c(β), s(αβ), s(αβ^{2}), s(α^{2}β), s(α^{2}β^{2})}. We consider expressing the elements s(αβα^{2}β^{2})+s(αβ^{2}α^{2}β) and s(αβα^{2}β^{2})s(αβ^{2}α^{2}β) as polynomials in the 10 elements of B. Since s(αβα^{2}β^{2}) + s(αβ^{2}α^{2}β) and s(αβα^{2}β^{2})s(αβ^{2}α^{2}β) have multi-degree (3, 3) and (6, 6), they must be expressed as linear combinations of monomials with multi-degree (3, 3) and (6, 6), respectively.

Here we consider the case s(αβα^{2}β^{2})s(αβ^{2}α^{2}β) as an example. In the case s(αβα^{2}β^{2}) + s(αβ^{2}α^{2}β), we also obtain the equality (12) in a similar way. First let us list up all monomials with multi-degree (6, 6) in the 10 elements of B. There are 305 monomials with multi-degree (6, 6). Pick up m pairs of matrices of size 3×3 at random, and calculate 305 monomials and s(αβα^{2}β^{2})s(αβ^{2}α^{2}β) for the m pairs of matrices. From these data, we obtain an m × 306 matrix. Call it A. Here we choose suitably large m. Let us consider a system of linear equations Ax = 0. If m pairs of matrices are chosen suitably, and if luck is with us, then the kernel of A has dimension 1. From the kernel of A we obtain the coefficients of the linear combination of 305 monomials in the expression of s(αβα^{2}β^{2})s(αβ^{2}α^{2}β). Thus we get the explicit forms of P (α, β) and Q(α, β).

Step 2. Calculation of the kernel of the large matrix.

We calculate the kernel of the large matrix A by computer. However because of restriction of memories, we need a technique for calculating the kernel. By picking up m pairs of matrices in M_{3}(ℤ) we have obtained the m × 306 matrix A. Considering the reduction mod p of A for a prime number p, we can calculate the kernel of A over ℤ/pℤ. If we choose suitably large p, then we can obtain the kernel of A.

Step 3. Proof of the claim.

We prove the equalities (12) and (13) by computer. For restriction of memories, we may need a trick. Remark that it suffices to show that (12) and (13) hold for any pairs of 3 × 3 matrices (X, Y) over the complex number field ℂ. We only have to prove the equalities for any pairs (X, Y) in a Zariski open subset of M_{3}(ℂ) × M_{3}(ℂ). Hence we may assume that the characteristic polynomial of X has no multiple root. Since the right hand sides of (12) and (13) are PGL_{3}-invariants, we may change (X, Y) to (P^{−1}XP, P^{−1}Y P). Therefore it is sufficient to prove the case that X is a diagonal matrix. We may also change (X, Y) to (aX, bY) because the both sides of (12) and (13) have multidegree (3, 3) and (6, 6), respectively. Furthermore we change X and Y as simply as we can, and we can prove the equalities by computer.

We can also verify the existence of the relations (12) and (13) by another method, although we can not obtain the explicit forms. Let us define the representation of GL_{3}(ℂ) in V := M_{3}(ℂ) by A ↦ P AP^{−1}. Then it induces a representation of GL_{3}(ℂ) in S^{a}(V) ⊗ S^{b}(V). The multiplicity of the trivial representation in S^{a}(V)⊗S^{b}(V) is equal to the dimension of the subspace ℂ[M_{3} ×M_{3}]_{(a,b)}^{PGL_{3}} of homogeneous elements of multi-degree (a, b) of ℂ[M_{3} × M_{3}]^{PGL^{3}}. By calculating the multiplicity we can check the existence of the relations (12) and (13). For example, we see that dim ℂ[M_{3} × M_{3}]_{(3,3)}^{PGL_{3}} = 25 by calculating the multiplicity. We see that the 10 elements in B in the proof of the above claim are algebraically independent over ℂ by considering the Jacobi matrix of B. Since the number of monomials of multi-degree (3, 3) in B is 24, we see that s(αβα^{2}β^{2}) and s(αβ^{2}α^{2}β) are linearly dependent in ℂ[M_{3}×M_{3}]_{(3,3)}^{PGL_{3}} modulo B. Therefore we can guess the existence of (12). We can also check the existence of (13) from dim ℂ[M_{3}×M_{3}]_{(6,6)}^{PGL_{3}} = 329.

We introduce the next lemma, which also allows us to verify the existence of the relation (13) without explicit description.

Lemma 2.2. Let R be a commutative ring. For 2n^{2} matrices A_{i}, B_{i} ∈ M_{n}(R) (1 ≤ i ≤ n^{2}), we have the following equality:

(14) det((n tr(A_{i}B_{j}) − tr(A_{i})tr(B_{j}))_{1≤i,j≤n^{2}}) = 0.

Proof. We define the bilinear form ψ by

ψ : M_{n}(R) × M_{n}(R) → R
(X, Y)  ↦ ntr(XY) − tr(X)tr(Y).

Let us consider the n^{2} × n^{2} matrix (ψ(E_{ij}, E_{kl}))_{(i,j),(k,l)}, where E_{ij} are the matrix units. For proving the statement, we only have to show that det(ψ(E_{ij}, E_{kl}))_{(i,j),(k,l)} = 0. We define the vector v ∈ R^{n^{2}} by v_{(k,l)} := δ_{kl}. Since ψ(E_{ij}, E_{kl}) = nδ_{il}δ_{jk}−δ_{ij}δ_{kl}, we have (ψ(E_{ij}, E_{kl}))_{(i,j),(k,l)}·v = 0. Hence we get det(ψ(E_{ij}, E_{kl}))_{(i,j),(k,l)} = 0.

For A, B ∈ M_{3}(R), we have an equality by substituting

\begin{aligned}A_{1} & = & B_{1} & = & A, \\ A_{2} & = & B_{2} & = & B, \\ A_{3} & = & B_{3} & = & A^{2}, \\ A_{4} & = & B_{4} & = & B^{2}, \\ A_{5} & = & B_{5} & = & AB, \\ A_{6} & = & B_{6} & = & BA, \\ A_{7} & = & B_{7} & = & ABA, \\ A_{8} & = & B_{8} & = & A^{2}B, \\ A_{9} & = & B_{9} & = & AB^{2} \end{aligned}

in (14). By using (12) we see that the equality has the form

♠ tr(ABA^{2}B^{2})^{2} + ♣ tr(ABA^{2}B^{2}) + ♡ = 0,

where ♠, ♣, ♡ are polynomials in B = {det(A), det(B), …, tr(A^{2}B^{2})}. We can check that ♠ ≠ 0 for general A, B ∈ M_{3}(R). Hence we can guess the existence of the equality

tr(ABA^{2}B^{2})^{2} + \frac{♣}{♠} tr(ABA^{2}B^{2}) + \frac{♡}{♠} = 0.

Therefore we can confirm the existence of the relation (13).

3. Appendix: the invariant rings of 2 × 2 matrices

In this appendix we deal with the invariant rings of 2 × 2 matrices, which have been investigated by many mathematicians ([10], [3], [4], [2], and so on). We review the structure of ℤ[M_{2} ×M_{2}]^{PGL^{2}} and ℤ[M_{2} × M_{2} × M_{2}]^{PGL^{2}}. The proofs of the following statements are due to [9] and [5].

Definition 3.1 ([9]). The Saito character ring of degree 2 for a group Γ is defined by

R_{2}(Γ) := ℤΓ^{ab}[s(γ) | γ ∈ Γ]/I(Γ),

where I(Γ) is the ideal of the polynomial ring ℤΓ^{ab}[s(γ) | γ ∈ Γ] over ℤΓ^{ab} generated by

s(e) − 2, s(γδ) − s(γ)s(δ) + d(γ)s(γ^{−1}δ)

for all γ, δ ∈ Γ. This definition is compatible with Definition 1.14.

Remark 3.2. In the Saito character ring of degree 2, we have

s(e) = 2, (15)

s(γδ) − s(γ)s(δ) + d(γ)s(γ^{−1}δ) = 0, (16)

for each γ, δ ∈ Γ.

We introduce several equalities in the Saito character ring of degree 2.

Lemma 3.3 ([9]). For α, β, γ ∈ Γ, we have the following equalities:

(17)  s(α) = d(α)s(α^{−1}),
(18)  s(αβ) = s(βα),
(19)  s(α) = s(β^{−1}αβ),
(20)  s(αβγ) − s(γβα) = (s(γ^{−1}βα) − s(αβγ^{−1}))d(γ).

Proof. Substitute δ = e in the equality (16) and use the equality (15). Then we obtain (17). By using (16) again, we have

s(α)s(β) = s(αβ) − d(α)s(α^{−1}β),
s(β)s(α) = s(βα) − d(β)s(β^{−1}α).

Since the equality (17) follows d(α)s(α^{−1}β) = d(β)s(β^{−1}α), we obtain (18) by taking the difference of the two equalities above. By using (18) we have s(β^{−1}αβ) = s(αββ^{−1}) = s(α). Multiplying s(γ) on the both sides of (18) and using the equality (16), we obtain (20).

Lemma 3.4 ([9]). In the Saito character ring R_{2}(Γ), we have the following equalities for α, β, γ, δ ∈ Γ. We call the equality (21) the three term relation.

(21) s(αβγ) + s(γβα) = s(α)s(βγ) + s(β)s(γα) + s(γ)s(αβ) − s(α)s(β)s(γ).

(22) 2s(αβγδ)

= s(α)s(βγδ) + s(β)s(γδα) + s(γ)s(δαβ) + s(δ)s(αβγ)

+ s(αβ)s(γδ) − s(αγ)s(βδ) + s(αδ)s(βγ)
− s(α)s(β)s(γδ) − s(γ)s(δ)s(αβ) − s(α)s(δ)s(βγ)
− s(β)s(γ)s(αδ) + s(α)s(β)s(γ)s(δ).

Proof. We can prove (21) by the following:

s(αβγ) + s(αγβ)

= s(αβ)s(γ) − d(αβ)s(β^{−1}α^{−1}γ) + s(α)s(γβ) − d(α)s(α^{−1}γβ)
= −d(α){d(β)s(β^{−1}α^{−1}γ) + s(βα^{−1}γ)} + s(γ)s(αβ) + s(α)s(βγ)
= −d(α)s(β)s(α^{−1}γ) + s(γ)s(αβ) + s(α)s(βγ)
= −s(β){s(α)s(γ) − s(αγ)} + s(γ)s(αβ) + s(α)s(βγ)
= s(α)s(βγ) + s(β)s(γα) + s(γ)s(αβ) − s(α)s(β)s(γ).

Using the three term relation, we have

(a) s(αβγδ) + s(αβδγ) = s(αβ)s(γδ) + s(γ)s(αβδ) + s(δ)s(αβγ) − s(αβ)s(γ)s(δ),

(b) s(αβδγ) + s(αδβγ) = s(δ)s(αβγ) + s(βδ)s(αγ) + s(β)s(αδγ) − s(αγ)s(β)s(δ),

(c) s(αδβγ) + s(αβγδ) = s(α)s(βγδ) + s(δ)s(αβγ) + s(βγ)s(αδ) − s(α)s(δ)s(βγ).

Calculate (a) − (b) + (c) and use the following equality:

s(αδγ) = −s(αγδ)+s(α)s(γδ)+s(γ)s(αδ)+s(δ)s(αγ)−s(α)s(γ)s(δ). Then we obtain (22). Lemma 3.5 ([5]). In the Saito character ring R_{2}(Γ), we have the following equality for α, β, γ ∈ Γ.

(23) s(αβγ)s(αγβ)

= d(β)d(γ)s(α)^{2} + d(γ)d(α)s(β)^{2} + d(α)d(β)s(γ)^{2}

+ d(γ)s(αβ)^{2} + d(α)s(βγ)^{2} + d(β)s(γα)^{2}
+ s(αβ)s(βγ)s(γα)
− d(γ)s(α)s(β)s(αβ) − d(α)s(β)s(γ)s(βγ)
− d(β)s(γ)s(α)s(γα) − 4d(α)d(β)d(γ).

Proof. Substituting δ = αγβ in (22), we obtain the following:

L.H.S. = 2s(αβγαγβ)
= 2s(αβγ)s(αγβ) − 2d(αβγ)s(γ^{−1}β^{−1}γβ),
R.H.S. = s(α)s(βγαγβ) + s(β)s(γαγβα) + s(γ)s(αγβαβ)
+s(αγβ)s(αβγ) + s(αβ)s(γαγβ) − s(αγ)s(βαγβ)

+s(α^{2}γβ)s(βγ) − s(α)s(β)s(γαγβ) − s(γ)s(αγβ)s(αβ)

−s(α)s(αγβ)s(βγ) − s(β)s(γ)s(α^{2}γβ)
+s(α)s(β)s(γ)s(αγβ).

Hence we get

s(αβγ)s(αγβ) = 2d(αβγ)s(γ^{−1}β^{−1}γβ) + s(α)s(βγαγβ)
+s(β)s(γαγβα) + s(γ)s(αγβαβ) + s(αβ)s(γαγβ)
−s(αγ)s(βαγβ) + s(α^{2}γβ)s(βγ) − s(α)s(β)s(γαγβ)
−s(γ)s(αγβ)s(αβ) − s(α)s(αγβ)s(βγ)
−s(β)s(γ)s(α^{2}γβ) + s(α)s(β)s(γ)s(αγβ).

Using several equalities, for examples,

s(γ^{−1}β^{−1}γβ) = d(β^{−1}γ^{−1})s(βγ)^{2} − d(β^{−1}γ^{−1})s(β)s(γ)s(βγ)
+d(β^{−1})s(β)^{2} + d(γ^{−1})s(γ)^{2} − 2,

we obtain (23).

These equalities were already found by Vogt [10] for representations in SL_{2}. From the equalities above, we can determine the structure of the invariant rings.

ℤ[M_{2} × M_{2}]^{PGL^{2}} ≅ ℤ[tr(A), tr(B), tr(AB), det(A), det(B)].

ℤ[M_{2} × M_{2} × M_{2}]^{PGL^{2}} ≅

ℤ [ \begin{matrix}tr(A), tr(B), tr(C), \\ tr(AB), tr(BC), tr(CA), \\ det(A), det(B), det(C), \\ tr(ABC) \end{matrix} ] /(tr(ABC)^{2} − P tr(ABC) + Q),

where

P = tr(A)tr(BC) + tr(B)tr(CA) + tr(C)tr(AB) − tr(A)tr(B)tr(C) and Q =

det(B) det(C)tr(A)^{2} + det(C) det(A)tr(B)^{2} + det(A) det(B)tr(C)^{2}

+ det(C)tr(AB)^{2} + det(A)tr(BC)^{2} + det(B)tr(CA)^{2}
+ tr(AB)tr(BC)tr(CA) − det(C)tr(A)tr(B)tr(AB)
− det(A)tr(B)tr(C)tr(BC) − det(B)tr(C)tr(A)tr(CA)
− 4 det(A) det(B) det(C).

Note that

R_{2}(F_{2}) ≅ ℤ[M_{2} × M_{2}]^{PGL^{2}}[\frac{1}{det(A)}, \frac{1}{det(B)}],

R_{2}(F_{3}) ≅ ℤ[M_{2} × M_{2} × M_{2}]^{PGL^{2}}[\frac{1}{det(A)}, \frac{1}{det(B)}, \frac{1}{det(C)}].

Here we denote by A, B (or A, B, C) the universal matrices. The spectra of ℤ[M_{2} × M_{2}]^{PGL^{2}} and ℤ[M_{2} × M_{2} × M_{2}]^{PGL^{2}} are isomorphic to 𝔸_{ℤ}^{5} and a hypersurface of 𝔸_{ℤ}^{10}, respectively.

References

[1] S. Donkin. Invariants of several matrices. Invent. math., Vol. 110, pp. 389–401, 1992.

[2] F.González-Acuña, José María Montesinos-Amilibia. On the character variety of group representations in SL(2, ℂ) and PSL(2, ℂ). Math. Z., Vol. 214, No. 4, pp. 627–652, 1993.

[3] Horowitz R. D. Characters of free groups represented in the two-dimensional linear group. Comm. Pure Appl. Math., Vol. 25, pp. 635–649, 1972.

[4] W. Magnus. Rings of Fricke characters and automorphism groups of free groups. Math. Z., Vol. 170, No. 1, pp. 91–103, 1980.

[5] K. Nakamoto. On the Structure of the Character Rings. Master’s thesis, Kyoto University, 1995. (in Japanese).

[6] K. Nakamoto. Representation varieties and character varieties. Publ. of RIMS, to appear.

[7] Y. Omoda. The invariant theory of McKay quivers. Master’s thesis, Kyoto University, 1996. (in Japanese).

[8] C. Procesi. Invariant theory of n×n matrices. Adv. Math., Vol. 19, pp. 306–381, 1976.

[9] K. Saito. Representation Varieties of a Finitely Generated Group into SL_{2} or GL_{2}. RIMS-958, December 1993.

[10] H. Vogt. Sur les invariants, fondamentaux des équations différentielles linéaires du second ordre. Ann. Sci. Ecole Norm. Sup., Vol. (3) 6, pp. Suppl. 3–72, 1889. Thèse, Paris.