The Tutte Group of Projective Spaces

Andreas Dress and \underline{Walter Wenzel}, Bielefeld

The Tutte group of a matroid offers an algebraic approach to matroid theory. In particular, this group reflects some important geometric properties of matroids in algebraic terms.

In the sequel let M denote a matroid defined on some possibly infinite set E of finite rank n with H = H_{M} as its set of hyperplanes and L = L_{M} as its set of hyperlines.

Definition of the extended Tutte group: Let 𝔽_{M}^{H} denote the free abelian group generated by the symbols ε and X_{H,a} for H ∈ H, a ∈ E \ H, and let 𝕂_{M}^{H} denote the subgroup of 𝔽_{M}^{H} generated by ε^{2} and all elements of the form

ε · X_{H_{1},a_{2}} · X_{H_{1},a_{3}}^{−1} · X_{H_{2},a_{3}} · X_{H_{2},a_{1}}^{−1} · X_{H_{3},a_{1}} · X_{H_{3},a_{2}}^{−1}

for H_{1}, H_{2}, H_{3} ∈ H, L := H_{1} ∩ H_{2} ∩ H_{3} = H_{i} ∩ H_{j} ∈ L for i ≠ j and a_{i} ∈ H_{i} \ L for 1 ≤ i ≤ 3. Then the (extended) Tutte group 𝕋_{M}^{H} of M is defined by

𝕋_{M}^{H} := 𝔽_{M}^{H}/𝕂_{M}^{H}.

Let ε_{M} denote the image of ε and H(a) the image of X_{H,a} in 𝕋_{M}^{H}, respectively.

Definition of the inner Tutte group: Define the homomorphism Φ : 𝕋_{M}^{H} → ℤ^{H} × ℤ^{E} by

\begin{aligned}Φ(ε_{M}) & := 0, \\ Φ(H(a)) & := (δ_{H}, δ_{a}) for H ∈ H, a ∈ E \ H. \end{aligned}

𝕋_{M}^{(0)} := kerΦ is the inner Tutte group of M.

Proposition 1: Assume M is representable over the skewfield K; that is, there exists a family of maps f_{H} : E → K (H ∈ H) with

(H1) f_{H}^{−1}({0}) = H for all H ∈ H;

(H2) if H_{1}, H_{2}, H_{3} ∈ H are pairwise distinct and L := H_{1} ∩ H_{2} ∩ H_{3} ∈ L, then there exist c_{1}, c_{2}, c_{3} ∈ K^{⋆} such that for all e ∈ E we have

f_{H_{1}}(e) · c_{1} + f_{H_{2}}(e) · c_{2} + f_{H_{3}}(e) · c_{3} = 0.

Put K_{C} := K^{⋆}/[K^{⋆}, K^{⋆}]. Then the homomorphism ψ : 𝔽_{M}^{H} → K^{⋆} given by

\begin{matrix}ψ(ε) := −1, \\ ψ(X_{H,a}) := f_{H}(a) for H ∈ H, a ∈ E \ H \end{matrix}

induces a homomorphism φ = \overline{ψ} : 𝕋_{M}^{H} → K_{C}.

Proposition 2: Assume H_{1}, H_{2}, H_{3} ∈ H with H_{1} ∩ H_{2} ∩ H_{3} ∈ L and H_{3} ≠ H_{i} for i ∈ {1, 2}. Then for a, b ∈ H_{3} \ (H_{1} ∪ H_{2}) we have

H_{1}(a) · H_{2}(a)^{−1} = H_{1}(b) · H_{2}(b)^{−1}.

Definition of cross ratios in Tutte groups: If H_{1}, H_{2}, H_{3} are as in Proposition 2, put

| \begin{matrix}H_{1} & & H_{2} \\ & H_{3} \end{matrix} | := H_{1}(a) · H_{2}(a)^{−1} for a ∈ H_{3} \ (H_{1} ∪ H_{2}).

Assume H_{1}, H_{2}, H_{3}, H_{4} ∈ H with H_{1} ∩ H_{2} ∩ H_{3} ∩ H_{4} ∈ L and H_{1}, H_{2} ≠ H_{3}, H_{4},

i.e. \{H_{1}, H_{2}\} ∩ \{H_{3}, H_{4}\} = ∅. The cross ratio [ \begin{matrix}H_{1} & H_{2} \\ H_{3} & H_{4} \end{matrix} ] ∈ 𝕋_{M}^{(0)} is defined

by

[ \begin{matrix}H_{1} & H_{2} \\ H_{3} & H_{4} \end{matrix} ] := | \begin{matrix}H_{1} & & H_{2} \\ & H_{3} \end{matrix} | · | \begin{matrix}H_{2} & & H_{1} \\ & H_{4} \end{matrix} | .

By applying the more elementary part of Tutte’s homotopy theory one proves Proposition 3:

i) 𝕋_{M}^{(0)} is the subgroup of 𝕋_{M}^{H} generated by ε_{M} and all cross ratios [ \begin{matrix}H_{1} & H_{2} \\ H_{3} & H_{4} \end{matrix} ]

for which H_{1}, H_{2}, H_{3}, H_{4} are pairwise distinct hyperplanes in M with H_{1} ∩ H_{2} ∩ H_{3} ∩ H_{4} ∈ L.

ii) 𝕋_{M}^{(0)} is generated by all cross ratios [ \begin{matrix}H_{1} & H_{2} \\ H_{3} & H_{4} \end{matrix} ] as in i) if and only if M

is not regular.

Definition: Put

\begin{aligned}H^{(4)} := \{(H_{1}, H_{2}, H_{3}, H_{4}) ∈ H^{4} | & H_{1} ∩ H_{2} ∩ H_{3} ∩ H_{4} = H_{i} ∩ H_{j} ∈ L \\ & for i ∈ \{1, 2\}, j ∈ \{3, 4\} \}. \end{aligned}

i) (H_{1}, H_{2}, H_{3}, H_{4}), (H′_{1}, H′_{2}, H′_{3}, H′_{4}) ∈ H^{(4)} are called projective neighbours of each other, if ⋂_{i=1}^{4} H_{i} ≠ ⋂_{i=1}^{4} H′_{i} and there exists some H ∈ H with

(I) ⋂_{i=1}^{4} H_{i} ⊊ H, ⋂_{i=1}^{4} H′_{i} ⊊ H and

(II) for 1 ≤ i ≤ 4 we have H_{i} = H′_{i} or H_{i} ∩ H′_{i} = H_{i} ∩ H = H′_{i} ∩ H ∈ L.

If this is the case, we write (H_{1}, H_{2}, H_{3}, H_{4}) ∧ (H′_{1}, H′_{2}, H′_{3}, H′_{4}).

ii) G = (H_{1}, H_{2}, H_{3}, H_{4}) and G′ = (H′_{1}, H′_{2}, H′_{3}, H′_{4}) ∈ H^{(4)} are called projectively equivalent, if there exist k ≥ 0 and G_{0}, G_{1}, …, G_{k} ∈ H^{(4)} such that

G_{0} = G, G_{k} = G′, G_{i−1} ∧ G_{i} for 1 ≤ i ≤ k.

In this case we write (H_{1}, H_{2}, H_{3}, H_{4})∼^{pr}(H′_{1}, H′_{2}, H′_{3}, H′_{4}).

Proposition 4:

Assume (H_{1}, H_{2}, H_{3}, H_{4})∼^{pr}(H′_{1}, H′_{2}, H′_{3}, H′_{4}). Then we have

[ \begin{matrix}H_{1} & H_{2} \\ H_{3} & H_{4} \end{matrix} ] = [ \begin{matrix}H′_{1} & H′_{2} \\ H′_{3} & H′_{4} \end{matrix} ] .

From now on we assume that M is a projective space of dimension m ≥ 2.

Proposition 5: Assume H_{1}, H_{2} ∈ H are distinct, put

H_{1,2} := {H ∈ H | H_{1} ∩ H_{2} ⊆ H}

and let G_{(H_{1},H_{2})} denote the group of projectivities p : H_{1,2} → H_{1,2} with p(H_{i}) = H_{i} for i ∈ {1, 2}. Then η : G_{(H_{1},H_{2})} ↠ 𝕋_{M}^{(0)} defined by

η(p) := [ \begin{matrix}H_{1} & H_{2} \\ H_{3} & p(H_{3}) \end{matrix} ] for H_{3} ∈ H_{1,2} \ \{H_{1}, H_{2}\}

is a well defined group epimorphism; that is, η does not depend on H_{3}.

Proposition 6: If H_{1}, H_{2}, H_{3} ∈ H are pairwise distinct with L := H_{1} ∩ H_{2} ∩ H_{3} ∈ L, then

𝕋_{M}^{(0)} = \{[ \begin{matrix}H_{1} & H_{2} \\ H_{3} & H_{4} \end{matrix} ] | L ⊆ H_{4} ∈ H \ \{H_{1}, H_{2}\}\} .

By applying Proposition 1 and Proposition 6 one proves

Proposition 7: If M is the projective space over the skewfield K of dimension m ≥ 2, then 𝕋_{M}^{(0)} ≅ K_{C}. If, in particular, K is a field, then 𝕋_{M}^{(0)} ≅ K^{⋆}.

In general it follows from [G], Theorem 2, that in case M is a finite nondesarguesian projective plane its inner Tutte group 𝕋_{M}^{(0)} is necessarily trivial.

References

[B] R. Baer: Linear Algebra and Projective Geometry. New York: Academic Press 1952.

[DW1] A. W. M. Dress and W. Wenzel: Geometric Algebra for Combinatorial Geometries; to appear in Advances in Mathematics 76 (1989).

[DW2] A. W. M. Dress and W. Wenzel: On Combinatorial and Projective Geometry; in preparation.

[G] Th. Grundhöfer: The Groups of Projectivities of Finite Projective and Affine Planes. Ars Combinatoria 25A (1988), 269 – 275.

[HP] D. R. Hughes and F. C. Piper: Projective Planes. Springer Verlag New York, Heidelberg, Berlin 1972.

[T] W. T. Tutte: Lectures on Matroids. National Bureau of Standards Journal of Research 69B (1965), 1 – 47.

[W] W. Wenzel: A Group Theoretic Interpretation of Tutte’s Homotopy Theory; to appear in Advances in Mathematics 76 (1989).

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