Alternating forms and transitive locally grid geometries

J.I. Hall Department of Mathematics Michigan State University East Lansing, MI 48824 USA

Version of: 10 March 2006

1 Introduction

Let V be a vector space over K. For each nonnegative integer k let P_{k}(V) be the set of k-subspaces of V. For positive d (≤ dimV), let L_{d}(V) be the set of pairs (U, W) with U ∈ P_{d−1}(V), W ∈ P_{d+1}(V), and U ≤ W. We also set B_{d}^{−}(V) = P_{d−1}(V), B_{d}^{+}(V) = P_{d+1}(V), and B_{d}(V) = B_{d}^{−}(V) ∪ B_{d}^{+}(V).

In the case V = K^{4} and d = 2, we can view P_{2}(V) and L_{2}(V) as the points and lines of the Klein quadric. Then (B_{2}^{−}(V), P_{2}(V), B_{2}^{+}(V)) is the associated rank 3 geometry of type D_{3} (= A_{3}) and (P_{2}(V), L_{2}(V), B_{2}(V)) the associated rank 3 polar space of type C_{3}.

We look at several geometries related to these for the Klein quadric. Let A_{d}(V) = (P_{d}(V), L_{d}(V)). (Here and throughout, d is some positive integer but dim_{K}V may be infinite unless stated otherwise.) A_{d}(V) is a partial linear space with point set P_{d}(V) and line set L_{d}(V) whose members (U, W) we often identify with the 1 + |K| distinct d-spaces (that is, incident points) in between the (d − 1)-space U and the (d + 1)-space W. A_{d}(V) is called a Grassmann space and, sometimes, a d-Grassmann space.

We associate two rank 3 geometries with A_{d}(V). The first is the Grassmannian geometry (or d-Grassmannian geometry)

G_{d}(V) = (B_{d}^{−}(V), P_{d}(V), B_{d}^{+}(V))

with diagram

[figure: P G^{∗}; P G] .

If d is 1 or dim_{K}V − 1, then B_{d}^{−}(V) or B_{d}^{+}(V) is empty (respectively), and we have the rank 2 geometry of points and lines for a projective space over K or K^{op}.

The second geometry is

CG_{d}(V) = (P_{d}(V), L_{d}(V), B_{d}(V))

with diagram

[figure: P G; G] .

Again if d is 1 or dim_{K}V −1, one of the B_{d}^{ϵ}(V) is empty; so the line set and block set of CG_{d}(V) are identical, being equal to the line set of G_{d}(V). For this reason, we will only consider the geometry CG_{d}(V) for d not equal to 1 or dim_{K}V − 1. With this restriction in mind, we may (when convenient) identity CG_{d}(V) with the collinearity graph Γ of A_{d}(V) and G_{d}(V), since the three types of objects correspond to three identifiable classes of cliques in Γ. (If d is 1 or dim_{K}V − 1, the collinearity graph Γ is complete.) The vertices of Γ are the points of P_{d}(V) with P_{1} and P_{2} adjacent precisely when collinear in A_{d}(V), that is, when P_{1}∩P_{2} is a (d−1)-space or, equivalently, ⟨P_{1}, P_{2}⟩ is a (d+1)-space. The maximal cliques in this graph are precisely the members of B_{d}(V) (each identified with the set of those points incident to it). Two distinct maximal cliques intersect trivially, in a vertex, or in a line of L_{d}(V). The diagram indicates that the residue of a point in the geometry CG_{d}(V) is a grid graph. More precisely, the collinearity graph Γ is locally a |K|-clique extension of a u × v grid, where 1 + |K|u is the number of d-spaces containing a fixed (d − 1)-space (from B_{d}^{−}(V), giving maximal “minus” cliques) and 1+|K|v is the number of d-spaces contained in a fixed (d+1)-space (from B_{d}^{+}(V), giving maximal “plus” cliques). A plus clique and minus clique are either disjoint or they intersect in the |K| + 1 points of a line.

A geometric hyperplane of A_{d}(V) = (P_{d}(V), L_{d}(V)) is a proper subset H of P_{d}(V) such that, for every line ℓ ∈ L_{d}(V), either ℓ ⊆ H or |ℓ ∩ H| = 1. If d is 1, then a geometric hyperplane is a hyperplane in the usually sense, and dually when d = dim_{K}V − 1. It is convenient to allow the possibility d = dim_{K}V, where A_{d}(V) has a single point and H is empty. A geometric hyperplane is always a subspace of the partial linear space A_{d}(V).

With H we associate the rank 3 geometry G_{d}(V)_{H} = (B_{H}^{−}, P_{H}, B_{H}^{+}), which has been called an affine Grassmannian. The geometry G_{d}(V)_{H} consists of point set P_{H} = P_{d}(V)\H, the hyperplane complement consisting of all d-spaces not in H, together with the set B_{H}^{−} of all (d − 1)-spaces contained in a point of P_{H} and the set B_{H}^{+} of all (d + 1)-spaces containing a point of P_{H}. For d equal to 1 or dim_{K}V − 1 this gives the usual affine geometries for V and its dual.

When d is not 1 or dim_{K}V − 1, the geometry G_{d}(V)_{H} has diagram

[figure: AG^{∗}; AG] .

Its collinearity graph CG_{d}(V)_{H} is locally a (|K| − 1)-clique extension of a u × v grid graph. The lines of L_{H} contain only |K| points of P_{H}, since each meets H in a point. Of particular note is the case K = 𝔽_{2}. There the affine Grassmannian G_{d}(V)_{H} is a (c^{∗}, c)-geometry, belonging to the diagram

[figure: c^{∗}; c] ,

and the collinearity graph CG_{d}(V)_{H} is a locally grid graph.

There are earlier papers on affine Grassmannians over arbitrary fields and division rings [2, 7, 12, 20]. (Note that the “affine Grassmannian” terminology has also been used elsewhere with a related but different definition.) The most fundamental and deep result in the area is due to Shult:

( 1.1) Theorem. (Shult [20, Theorem 2].) Let V be a finite dimensional vector space over a field, and let H be a geometric hyperplane in A_{d}(V). Then for some nonzero alternating d-linear form f on V, the set H = H_{f} consists of all the f-degenerate d-subspaces of V.

Thus the study of geometric hyperplanes of A_{d}(V) is equivalent to the study of alternating bilinear forms on V.

We say that the subgroup G of Aut(G_{d}(V)_{H}) is transitive on G_{d}(V)_{H} if G is transitive on P_{H}. The subgroup G of Aut(G_{d}(V)_{H}) is flag-transitive on G_{d}(V)_{H} if G is transitive on incident triples (U, P, W) (chambers or maximal flags) with U ∈ B_{H}^{−}, P ∈ P_{H}, and W ∈ B_{H}^{+}. Similarly the subgroup G of Aut(CG_{d}(V)_{H}) is flag-transitive on CG_{d}(V)_{H} if G is transitive on incident triples (P, ℓ, B) with point P ∈ P_{H}, line ℓ ∈ L_{H}, and maximal clique B ∈ B_{H}^{−} ∪ B_{H}^{+}. If the chamber (P, ℓ, B) has B ∈ B_{H}^{−}, then it is a −-chamber, whereas when B ∈ B_{H}^{+} it is a +-chamber. The geometry is transitive (respectively, flag-transitive) if its full automorphism group is transitive (respectively, flag-transitive) on it.

The focus of this paper is an interesting class of transitive affine Grassmannians.

( 1.2) Theorem. Let s: V × V ⟶ K be a symplectic form on V. Let H = H_{s,d} be the set of all d-subspaces P of V for which s|_{P} is degenerate. Assume additionally that d ≥ rank(V, s) ≠ 0, so that H does not consist of all d-subspaces of V.

Then H is a geometric hyperplane of A_{d}(V) = (P_{d}(V), L_{d}(V)), and the associated affine Grassmannian G_{d}(V)_{H} is connected and transitive. Indeed G_{d}(V)_{H} is flag-transitive if and only if either

(a) s is nondegenerate or

(b) d = rank(V, s).

We then have

( 1.3) Corollary. Let 1 < d < dim_{K}V − 1. The locally grid geometry CG_{d}(V)_{H} is connected and transitive. Indeed CG_{d}(V)_{H} is flag-transitive if and only if 2d = dim_{K}V and

(a) s is nondegenerate or

(b) d = rank(V, s).

These are proved (in slightly greater detail) in Section 3. We refer to all the geometric hyperplanes of Theorem 1.2 as being of Pfaffian type for reasons that will become clear in Section 4 below. Similarly the related affine Grassmannians and locally grid geometries will be called Pfaffian.

The affine Grassmannians of Theorem 1.2(b) and Corollary 1.3(b) with d equal to rank(V, s) are examples of attenuated spaces [21, 22]. If R is a fixed subspace of codimension d in V, then the set of all d-subspaces of V that meet R nontrivially is a geometric hyperplane H_{R} of A_{d}(V) called an attenuated hyperplane. The associated affine Grassmannian and locally grid geometry will also be called attenuated. The examples of Theorem 1.2(b) come from the attenuated hyperplane H_{R} for R = Rad(V, s) of codimension d (so attenuated affine Grassmannians are Pfaffian if and only if d is even). If d is 1 or dim_{K}V −1, then all geometric hyperplanes are actual hyperplanes of V or V^{∗} (respectively) and so are attenuated.

The author first noted the constructions and results of Theorem 1.2 and Corollary 1.3 early in the 1980’s in the context of locally grid graphs, which corresponds to the case K = 𝔽_{2}. Also in 1981 Yoshimi Egawa [9] pointed out that attenuated spaces over 𝔽_{2} (presented in a slightly different form) give rise to locally grid graphs.

Meixner and Pasini’s census of flag-transitive locally grid graphs [18] lists attenuated spaces but not the Pfaffian examples. Shult’s work and surveys [21, 22] on (c^{∗}, c)-geometries (and locally grid graphs) mention affine Grassmannians over 𝔽_{2} as a source, but only the attenuated examples are discussed specifically.

Shult suggests that attenuated spaces might provide the only examples of flag-transitive affine Grassmannians. As we have seen above, that is not the case. But Shult’s feeling that the condition is highly restrictive is certainly correct. In fact, the finite attenuated and nondegenerate Pfaffian examples are characterized by flag-transitivity.

( 1.4) Theorem. Let H be a geometric hyperplane of A_{d}(V) with V = 𝔽_{q}^{n}. Then G_{d}(V)_{H} is flag-transitive if and only if we have one of

(a) H = H_{R}, an attenuated hyperplane with respect to some R of codimension d in V;

(b) d is even and H = H_{s,d} for some nondegenerate symplectic s form on V.

The theorem is related to Witt’s theorem [24, 7.4], since it classifies alternating forms admitting semisimilarity groups that are transitive on all subconfigurations of a certain isometry type.

( 1.5) Corollary. Let 1 < d < dim_{K}V −1. Let H be a geometric hyperplane of A_{d}(V) with V = 𝔽_{q}^{n}. Then CG_{d}(V)_{H} is flag-transitive if and only if 2d = n and we have one of

(a) H = H_{R}, an attenuated hyperplane with respect to some R of codimension d in V;

(b) d is even and H = H_{s,d} for some nondegenerate symplectic s form on V.

Of particular interest is the locally grid graph case q = 2.

In Section 2 we give various properties of alternating bilinear forms and the related geometric hyperplanes. Section 3 contains a proof of Theorem 1.2 and Corollary 1.3. In Section 4 we construct alternating d-linear forms that give rise to the attenuated and Pfaffian examples. Section 5 is devoted to the proof of Theorem 1.4 and Corollary 1.5.

2 Alternating forms and related hyperplanes

An alternating d-linear form f on the vector space V over the field K is a map f: V^{d} ⟶ K that is linear in each variable and is 0 whenever two arguments are equal. We also view an alternating d-linear form as an element of (⋀^{d} V)^{∗}. A symplectic form on the K-space V is an alternating 2-linear form. The radical of f on V is the subspace

Rad(V, f) = { v ∈ V | f(v_{1}, …, v_{d−1}, v) = 0, for all v_{1}, …, v_{d−1} ∈ V } .

The rank of V, rank(V, f), is the codimension of its radical. When d = 2 the rank is always even; for this and other well-known geometry, see [24].

Alternating d-linear forms with d > 2 are less familiar; see [1, 20]. After Aschbacher [1], for each subspace P of V we set

P^{θ} = { v ∈ V | f(p_{1}, …, p_{d−1}, v) = 0, for all p_{1}, …, p_{d−1} ∈ P } .

Especially V^{θ} is Rad(V, f). More generally P ∩P^{θ} = Rad(P, f|_{P}). The subspace P (possibly V itself) is f-degenerate (or just degenerate if the context is clear) when its radical Rad(P, f|_{P}) is nonzero. If Rad(P, f|_{P}) = 0, then P is fnondegenerate or nondegenerate.

The zero form is identically 0 on V^{d}.

( 2.1) Lemma. Let f be an alternating d-linear form on the K-space V, and let P be a subspace of dimension at most d. Then either f restricted to P is the zero form and P ≤ P^{θ} or dim_{K}P = d, P is nondegenerate, and V = P ⊕ P^{θ}.

Proof. The arguments of [1, (1.4-5)] for d = 3 go over to the general case. Let v_{1}, …, v_{d} have span W with basis { w_{j} | j ∈ J }. Then f(v_{1}, …, v_{d})

is a K-linear combination of the f(w_{j_{1}}, …, w_{j_{d}}) with j_{i} ≤ j_{i+1}. In particular f(v_{1}, …, v_{d}) = 0 if dim_{K}W < d. Indeed, for any subspace W of dimension k, the codimension in V of W^{θ} is at most \binom{k}{d−1}.

If dim_{K}P < d or P ∩ P^{θ} ≠ 0, then (with an appropriate choice of the w_{j}) each f(w_{j_{1}}, …, w_{j_{d}}) is 0; so the restriction of f to P is zero and P ≤ P^{θ}. Otherwise P ∩ P^{θ} = 0, and nondegenerate P has dimension d = \binom{d}{d−1}; so we have V = P ⊕ P^{θ}.

Given an alternating d-linear form f on V and a map A ∈ ΓL(V), semilinear with respect to the automorphism α of the field F, we can define a new alternating d-linear form f^{A} on V by

f^{A}(v_{1}, …, v_{i}, …, v_{n}) = f(v_{1}A^{−1}, …, v_{i}A^{−1}, …, v_{n}A^{−1})^{α} .

The semilinear map A is then a semisimilarity of (V, f) if there is a nonzero scalar c with cf^{A} = f. It is an isometry if additionally c = 1 and α = 1.

( 2.2) Proposition. Let f be a nonzero alternating d-linear form on V, and let H = H_{f} be the associated geometric hyperplane of A_{d}(V). Then the following groups are all equal:

(a) the stablizer ΓL(V)_{H} of H in ΓL(V);

(b) the group of all semisimilarites of (V, f);

(c) the group of all A ∈ ΓL(V) for which

f(v_{1}, …, v_{i}, …, v_{n}) = 0 ⟺ f(v_{1}A, …, v_{i}A, …, v_{n}A) = 0 ,

for all v_{1}, …, v_{n} ∈ V.

Proof. For any A ∈ ΓL(V),

f(v_{1}, …, v_{i}, …, v_{n}) = 0 ⟺ f^{A}(v_{1}A, …, v_{i}A, …, v_{n}A) = 0 .

Especially semisimilarites of (V, f) belong to the group of (c). In turn, semilinear maps A as in (c) take f-degenerate d-spaces to f-degenerate d-spaces and so are contained in the stabilizer ΓL(V)_{H}.

It remains to prove that the stabilizer consists of semisimilarities. Let A ∈ ΓL(V)_{H}. The implication of the previous paragraph shows that H_{f} = H_{f^{A}}. By [20, Cor. 2.1.1] we have f = cf^{A}, for some nonzero constant c. Therefore A is a semisimilarity, as desired.

The induced semisimilarity group PΓL(V)_{H} is clearly a subgroup of the automorphism groups Aut(G_{d}(V)_{H}) and Aut(CG_{d}(V)_{H}), and usually we have equality.

( 2.3) Theorem. Let V be a K-space, and let H be geometric hyperplane of A_{d}(V). Assume also that 2 < d < dim_{K}V − 2 or that |K| > 2. Then

Aut(G_{d}(V)_{H}) = Aut(G_{d}(V))_{H} = PΓL(V)_{H} .

This is due to Blok [2, 3] and answers a question of Shult [22]. For K = 𝔽_{2} there are genuinely exceptional cases when d is any of 1, 2, dim_{K}V −2, or dim_{K}V −1; see [3]. (The equality Aut(G_{d}(V)) = PΓL(V) is Chow’s Theorem [6].)

Theorem 2.3 allows us largely to restrict our attention to automorphisms of affine Grassmannians that are induced by semilinearities of the parent vector space, that is, semisimilarities of (V, f).

Assume that V has finite dimension n > 2 over the field K. Then V possesses many dualities. These are bijections that take i-subspaces to j-subspaces, for i + j = n, and that respect inclusion. If δ is a duality, then δ^{2} belongs to the group PΓL(V); so PΓL(V) has index 2 in PΓL^{∗}(V), its extension by all dualities. Correspondingly the induced stabilizer PΓL^{∗}(V)_{H} can be twice as big as PΓL(V)_{H}.

( 2.4) Lemma. Let dim_{K}V = n > d. Further let δ be a duality of V and H a geometric hyperplane of A_{d}(V).

(1) H^{δ} is a geometric hyperplane of A_{n−d}(V).

(2) If H = H^{δ}, then n = 2d and δ induces an automorphism of CG_{d}(V)_{H} that switches B_{H}^{−} and B_{H}^{+}. In this case PΓL^{∗}(V)_{H}\ PΓL(V)_{H} is the set of all such stabilizing dualities.

Proof. (1) Let A, B, C be subspaces of V with A ≤ B ≤ C. Then A^{δ} ≥ B^{δ} ≥ C^{δ}. Thus (A, B, C) ∈ A_{d}(V) if and only if (C^{δ}, B^{δ}, A^{δ}) ∈ A_{n−d}(V).

(2) If H = H^{δ}, then A_{d}(V) = A_{n−d}(V) and d = n − d. Thus δ switches B^{−} and B^{+} and fixes H, so (B_{H}^{−})^{δ} = B_{H}^{+}. The stabilizing dualities are those of the coset δ PΓL(V)_{H}.

The following important observation is essentially due to Shult [20] and has several helpful consequences.

( 2.5) Proposition. (1) If H and J are geometric hyperplanes of A_{d}(V) with H ⊆ J, then H = J.

(2) For any geometric hyperplane H of A_{d}(V), the associated affine Grassmannian G_{d}(V)_{H} and locally grid geometry CG_{d}(V)_{H} are connected.

Proof. It is a well-known and elementary fact that a geometric hyperplane is maximal if and only its complement is connected. For finite dimensional V a full proof of (2) can be found in [3, Lemma 3.3], and the infinite dimensional case follows easily.

( 2.6) Theorem. Let H be a geometric hyperplane of A_{d}(V). Then PΓL(V)_{H} is flag-transitive on G_{d}(V)_{H} but reducible on V if and only if we have H = H_{R}, an attenuated hyperplane with respect to some R of codimension d in V.

Proof. Let R be a nontrivial invariant subspace.

Transitivity on incident pairs from (B_{H}^{−}, P_{H}) implies, for P ∈ P_{H}, that either P ≤ R or P ∩ R = 0. If P ≤ R, then ⟨P_{H}⟩ ≤ R. Since R is proper in V there are (d+1)-spaces W with W ∩R = P. But then, for any hyperplane U of P, the line (U, W) meets P_{H} in the unique point P, an impossibility. Thus P ∩ R = 0.

Transitivity on B_{H}^{+} and the existence of members of B_{H}^{+} meeting R nontrivially imply that everything in B_{H}^{+} meets R nontrivially. This forces the codimension of R to be d.

Now H contains H_{R}, so we have equality by Proposition 2.5.

( 2.7) Lemma. Let 1 < d < dim_{K}V − 1, and let H be a geometric hyperplane of A_{d}(V). Then Aut(G_{d}(V)_{H}) is a subgroup of of index at most 2 in Aut(CG_{d}(V)_{H}). If the index is equal to 2, then 2d = n and the automorphisms of Aut(CG_{d}(V)_{H})\Aut(G_{d}(V)_{H}) switch B_{H}^{−} and B_{H}^{+}.

Proof. The lines of G_{d}(V)_{H} and CG_{d}(V)_{H} are exactly those cliques of CG_{d}(V)_{H} of size at least 2 that occur as the intersection of two distinct maximal cliques. In particular Aut(G_{d}(V)_{H}) ≤ Aut(CG_{d}(V)_{H}).

The locally grid geometry CG_{d}(V)_{H} is connected by Proposition 2.5. On the maximal clique set B_{H} of CG_{d}(V)_{H} define a relation by B_{1} ∼ B_{2} when B_{1} ∩ B_{2} contains a unique point of P_{H}. B_{H} then has exactly two connected components under ∼, namely B_{H}^{−} and B_{H}^{+}. Any automorphism g of CG_{d}(V)_{H} must therefore either switch these two components or fix both. In the second case, g induces an automorphism of G_{d}(V)_{H}.

In the first case, there is a maximal clique B^{−} ∈ B^{−} with (B^{−})^{g} = B^{+} ∈ B_{H}^{+}. If K = 𝔽_{q}, we have q^{d} = |B^{−}| and |B^{+}| = q^{n−d}, where n = dim_{K}V. If K is not finite, then the lines of L give B^{−} the structure of an affine space K^{d} and B^{+} the structure of K^{n−d}. In either event, dim_{K}V = n is finite and d = n − d, as desired.

3 Proof of Theorem 1.2

This section is mainly concerned with proving of Theorem 1.2 and Corollary 1.3. Theorem 3.5 presents similar results for attenuated hyperplanes and the associated attenuated spaces. Also Proposition 3.6 gives the complete classification of hyperplanes in the cases where d is one of 1, 2, dimV − 2, or dimV − 1.

The result which motivated this paper is

( 3.1) Proposition. Let V be a vector space over the field K and s a symplectic form on V. Let d be even with 0 < d ≤ rank(V, s). Then

H_{s,d} = {P ≤ V | dim_{K}P = d, Rad(P, s|_{P}) ≠ 0 }

is a geometric hyperplane of the d-Grassmann space A_{d}(V).

Proof. As even d ≤ rank(V, s), H_{s,d} is not all of P_{d}(V). Let U ≤ P ≤ W be a chamber of the Grassmannian G_{d}(V). If the line (U, W) is not entirely within H, then we may choose P to be nondegenerate (under s|_{P}). In that case Rad(W, s|_{W}) = ⟨r⟩ has dimension 1, and any d-space complement to ⟨r⟩ in W is nondegenerate. In particular, of the |K| + 1 distinct d-spaces ⟨U, w⟩ ≤ W, only ⟨U, r⟩ is degenerate.

( 3.2) Lemma. Let H = H_{s,d} be as in Proposition 3.1. Then in its action on G_{d}(V)_{H} the group P GL(V)_{H} has chamber orbits

{(U, P, W) | U ∈ B_{H}^{−}, P ∈ P_{H}, W ∈ B_{H}^{+}, W ∩ Rad(V, s) = 0 }

and

{(U, P, W) | U ∈ B_{H}^{−}, P ∈ P_{H}, W ∈ B_{H}^{+}, W ∩ Rad(V, s) ≠ 0 } .

In particular PΓL(V)_{H} is transitive on the points of G_{d}(V)_{H} (and CG_{d}(V)_{H}). Indeed PΓL(V)_{H} acts flag-transitively on G_{d}(V)_{H} if and only if we have one of

(a) s is nondegenerate or

(b) d = rank(V, s), the codimension of Rad(V, s).

Proof. Clearly Sp(V, s) ≤ ΓL(V)_{H}.

Let P ∈ P_{H} and X = P^{⊥}, so that V = P ⊕ X.

If P_{1} and P_{2} are two points, then ⟨P_{1}, P_{2}⟩ can be embedded in a finite dimensional subspace Y of V with Rad(Y, s|_{Y}) = Y ∩ Rad(V, s). A symplectic basis for each of P_{1} and P_{2} can be extended to symplectic bases for Y. Thus there is a g ∈ Sp(Y, s|_{Y}) taking P_{1} to P_{2}. As Y has a perpendicular complement in V, all members of Sp(Y, s|_{Y}) extend to elements of Sp(V, s).

We now need only prove that the stabilizer of P is transitive on the chambers U ≤ P ≤ W.

Any (d − 1)-space U ≤ P equals u^{⊥} ∩ P for a unique 1-space ⟨u⟩ ≤ P. Any (d + 1)-space W ≥ P equals ⟨P, x⟩ for the unique 1-space ⟨x⟩ = X ∩ W. Thus the orbits of the stabilizer of P in ΓL(V)_{H} are the same as the orbits on 1spaces within P (for B_{H}^{−}) and within X = P^{⊥} (for B_{H}^{+}). As before Sp(P, s|_{P}) × Sp(X, s|_{X}) ≤ ΓL(V)_{H}. The factors act independently on P and X. Sp(P, s|_{P}) is then transitive on those U ∈ B_{H}^{−} incident to P. Sp(X, s|_{X}) is transitive on those 1-spaces of X contained in Rad(X, s|_{S}) = Rad(V, s) and on those 1-spaces of X outside of Rad(X, s|_{X}). This gives the two orbits described. ΓL(V)_{H} acts flag-transitively precisely when one of these two orbits is empty. The second is empty if and only if Rad(V, s) = 0, that is, when s is nondegenerate as in (a). The first is empty when every (d + 1)-space that contains the nondegenerate d-space P meets Rad(V, s). This is the case if and only if d is equal to the codimension of Rad(V, s) as in (b).

No element of PΓL(V)_{H} can switch B_{H}^{−} and B_{H}^{+}; so, for 1 < d < dim_{K}V − 1, the group PΓL(V)_{H} can never act flag-transitively on CG_{d}(V)_{H}. But the group PΓL^{∗}(V)_{H} sometimes can.

( 3.3) Lemma. Let H be a geometric hyperplane of A_{d}(V) with 1 < d < dim_{K}V −1. The group PΓL(V)_{H} acts flag-transitively on G_{d}(V)_{H} if and only if it is transitive on the set of −-chambers and the set of +-chambers of CG_{d}(V)_{H}.

Proof. First suppose that PΓL(V)_{H} is flag-transitive on G_{d}(V)_{H}. Let (P_{1}, ℓ_{1}, B_{1}) and (P_{2}, ℓ_{2}, B_{2}) be two chambers of CG_{d}(V)_{H}, with B_{1}, B_{2} ∈ B^{+}, say (a similar argument handling the case B_{1}, B_{2} ∈ B^{−}). Let B_{1}^{−} and B_{2}^{−} from B^{−} be chosen so that each line ℓ_{i} is contained in the line of CG_{d}(V) determined by the incident pair (B_{i}^{−}, B_{i}). By flag-transitivity, there is a g ∈ PΓL(V)_{H} with (B_{1}^{−}, P_{1}, B_{1})^{g} = (B_{2}^{−}, P_{2}, B_{2}). This forces ℓ_{1}^{g} = ℓ_{2}. Therefore g is an element of PΓL(V)_{H} with (P_{1}, ℓ_{1}, B_{1})^{g} = (P_{2}, ℓ_{2}, B_{2}), as desired.

Now suppose that PΓL(V)_{H} is transitive on the set of −-chambers and the set of +-chambers of CG_{d}(V)_{H}. Let (U_{1}, P_{1}, W_{1}) and (U_{2}, P_{2}, W_{2}) be two chambers of G_{d}(V)_{H}. By assumption, there is a g ∈ PΓL(V)_{H} with (P_{1}, W_{1})^{g} = (P_{2}, W_{2}), so we may take P = P_{1} = P_{2} and W = W_{1} = W_{2}. Each U_{i} ∈ B_{H}^{−}, with (U_{i}, P, W) a chamber of G_{d}(V)_{H}, meets W in a line ℓ_{i} = U_{i} ∩W, which contains the point P; and ℓ_{i} = ℓ_{j} if and only if i = j. Again by flag-transitivity, there is an f ∈ PΓL(V)_{H} with (P, ℓ_{1}, W)^{f} = (P, ℓ_{2}, W). But then (U_{1}, P, W)^{f} = (U_{2}, P, W), as desired.

( 3.4) Corollary. Let H = H_{s,d} be as in Proposition 3.1 with 1 < d < dim_{K}V −1. The group PΓL^{∗}(V)_{H} acts flag-transitively on CG_{d}(V)_{H} if and only if PΓL(V)_{H} is flag-transitive on G_{d}(V)_{H} and dim_{K}V = 2d.

Proof. If PΓL^{∗}(V)_{H} is flag-transitive on CG_{d}(V)_{H}, then PΓL(V)_{H}, of index 2, has at most two orbits on the chambers of CG_{d}(V)_{H}. These must be the +- chambers and the −-chambers; so, by the lemma, PΓL(V)_{H} is flag-transitive on G_{d}(V)_{H}. As PΓL(V)_{H} has two orbits, it is proper in PΓL^{∗}(V)_{H}; therefore we have 2d = n by Lemma 2.4.

To prove the converse, we consider the two cases of Lemma 3.2 in turn. If, as in Lemma 3.2(a), s is nondegenerate, then the polarity X ↦ X^{⊥} induces an automorphism of CG_{d}(V)_{H}. Indeed it maps the chamber (U, P, W) to the chamber (W^{⊥}, P^{⊥}, U^{⊥}).

In Lemma 3.2(b), let R = Rad(V, s). Then there is a duality δ of V (here definitely not associated with s) that takes R to itself. But then δ induces an automorphism of CG_{d}(V)_{H} that interchanges B_{H}^{−} and B_{H}^{+}, as desired.

The dual of an attenuated affine Grassmannian is again attenuated. As seen in the proof of the previous corollary, when the parent symplectic form is nondegenerate on V, a dual Pfaffian affine Grassmannian is Pfaffian. On the other hand, Lemma 3.2 proves that the dual of a Pfaffian affine Grassmannian is not Pfaffian when the form is degenerate of rank larger than d.

The results on Pfaffian geometric hyperplanes are, in particular, valid for attenuated hyperplanes for which the subspace R has even codimension in the space V. This restriction on codimension is spurious.

( 3.5) Theorem. Let R be a nonzero subspace of finite codimension d (0 < d < dimV) in the space V over the field K. Set

H = H_{R} = {P ≤ V | dim_{K}P = d, P ∩ R ≠ 0 } .

(1) The set H_{R} is a geometric hyperplane of the d-Grassmann space A_{d}(V).

(2) The affine Grassmannian G_{d}(V)_{H} is connected and admits PGL(V)_{R} = PGL(V)_{H} acting flag-transitively.

(3) The locally grid geometry CG_{d}(V)_{H} is connected and transitive, and it admits PΓL^{∗}(V)_{R} = PΓL^{∗}(V)_{H} acting flag-transitively if and only if dim_{K}V = 2d.

Proof. (1) Let P_{1} and P_{2} be two d-spaces that are disjoint from R with dim(P_{1} ∩ P_{2}) = d − 1 and dim⟨P_{1}, P_{2}⟩ = d + 1. Then ⟨P_{1}, P_{2}⟩ ∩ R = T has dimension 1, and Q = ⟨P_{1} ∩ P_{2}, T ⟩ is the unique d-subspace of ⟨P_{1}, P_{2}⟩ that meets R nontrivially. Therefore H is a geometric hyperplane.

(2) Connectivity is true in general by Proposition 2.5.

The definition of H uniquely determines R, so GL(V)_{H} = GL(V)_{R}. Let v̄_{1}, … v̄_{d} be a basis of V̄ = V/R, and assume that the two d-spaces P_{1} and P_{2} of G_{d}(V)_{H} have bases x_{1}, …, x_{d} and y_{1}, … y_{d}, respectively, with x̄_{i} = v̄_{i} = ȳ_{i}, for all i. Then the map x_{i} ↦ y_{i}, for all i, extends to a member of GL(V)_{R} that is trivial on R and V̄ and takes P_{1} to P_{2}. As GL(V)_{R,P} ≃ GL(V̄) × GL(R) is transitive on the pairs of (d−1)- and (d+1)-spaces incident to the fixed d-space P of G_{d}(V)_{H}, we have the desired flag-transitivity.

(3) This follows directly from (2) as in Corollary 3.4.

Remark. Parts (1) and (2) of the theorem remain valid over arbitrary division rings without change to either statement or proof. Part (3) must be changed to say that PΓL^{∗}(V)_{H} is flag-transitive on CG_{d}(V)_{H} if and only if dim_{K}V = 2d and K has an anti-isomorphism. The backwards implication follows as before, but the direct part requires additional proof.

As we have seen in Theorem 2.3, the cases in which d is one of 1, 2, dim_{K}V −2, or dim_{K}V − 1 are exceptional. But these are the cases that admit full classification of the geometric hyperplanes.

( 3.6) Proposition. Let V be a vector space over K and d a positive integer. Let H be a geometric hyperplane of A_{d}(V).

(1) If d = 1 or d = dim_{K}V − 1, then H is an attenuated hyperplane and Aut(G_{d}(V)_{H}) is flag-transitive on G_{d}(V)_{H}.

(2) Let d = 2. Then H = H_{s,2} is a Pfaffian hyperplane for some symplectic form s. The group Aut(G_{d}(V)_{H}) is flag-transitive on G_{d}(V)_{H} if and only if either rank(V, s) = 2 or rank(V, s) = dimV. Furthermore Aut(CG_{d}(V)_{H}) is flag-transitive on CG_{d}(V)_{H} if and only if either rank(V, s) = 2 or rank(V, s) = dimV = 4.

(3) Let d = dim_{K}V −2. Then H is the dual of a Pfaffian hyperplane H_{s,2} for some symplectic form s. Again Aut(G_{d}(V)_{H}) is flag-transitive on G_{d}(V)_{H} if and only if either rank(V, s) = 2 or rank(V, s) = dimV. Furthermore Aut(CG_{d}(V)_{H}) is flag-transitive on CG_{d}(V)_{H} if and only if either rank(V, s) = 2 or rank(V, s) = dimV = 4.

Proof. Part (1) is immediate, and (3) follows directly from (2) by duality. When d = 2, as in (2), Shult’s Theorem 1.1 implies that H = H_{s,2} for some

symplectic form s. The examples given are indeed flag-transitive by Lemma 3.2 and Corollary 3.3. It remains to prove that these are the only flag-transitive examples.

Suppose that rank(V, s) > 2 and that G_{d}(V)_{H} is flag-transitive. Assume that s is degenerate. By Lemma 3.2, PΓL(V)_{H} has two orbits on chambers—that of the radical chambers (B^{−}, P, B^{+}) with B^{+} ∩ Rad(V, s) ≠ 0 and that of the nonradical chambers. We must show that Aut(G_{d}(V)_{H}) does not fuse these two orbits. But this is clear, since every nonradical chamber is in a nondegenerate Pfaffian subgeometry G_{2}(W)_{H′}, for an s-nondegenerate 4-space W containing B^{+} and H′ = H ∩ P_{2}(W), while no radical chamber is in a subgeometry with this isomorphism type. The contradiction proves s to be nondegenerate.

The previous paragraph and Lemma 2.7 show that the only flag-transitive CG_{d}(V)_{H} are the stated examples.

Proof of Theorem 1.2 and Corollary 1.3:

Proposition 2.5(2) shows that the geometries G_{d}(V)_{H} of Theorem 1.2 and CG_{d}(V)_{H} of Corollary 1.3 are both connected. All other parts of Theorem 1.2 and Corollary 1.3 then follow from the various results of this section.

4 Alternating forms and Pfaffians

Shult’s Theorem 1.1 encourages us to look for alternating forms on V that give rise to attenuated and Pfaffian geometric hyperplanes.

First consider the case of the attenuated hyperplane H_{R}. Let P be a dspace complement to R in V, and let e_{1}, …, e_{d} be a K-basis of P. For each v_{i} ∈ V, we have v_{i} = ∑_{j=1}^{d} a_{i,j}e_{j} + r_{i}, for unique r_{i} ∈ R and a_{i,j} ∈ K. Then f(v_{1}, …, v_{d}) = det([a_{i,j}]_{i,j}) is an alternating d-linear form. The form f is 0 precisely when the span Q = ⟨v_{1}, …, v_{d}⟩ projects onto a proper subspace of P, that is, when Q intersects R nontrivially and so belongs to H_{R}. The projection λ_{j} of each v_{i} onto its j-th coordinate a_{i,j} is a linear functional on V. We can think of the construction of f in terms of the canonical embedding of ⋀^{d} V^{∗} in (⋀^{d} V)^{∗} given by

(v_{1} ∧ ⋯ ∧ v_{i} ∧ ⋯ ∧ v_{d}) · (λ_{1} ∧ ⋯ ∧ λ_{j} ∧ ⋯ ∧ λ_{d}) = det([v_{i}λ_{j}]_{i,j}) .

( 4.1) Proposition. Let H_{R} be an attenuated hyperplane of A_{d}(V). Let λ_{1}, …, λ_{d} be a basis of ker_{V^{∗}} R. Then λ_{R} = ∧_{j=1}^{d}λ_{j} is an alternating dlinear form on V for which the λ_{R}-degenerate d-spaces of V are exactly those of H_{R}. Conversely, for each pure vector λ of ∧^{d}V^{∗} (thought of as an element of (∧^{d}V)^{∗}), the pure vectors of the hyperplane ker λ give an attenuated hyperplane of A_{d}(V).

We next want to find an alternating form that produces the Pfaffian examples. This is particularly easy when K = 𝔽_{2}.

( 4.2) Proposition. Let s be a symplectic form on the 𝔽_{2}-space V. The map δ(v_{1}, …, v_{d}) = det([s(v_{i}, v_{j})]_{i,j}) is an alternating d-linear form on V with δ(v_{1}, …, v_{d}) ≠ 0 if and only if ⟨v_{1}, …, v_{d}⟩ is an s-nondegenerate d-space of V.

Proof. Since, in Theorem 4.3 below, we will prove the appropriate generalization to arbitrary fields, we only sketch the proof here.

Consider the map D: V^{d} × V^{d} ⟶ K given by

D(v_{1}, …, v_{d}; w_{1} …, w_{d}) = det([s(v_{i}, w_{j})]_{i,j})

and the related

δ(v_{1}, …, v_{d}) = D(v_{1}, …, v_{d}; v_{1} …, v_{d}) .

Over the field K (initially arbitrary), we use the basic properties of determinants and the Gram matrix [s(v_{i}, v_{j})]_{i,j} to see that the maps D and δ have the properties:

(i) D(v_{1}, …, v_{d}; w_{1} …, w_{d}) = 0 if v_{i} = v_{j} or w_{i} = w_{j} for some pair of indices i ≠ j;

(ii) D(v_{1}, …, v_{d}; w_{1} …, w_{d}) is linear in each variable.

(iii) δ(v_{1}, …, v_{d}) ≠ 0 if and only if ⟨v_{1}, …, v_{d}⟩ is an s-nondegenerate d-space of V;

By (i) and (ii) D induces a symmetric bilinear form (also denoted D) on ∧^{d}V × ∧^{d}V via

D(v_{1} ∧ ⋯ ∧ v_{d}, w_{1} ∧ ⋯ ∧ w_{d}) = D(v_{1}, …, v_{d}; w_{1} …, w_{d}) ,

and δ is the quadratic form coming from the diagonal of D. By (iii) the pure vectors of ∧^{d}V that are nonsingular for δ are exactly those coming from d-spaces that are nondegenerate for s.

In the special case K = 𝔽_{2}, the quadratic form δ is in fact a linear form, giving the result.

Degeneracy of a form is detected by whether or not its discriminant, the determinant of a Gram matrix, is zero. In the proof of the previous proposition, we have seen that the discriminant provides a symmetric bilinear and diagonal quadratic form on the exterior power that indicate degeneracy. To find a linear form, as desired, we wish to “take the square root” of this quadratic form. In characteristic 2 the quadratic form is additive and thus over 𝔽_{2} is its own square root, but this is an anomaly. We must instead look for a general way of finding a square root for the discriminant. Luckily, for symplectic forms there is such a general function, namely the Pfaffian. Our presentation of Pfaffians is similar to that of Chevalley [5, p. 57] which uses exponentials in exterior algebras.

As before, let V be a K-vector space and s a symplectic form on V. For every tuple (v_{1}, …, v_{d}) from V, we define an element of the exterior algebra ⋀ V by setting

Pf(v_{1}, …, v_{d}) = ∏_{{i,j}∈U}(1 + s(v_{i}, v_{j})v_{i}v_{j}) ,

the product being over the set U of all unordered pairs {i, j} from {1, …, d}. (Note that s(v_{i}, v_{j})v_{i}v_{j} = s(v_{j}, v_{i})v_{j}v_{i}.) Every term of Pf has even degree, so the calculation of Pf can be done within the commutative subalgebra ⋀_{even} V of ⋀ V.

We now define a map pf : V^{d} ⟶ K by looking at projection onto the dgraded component of Pf:

pf(v_{1}, …, v_{d})v_{1} ⋯ v_{d} = Pf(v_{1}, …, v_{d})|_{d} .

Especially pf(v_{1}, v_{1}) = s(v_{1}, v_{2}). Here it is understood that pf(v_{1}, …, v_{d}) is 0 whenever v_{1} ⋯ v_{d} is the 0 element of ⋀ V. (In particular, since Pf ∈ ⋀_{even} V, pf is identically 0 for odd d.) We call pf(v_{1}, …, v_{d}) the Pfaffian of the ordered set of vectors v_{1}, …, v_{d} ∈ V.

The remainder of this section is devoted to a proof of ( 4.3) Theorem. The map pf is an alternating d-linear form on V with pf(v_{1}, …, v_{d}) ≠ 0 if and only if ⟨v_{1}, …, v_{d}⟩ is an s-nondegenerate d-space of V.

The map pf is indeed alternating.

( 4.4) Proposition. (1) If v_{i} = v_{j} for i ≠ j, then pf(v_{1}, …, v_{d}) = 0. (2) If σ ∈ Sym(d), then pf(v_{1}, …, v_{d}) = sgn(σ) pf(v_{σ(1)}, …, v_{σ(d)}).

Proof. (1) is clear, since in this case v_{1} ⋯ v_{d} is 0 in ⋀ V.

For (2) we start with Pf(v_{σ(1)}, …, v_{σ(d)}) = Pf(v_{1}, …, v_{d}) from the definition and remarks following it. Therefore, the d-graded piece is

pf(v_{σ(1)}, …, v_{σ(d)})v_{σ(1)} ⋯ v_{σ(d)} = pf(v_{1}, …, v_{d})v_{1} ⋯ v_{d}
= pf(v_{1}, …, v_{d})sgn(σ) v_{σ(1)} ⋯ v_{σ(d)} ,

as desired.

We first prove Cayley’s theorem, showing that the Pfaffian provides the desired square root of the discriminant. This is the fundamental property of the Pfaffian and is crucial in our proof of Theorem 4.3.

( 4.5) Theorem. pf(v_{1}, …, v_{d})^{2} = det[s(v_{i}, v_{j})]_{i,j} .

Proof. If {v_{1}, …, v_{d}} is linearly dependent, then the lefthand side is 0 by definition. But in that case, the Gram matrix [s(v_{i}, v_{j})]_{i,j} will be that of a form with rank less than d, so its determinant will also be 0. Therefore we may assume that v_{1}, …, v_{d} form a linearly independent set in V. Both sides are also 0 if d is odd, so we may assume d = 2m.

We begin with a standard reduction ([11, Lemma 7.2.1],[23, Prop.2.2]) in the calculation of the discriminant

det[s(v_{i}, v_{j})]_{i,j} = ∑_{σ∈Sym(d)} sgn(σ) ∏_{i} s(v_{i}, v_{σ(i)}) .

Write Sym(d) = F ∪ O ∪ E, where F consists of those permutations with a fixed point, O consists of those permutations with no fixed points but with orbits of odd length, and E consists of those permutations with all orbits of even length.

For σ ∈ F, ∏_{i} s(v_{i}, v_{σ(i)}) = 0 as s(v_{j}, v_{j}) = 0 for a fixed point j. For σ ∈ O, among the odd cycles of σ, let c be the cycle containing the point of {1, … d} of smallest value. Set σ′ = c^{−2}σ. Then

(i) σ ≠ σ′ ∈ O and (σ′)′ = σ;

(ii) sgn(σ) = sgn(σ′);

(iii) ∏_{i} s(v_{i}, v_{σ(i)}) = − ∏_{i} s(v_{i}, v_{σ′(i)}). We conclude that

det[s(v_{i}, v_{j})]_{i,j} = ∑_{σ∈E} sgn(σ) ∏_{i} s(v_{i}, v_{σ(i)}) .  (1)

We will prove that squaring the Pfaffian gives this calculation of the discriminant.

A 1-factor of the complete graph on {1, 2, …, d = 2m} is an m edge subgraph of valency 1, one edge on each vertex. We let I be the set of all such 1-factors. Starting from the definition of the Pfaffian, we find in ⋀_{even}

pf(v_{1}, …, v_{d})v_{1} ⋯ v_{d} = Pf(v_{1}, …, v_{d})|_{d}
= ( ∏_{{i,j}∈U}(1 + s(v_{i}, v_{j})v_{i}v_{j}))|_{d}
= ∑_{I∈I} ∏_{{i,j}∈I} s(v_{i}, v_{j})v_{i}v_{j} .  (2)

Let R have dimension d. We replace V by V ⊕R and extend the form s so that R is in its radical. Assume that r_{1}, …, r_{d} are linearly independent elements of R, and set w_{i} = v_{i} +r_{i}, for i = 1, …, d. Then {v_{1}, …, v_{d}, w_{1}, …, w_{d}} is linearly independent and

s(v_{i}, v_{j}) = s(v_{i}, w_{j}) = s(w_{i}, v_{j}) = s(w_{i}, w_{j}) ,

for all i, j. By definition and (2) we have

pf(v_{1}, …, v_{d})^{2}v_{1} ⋯ v_{d}w_{1} ⋯ w_{d} =
= pf(v_{1}, …, v_{d})v_{1} ⋯ v_{d} · pf(w_{1}, …, w_{d})w_{1} ⋯ w_{d}
= (∑_{I∈I} ∏_{{i,j}∈I} s(v_{i}, v_{j})v_{i}v_{j})(∑_{J∈I} ∏_{{k,l}∈J} s(w_{k}, w_{l})w_{k}w_{l}) . (3)

We now need a bijection between the permutations of E and ordered pairs of 1-factors:

E ⟷ I_{G} × I_{W} given by σ ⟷ (I, J) ,

where, in order to distinguish 1-factors, we call the I from I_{G} (= I) green and the J from I_{W} (= I) white.

First, let I (green) and J (white) be two 1-factors. The graph I ∪ J has degree 2 and so is a union of cycles. Each cycle has edges of alternating colors (green, white, green, white,...) and so has even length. To find a unique σ ∈ E, we only need to orient each cycle c of σ (for instance, to distinguish (1, 2, 3, 4) from (4, 3, 2, 1)). This we do by finding the smallest value i_{c} in c and then orienting the cycle in the direction of the green edge on i_{c}.

Conversely, if we begin with σ ∈ E, then each cycle c breaks into two partial 1-factors. To find unique corresponding I ∈ I_{G} and J ∈ I_{W} we must color each partial 1-factor. We do this by coloring each edge i_{c} ∼ σ(i_{c}) green. This gives the bijection.

From (1) and (3), we will be done when we have proven

sgn(σ) (∏_{i} s(v_{i}, v_{σ(i)})) v_{1} ⋯ v_{d} w_{1} ⋯ w_{d} =

∏_{{i,j}∈I} s(v_{i}, v_{j})v_{i}v_{j} ∏_{{k,l}∈J} s(w_{k}, w_{l})w_{k}w_{l} ,  (4)

for any fixed σ ⟷ (I, J). To do this, we rewrite the righthand side.

Let z_{i} = v_{i} and z′_{σ(i)} = v_{σ(i)}, if the edge i ∼ σ(i) is green. Let z_{i} = w_{i} and z′_{σ(i)} = w_{σ(i)}, if the edge i ∼ σ(i) is white. Then

∏_{{i,j}∈I} s(v_{i}, v_{j})v_{i}v_{j} ∏_{{k,l}∈J} s(w_{k}, w_{l})w_{k}w_{l} = ∏_{i} s(z_{i}, z′_{σ(i)}) ∏_{i} z_{i} z′_{σ(i)} .

Since s(z_{i}, z′_{σ(i)}) = s(v_{i}, v_{σ(i)}), the proof of (4) is reduced to verification of

sgn(σ) v_{1} ⋯ v_{d} w_{1} ⋯ w_{d} = ∏_{i} z_{i} z′_{σ(i)} .  (5)

We rewrite ∏_{i} z_{i} z′_{σ(i)} using a suggestive two line notation:

∏_{i} z_{i} z′_{σ(i)} = ∏_{i} \begin{matrix}z_{i} \\ z′_{σ(i)} \end{matrix} = \begin{matrix}z_{1} \\ z′_{σ(1)} \end{matrix} ⋯ \begin{matrix}z_{i} \\ z′_{σ(i)} \end{matrix} ⋯ \begin{matrix}z_{d} \\ z′_{σ(d)} \end{matrix} .

For instance, the product abhkxy would be written \begin{matrix}a \\ b \end{matrix} \begin{matrix}h \\ k \end{matrix} \begin{matrix}x \\ y \end{matrix} . In the exterior

algebra ⋀ V, the identity

z_{j} z′_{p} z_{j+1} z′_{q} = −z_{j} z′_{q} z_{j+1} z′_{p}

thus becomes

\begin{matrix}z_{j} \\ z′_{p} \end{matrix} \begin{matrix}z_{j+1} \\ z′_{q} \end{matrix} = − \begin{matrix}z_{j} \\ z′_{q} \end{matrix} \begin{matrix}z_{j+1} \\ z′_{p} \end{matrix} .

Therefore

∏_{i} z_{i} z′_{σ(i)} = \begin{matrix}z_{1} \\ z′_{σ(1)} \end{matrix} ⋯ \begin{matrix}z_{i} \\ z′_{σ(i)} \end{matrix} ⋯ \begin{matrix}z_{d} \\ z′_{σ(d)} \end{matrix} = sgn(σ) \begin{matrix}z_{1} \\ z′_{1} \end{matrix} ⋯ \begin{matrix}z_{i} \\ z′_{i} \end{matrix} ⋯ \begin{matrix}z_{d} \\ z′_{d} \end{matrix} .

Here always {z_{i}, z′_{i}} = {v_{i}, w_{i}}. Half the edges of I ∪ J are white, so for exactly m = d/2 values of i we have z_{i} = w_{i} and z′_{i} = v_{i}. Hence

∏_{i} z_{i} z′_{σ(i)} = sgn(σ)(−1)^{m} \begin{matrix}v_{1} \\ w_{1} \end{matrix} ⋯ \begin{matrix}v_{i} \\ w_{i} \end{matrix} ⋯ \begin{matrix}v_{d} \\ w_{d} \end{matrix}
= sgn(σ)(−1)^{m}v_{1} w_{1} ⋯ v_{i} w_{i} ⋯ v_{d} w_{d}
= sgn(σ)(−1)^{m}(−1)^{2m(2m−1)/2}v_{1} ⋯ v_{d} w_{1} ⋯ w_{d}
= sgn(σ)v_{1} ⋯ v_{d} w_{1} ⋯ w_{d} ,

as desired for (5). This completes the proof of Theorem 4.5.

( 4.6) Corollary. Let t_{1}, …, t_{d} ∈ Rad(V, s). Then

pf(v_{1}, …, v_{d}) = pf(v_{1} + t_{1}, …, v_{d} + t_{d}) .

Proof. If {v_{1}, …, v_{d}} and {v_{1} + t_{1}, …, v_{d} + t_{d}} are both linearly independent sets, then this is clear from the definition (and we have used this observation in our proof of the theorem). If {v_{1}, …, v_{d}} and {v_{1} + t_{1}, …, v_{d} + t_{d}} are both linearly dependent, then both sides are 0 by definition and again the result is clear.

Suppose now that {v_{1}, …, v_{d}} is linearly dependent but {v_{1} +t_{1}, …, v_{d} +t_{d}} is linearly independent. Then pf(v_{1}, …, v_{d}) = 0 by definition. On the other hand, we must use the exponential to calculate pf(v_{1} + t_{1}, …, v_{d} + t_{d}). By the theorem, its square satisfies

pf(v_{1} + t_{1}, …, v_{d} + t_{d})^{2} = det[s(v_{i} + t_{i}, v_{j} + t_{j})]_{i,j} = det[s(v_{i}, v_{j})]_{i,j} .

Since the v_{i} are linearly dependent, this last is again 0, completing the proof of the corollary.

( 4.7) Proposition. For d > 2

pf(v_{1}, …, v_{d−1}, v_{d}) = s (∑_{k=1}^{d−1} α_{k}v_{k}, v_{d}) ,

where α_{k} = (−1)^{k+1} pf(v_{1}, …, v̂_{k}, …, v_{d−1}). Here (v_{1}, …, v̂_{k}, …, v_{d−1}) is the (d − 2)-tuple that results from deleting v_{k} from (v_{1}, …, v_{d−1}).

Proof. By the previous corollary, we may assume that v_{1}, …, v_{d} is a linearly independent collection of vectors. If d is odd, then the result is trivial; so we assume that d is even. We have

pf(v_{1}, …, v_{d})v_{1} ⋯ v_{d} =( ∏_{{i,j}∈U}(1 + s(v_{i}, v_{j})v_{i}v_{j}))|_{d}

= ( ∏_{d∉{i,j}}(1 + s(v_{i}, v_{j})v_{i}v_{j}) ∏_{k=1}^{d−1}(1 + s(v_{k}, v_{d})v_{k}v_{d}))|_{d}

= ∑_{k=1}^{d−1}( ∏_{k,d∉{i,j}}(1 + s(v_{i}, v_{j})v_{i}v_{j}))|_{d−2} s(v_{k}, v_{d})v_{k}v_{d}

= ∑_{k=1}^{d−1} pf(v_{1}, …, v̂_{k}, …, v_{d−1})v_{1} ⋯ v̂_{k} ⋯ v_{d−1} · s(v_{k}, v_{d})v_{k}v_{d}

= ∑_{k=1}^{d−1}(−1)^{d−1−k} pf(v_{1}, …, v̂_{k}, …, v_{d−1})s(v_{k}, v_{d}) v_{1} ⋯ v_{k} ⋯ v_{d}

= ∑_{k=1}^{d−1} s((−1)^{d−1−k} pf(v_{1}, …, v̂_{k}, …, v_{d−1})v_{k}, v_{d}) v_{1} ⋯ v_{d};

hence pf(v_{1}, …, v_{d}) = s(∑_{k=1}^{d−1} α_{k}v_{k}, v_{d}) , as claimed.

Proof of Theorem 4.3:

As pf(v_{1}, v_{2}) = s(v_{1}, v_{2}), the map pf is linear in its last variable by Proposition 4.7. Therefore pf is linear in all variables by Proposition 4.4.2. It is alternating by Proposition 4.4. Finally, pf(v_{1}, …, v_{d}) is nonzero precisely when ⟨v_{1}, …, v_{d}⟩ is nondegenerate by Theorem 4.5.

Remarks. There are many treatments of Pfaffians. See Knuth [16] for historical discussion. Often 1-factors (perfect matchings) with appropriate sign conventions are used from the start [11, 13, 16, 23]. In enumerative applications [11, 23] it is appropriate to consider only characteristic 0 (from which the general result can be derived [10, 14]).

Dress and Wentzel [8] use exterior algebra and give a recursive definition of the Pfaffian, which is effectively the identity of Proposition 4.7 initialized by taking pf(v_{1}, v_{2}) = s(v_{1}, v_{2}). Their Theorem 2 proves the Pfaffian, under that definition, to be an alternating multilinear form.

Here we sought a direct construction of an alternating form having the properties of Theorem 4.3. Our version is motivated by that of Chevalley [5], which uses the exponential, and is similar to that of [4, 19], which use derivations, and that of [10, pp. 588-589], which uses divided powers. It is free from recursion, restrictions on characteristic, and additional sign conventions.

5 Transitivity of finite affine Grassmannians

This section is concerned with the proofs of Theorem 1.4 and Corollary 1.5.

Theorem 1.2 shows that the Pfaffian affine Grassmannians come from geometric hyperplanes H for which PΓL(V)_{H} is transitive on the d-spaces that are nondegenerate for a particular alternating d-linear form. This is a Witt type property for these forms. It seems possible that all such transitive affine Grassmannians could be classified. Indeed we may be seeing all of them in the Pfaffian and attenuated examples. In this section we prove that this is the case when we restrict attention to finite, flag-transitive affine Grassmannians.

We will need a result from the literature.

( 5.1) Theorem. Let G ≤ ΓL_{n}(q) be transitive on the 1-spaces of V = 𝔽_{q}^{n}. Then G is also transitive on the 1-spaces of V^{∗}, and we have one of:

(a) n ≤ 4;

(b) G = SL_{2}(13) with q = 3 and n = 6;

(c) G ≤ ΓL_{1}(q^{n});

(d) ΓL_{m}(q^{r}) ⊵ G ⊵ SL_{m}(q^{r}) with n = mr > 4 and m ≥ 2;

(e) ΓSp_{m}(q^{r}) ⊵ G ⊵ Sp_{m}(q^{r}) with n = mr > 4 and m even;

(f) ΓG_{2}(q^{r}) ⊵ G ⊵ G_{2}(q^{r})′ with q even and n = mr for m = 6. In cases (d), (e), and (f), V is the natural 𝔽_{q^{r}}-module of dimension m viewed as a 𝔽_{q}-module.

Proof. See [15, p. 68] or [17, p. 199].

Here ΓSp_{m}(q^{r}) is the full group of semisimilarities of a nondegenerate symplectic form on 𝔽_{q^{r}}^{m} and consists of Sp_{m}(q^{r}) extended by the automorphisms and scalars of 𝔽_{q^{r}}. Similarly ΓG_{2}(q^{r}) is G_{2}(q^{r}) extended by the automorphisms and scalars of 𝔽_{q^{r}}.

Proof of Corollary 1.5:

One direction is immediate from Corollary 1.3. Assume now that G = Aut(CG_{d}(V)_{H}) is flag-transitive on CG_{d}(V)_{H}. By Lemma 2.7, n = 2d and the subgroup H = Aut(G_{d}(V)_{H}) has index 2 in G, the elements of G\H exchanging B_{H}^{−} and B_{H}^{+}.

If d ∈ {1, 2, n − 2, n − 1} then Proposition 3.6 gives the corollary, so we may assume that 2 < d < n − 2. Therefore, by Theorem 2.3, we have H = Aut(G_{d}(V)_{H}) = PΓL(V)_{H}.

As G has a single chamber orbit on CG_{d}(V)_{H}, the index 2 subgroup H has at most two chamber orbits. Those must be that of the −-chambers and that of the +-chambers of CG_{d}(V)_{H}. By Lemma 3.3, H is flag-transitive on G_{d}(V)_{H}. Hence by Theorem 1.4, the geometric hyperplane H is either attenuated or nondegenerate Pfaffian, completing the corollary.

Remark. Except in its use of Theorem 1.4, our proof of Corollary 1.5 does not require K to be finite.

The rest of the section is devoted to our proof of Theorem 1.4. The examples are flag-transitive by Theorem 1.2, so we only need to prove that a flag-transitive finite affine Grassmannian is either attenuated or nondegenerate Pfaffian.

We let H be a geometric hyperplane of A_{d}(V), for V = 𝔽_{q}^{n}, and assume that Aut(G_{d}(V)_{H}) is transitive on the chambers of G_{d}(V)_{H}. Throughout we let G = ΓL(V)_{H} and q = p^{a}, where p is the characteristic of K = 𝔽_{q}.

We have some initial reductions.

( 5.2) Lemma. In proving Theorem 1.4 we may assume:

(1) H is not attenuated;

(2) H is the set of d-spaces that are degenerate for the alternating d-linear form f on V;

(3) 2 < d ≤ n/2 < n − 2, especially n ≥ 6;

(4) Aut(G_{d}(V)_{H}) = PΓL(V)_{H};

(5) G = ΓL(V)_{H} is irreducible on V and flag-transitive in its action on G_{d}(V)_{H}.

Proof. We may assume, as in (1), that H is not attenuated, with the goal of proving that H is a nondegenerate Pfaffian geometric hyperplane.

By Shult’s Theorem 1.1, H is the collection of d-spaces that are degenerate for an alternating d-linear form f as in (2).

We may take 2 < d < n − 2 by Proposition 3.6. The affine Grassmannian G_{d}(V)_{H} in A_{d}(V) is isomorphic to its dual in A_{n−d}(V), and the list of conclusions to the theorem is closed under duality; so we may also assume d ≤ n/2. This gives (3).

By (3) and Theorem 2.3 we have Aut(G_{d}(V)_{H}) = PΓL(V)_{H} as in (4).

By (4) and assumption, the action of G = ΓL(V)_{H} is flag-transitive. If G is reducible on V, then H is attenuated by Theorem 2.6, counter to (1).

( 5.3) Lemma. Let S = Stab_{G}(P) be the global stabilizer in G of P ∈ G_{d}(V)_{H}. Then S ≤ Stab_{G}(P^{θ}), and S is transitive on the 1-spaces of P. Indeed, S is transitive on the pairs (X, W) where X is a 1-space of P and W is a (d+1)-space containing P.

Proof. By Proposition 2.2, S ≤ Stab_{G}(P^{θ}). By flag-transitivity S is transitive on the (d − 1)-spaces B^{−} of P, so by duality it is transitive on the 1-spaces of P. Indeed, as S is transitive on the pairs (U, W) with U a (d − 1)- subspace of P and W and (d + 1)-space containing P, it is also transitive on pairs (X, W) as described.

( 5.4) Corollary. G is transitive on the 1-spaces of V.

Proof. The group G is irreducible on V by Lemma 5.2(5), so Rad(V, f) is trivial. That is, for every v ∈ V there is a P in P_{H} with v ∈ V. By the lemma, the stabilizer of P is transitive on the set of 1-spaces of P. Since G_{d}(V)_{H} is connected by Proposition 2.5, the group G is transitive on the full set of 1-spaces of V.

( 5.5) Lemma. There is no affine Grassmanian of A_{3}(𝔽_{3}^{6}) that admits SL_{2}(13) acting flag-transitively.

Proof. The number of chambers (B^{−}, P, B^{+}) would be 13^{2}|P_{H}|, but 13^{2} does not divide |SL_{2}(13)|.

( 5.6) Lemma. The subgroup N = G^{(∞)} is isomorphic to one of SL_{n}(q), Sp_{n}(q), or G_{2}(q)′. Indeed, with an appropriate choice of notation we have, for n = mr and m ≥ 2, one of:

(d) ΓL_{m}(q^{r}) ⊵ G ⊵ N = SL_{m}(q^{r});

(e) ΓSp_{m}(q^{r}) ⊵ G ⊵ N = Sp_{m}(q^{r}) with m even;

(f) ΓG_{2}(q^{r}) ⊵ G ⊵ N = G_{2}(q^{r})′ with q even and m = 6.

Proof. We must eliminate cases (a)-(c) of Theorem 5.1. As n ≥ 6, case (a) does not occur. Case (b) is disposed of in Lemma 5.5.

Now suppose as in (c) that G ≤ ΓL_{1}(q^{n}). Recall that q = p^{a}, for prime p; set b = na. Thus |G| divides (q^{n} − 1)b.

Let (B^{−}, P, B^{+}) be a chamber of G_{d}(V)_{H}. Then the stabilizer in G of B^{−} is transitive on the various P ∈ P_{H} containing it. These form a hyperplane complement in the quotient space V/B^{−}, so the order of G is divisible by q^{n−d}. Similarly the stabilizer of B^{+} is transitive on { P ∈ P_{H} | P ≤ B^{+} }, a hyperplane complement in the dual of B^{+}; so |G| is divisible by q^{d}. Thus q^{⌊n/2⌋} divides (q^{n} − 1)b, whence (p^{a})^{⌊n/2⌋} divides b. This in turn implies that p^{b−a} divides b^{2}. As b = na ≥ 6a, we conclude that p^{5b} divides b^{12}; and so 32^{b} ≤ b^{12}. This is false for b ≥ 7. As 6 ≤ n and b = na, we can only have a = 1 and n = b = 6. Since p^{b−a} must divide b^{2}, this yields a contradiction.

( 5.7) Lemma. We have r = 1 and n = m.

Proof. Set F = End_{K}(N) = 𝔽_{q^{r}}.

We first claim that P = F P for P ∈ P_{H}. Otherwise there is a 1-dimensional K-space X ≤ P with P < F X+P. In that case, there is a (d+1)-dimensional Kspace W_{0} ≤ F X +P. By Lemma 5.3, the stabilizer of X and P in G (≤ ΓL_{F} (V)) remains transitive on the set of (d + 1)-dimensional K-spaces W containing P. Therefore W ≤ F X + P always, and so V = F X + P. Thus

dim_{K}V = dim_{K}(F X + P) = dim_{K}F X + dim_{K}P − dim_{K}F X ∩ P .

Now dim_{K}F X = r ≤ n/2 by Lemma 5.6; and dim_{K}P = d ≤ n/2 by Lemma 5.2; and F X ∩ P ≥ X. Therefore our dimension calculation yields

n ≤ n/2 + n/2 − 1 ,

a contradiction. Thus P = F P, as claimed.

Let collinear P_{1}, P_{2} ∈ P_{H}. Then by the previous paragraph F acts on the 1-dimensional K-space P_{1}/P_{1} ∩ P_{2}. Thus K = F, r = 1, and n = m, as desired.

( 5.8) Lemma. There is a subgroup N ≃ Sp_{n}(q) normal in G and a nondegenerate N-invariant symplectic form s on V with H = H_{s,d}, as in the Pfaffian case (b) of Theorem 1.4. In particular d is even.

Proof. From Lemmas 5.6 and 5.7, we have three possibilities: G normalizes a subgroup N that is one of SL_{n}(q), Sp_{n}(q), or G_{2}(q)′ (where in the last case n = 6 and q is even) with, respectively, SL_{n}(q) ≤ G ≤ ΓL_{n}(q), Sp_{n}(q) ≤ G ≤ ΓSp_{n}(q), or G_{2}(q)′ ≤ G ≤ ΓG_{2}(q).

By Lemma 5.3, the stabilizer S ∩ N of P ∈ P_{H} also fixes P^{θ} and so leaves invariant the decomposition V = P ⊕ P^{θ}. In particular N is not SL_{n}(q), since there the stabilizer of any P is transitive on the set of complements.

If G normalizes N = G_{2}(q)′, then there is a nondegenerate symplectic form s on V for which G induces semisimilarities [1]. As 2 < d < n − 2 we must have d = 3. By Lemma 5.3, S is irreducible on P; so P is totally isotropic for s. The stabilizer S ∩ N of P is then parabolic and stabilizes no complement P^{θ} [1]. This is a contradiction.

Finally, if G normalizes N = Sp_{n}(q), then again there is a nondegenerate symplectic form s on V for which G induces semisimilarities. Again S is irreducible on P and fixes the complement P^{θ}, and so P is nondegenerate for s and has even dimension d. We conclude that H contains H_{s,d}. By Proposition 2.5(1) we have H = H_{s,d}, as desired.

This lemma completes the proof of Theorem 1.4.

Acknowledgements. The author has talked with many people about parts of this paper over the years and thanks them all for their help and interest. The list includes Rieuwert Blok, Francis Buekenhout, Hans Cuypers, Alice Devillers, Yoshimi Egawa, Anna Kasikova, Ulrich Meierfrankenfeld, Thomas Meixner, Antonio Pasini, Bruce Sagan, Jason Troye, and especially Ernie Shult.

References

[1] M. Aschbacher, Chevalley groups of type G_{2} as the group of a trilinear form, J. Algebra 109 (1987), 193–259.

[2] R. Blok, Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces, Adv. Geom. 5 (2005), 509–532.

[3] R. Blok and J.I. Hall, Extensions of isomorphisms for affine grassmannians over 𝔽_{2}, Adv. Geom., to appear.

[4] C. Chevalley, “Fundamental Concepts of Algebra,” Academic Press Inc., New York, 1956.

[5] C. Chevalley, “The Algebraic Theory of Spinors and Clifford Algebras, Collected Works, Volume 2,” eds.: P. Cartier, C. Chevalley, Springer-Verlag, 1997.

[6] W.-L. Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32–67.

[7] H. Cuypers, Affine Grassmannians, J. Comb. Th. (Ser. A) 70 (1995), 289– 304.

[8] A.W.M. Dress and W. Wenzel, A simple proof of an identity concerning Pfaffians of skew symmetric matrices, Adv. Math. 112 (1995), 120-134.

[9] Y. Egawa, personal communication, 1981.

[10] D. Eisenbud, “Commutative Algebra with a View Toward Algebraic Geometry,” Springer-Verlag, 1995.

[11] C.D. Godsil, “Algebraic Combinatorics,” Chapman and Hall, 1993.

[12] J.I. Hall and E.E. Shult, Geometric hyperplanes of non-embeddable Grassmannians, Europ. J. Combin. 14 (1993), 29–35.

[13] A.M. Hamel, Pfaffian identities: a combinatorial approach, J. Combin. Th. (Ser. A) 94 (2001), 205–217.

[14] N. Jacobson, “Basic Algebra I,” Second Edition, W. H. Freeman and Company, 1985.

[15] W.M. Kantor, Homogeneous designs and geometric lattices, J. Combin. Th. (Ser. A) 38 (1985), 66–74.

[16] D.E. Knuth, Overlapping Pfaffians, Electron. J. Combin. 3 (1996); reprinted in: D.E. Knuth, “Selected Papers on Discrete Mathematics,” CSLI Lecture Notes 106, CSLI Publications, Stanford, 2003, pp. 105–121.

[17] M.W. Liebeck, The classification of finite linear spaces with flag-transitive automorphism groups of affine type, J. Combin. Th. (Ser. A) 84 (1998), 196–235.

[18] T. Meixner and A. Pasini, A census of known flag-transitive extended grids, in: “Finite Geometries and Combinatorics,” eds. F. De Clerck et al, Cambridge Univ. Press, 1993, 249–268.

[19] D.G. Northcott, “Multilinear Algebra,” Cambridge University Press, Cambridge, 1984.

[20] E.E. Shult, Geometric hyperplanes of embeddable Grassmannians, J. Alg. 145 (1992), 55–82.

[21] E.E. Shult, Rank 3 geometries with simplicial residues, in: “Geometry, Combinatorial Designs and Related Structures (Spetses, 1996),” London Math. Soc. Lecture Note Series 245 (1997), 217–239.

[22] E.E. Shult, A survey of (c^{∗}, c)-geometries, unpublished manuscript (1996), 53 pages.

[23] J.R. Stembridge, Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990), 96–131.

[24] D.E. Taylor, “The Geometry of the Classical Groups,” Heldermann Verlag, Berlin, 1992.