Journal of Algebraic Combinatorics

, Volume 44, Issue 1, pp 59–79 | Cite as

Polarized non-abelian representations of slim near-polar spaces

Article

Abstract

In (Bull Belg Math Soc Simon Stevin 4:299–316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195–213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding.

Keywords

Near-polar space Universal/ Polarized non-abelian representation Universal projective embedding Minimal polarized embedding Extraspecial 2-group  Combinatorial group theory 

Mathematics Subject Classification

05B25 51A45 51A50 20F05 

1 Introduction

Projective embeddings of point-line geometries have been widely studied. A projective embedding is a map from the point set of a point-line geometry \(\mathcal {S}\) to the point set of a projective space \(\mathrm {PG}(V)\) mapping lines of \(\mathcal {S}\) to full lines of \(\mathrm {PG}(V)\). In case \(\mathcal {S}\) has three points per line, the underlying field of V is \(\mathbb {F}_2\). For such a geometry, a projective embedding can alternatively be viewed as a map \(p \mapsto \bar{v}_p\) from the point set of \(\mathcal {S}\) to the nontrivial elements of the additive group of V such that if \(\{ p_1,p_2,p_3 \}\) is a line of \(\mathcal {S}\), then \(\bar{v}_{p_3} = \bar{v}_{p_1} + \bar{v}_{p_2}\). This alternative point of view allows to generalize the notion of projective embeddings to so-called representations, where points of the slim geometry are no longer mapped to points of a projective space or to nonzero vectors of a vector space, but to involutions of a group R, the so-called representation group. If R is a non-abelian group, then the representation itself is also called non-abelian.

Non-abelian representations have been studied for a variety of geometries, including polar spaces and dense near polygons. In this paper, we study non-abelian representations for a class of parapolar spaces that include both the polar spaces and the dense near polygons. This class of parapolar spaces was introduced by Shult [15] and called near-polar spaces in [2].

In this paper, we restrict to those near-polar spaces that are slim and to a particular family of non-abelian representations, the so-called polarized ones. For polarized non-abelian representations of slim near-polar spaces, we derive quite some information about the representation groups. We show that these representation groups are closely related to extraspecial 2-groups, and obtain information about the centers of these groups. We also show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding (by taking a suitable quotient).

2 Preliminaries

2.1 Partial linear spaces and their projective embeddings

Let \(\mathcal {S}=(\mathcal {P},\mathcal {L},\mathrm {I})\) be a point-line geometry with nonempty point set \(\mathcal {P}\), line set \(\mathcal {L}\) and incidence relation \(\mathrm {I} \subseteq \mathcal {P} \times \mathcal {L}\).

We call \(\mathcal {S}\) a partial linear space if every two distinct points of \(\mathcal {S}\) are incident with at most one line. We call \(\mathcal {S}\)slim if every line of \(\mathcal {S}\) is incident with precisely three points. In the sequel, all considered point-line geometries will be partial linear spaces. We will often identify a line with the set of points incident with it. The incidence relation then corresponds to “containment.”

A subspace of \(\mathcal {S}\) is a set X of points with the property that if a line L has at least two of its points in X then all the points of L are in X. A hyperplane of \(\mathcal {S}\) is a subspace, distinct from \(\mathcal {P}\), meeting each line of \(\mathcal {S}\).

The distance \(\mathop {\text{ d }}\nolimits (x_1,x_2)\) between two points \(x_1\) and \(x_2\) of \(\mathcal {S}\) will be measured in the collinearity graph of \(\mathcal {S}\). A path of minimal length between two points of \(\mathcal {S}\) is called a geodesic. A subspace X of \(\mathcal {S}\) is called convex if every point on a geodesic between two points of X is also contained in X. If \(x_1\) and \(x_2\) are two points of \(\mathcal {S}\), then the intersection of all convex subspaces containing \(\{ x_1,x_2 \}\) is denoted by \(\langle x_1,x_2\rangle \). (This is well defined since \(\mathcal {P}\) is a convex subspace.) The set \(\langle x_1,x_2\rangle \) itself is a convex subspace and hence it is the smallest convex subspace of \(\mathcal {S}\) containing \(\{ x_1,x_2 \}\). The subspace \(\langle x_1,x_2\rangle \) is called the convex closure of \(x_1\) and \(x_2\).

A full projective embedding of \(\mathcal {S}\) is a map e from \(\mathcal {P}\) to the point set of a projective space \(\Sigma \) satisfying: (i) \(\langle e(\mathcal {P})\rangle _\Sigma = \Sigma \); and (ii) \(e(L) := \{e(x) \mid x \in L\}\) is a full line of \(\Sigma \) for every line L of \(\mathcal {S}\). If e is moreover injective, then the full projective embedding e is called faithful. A full projective embedding e from \(\mathcal {S}\) into a projective space \(\Sigma \) will briefly be denoted by \(e: \mathcal {S} \rightarrow \Sigma \).

If N is the maximum dimension of a projective space into which \(\mathcal {S}\) is fully embeddable, then the number \(N+1\) is called the embedding rank of \(\mathcal {S}\) and is denoted by \(er(\mathcal {S})\). The number \(er(\mathcal {S})\) is only defined when \(\mathcal {S}\) is fully embeddable.

Two full projective embeddings \(e_1: \mathcal {S} \rightarrow \Sigma _1\) and \(e_2:\mathcal {S} \rightarrow \Sigma _2\) of \(\mathcal {S}\) are called isomorphic (denoted by \(e_1 \cong e_2\)) if there exists an isomorphism \(\theta \) from \(\Sigma _1\) to \(\Sigma _2\) such that \(e_2 = \theta \circ e_1\).

Let \(e:\mathcal {S} \rightarrow \Sigma \) be a full projective embedding of \(\mathcal {S}\) and suppose \(\alpha \) is a subspace of \(\Sigma \) satisfying the following two properties:
  1. (Q1)

    \(e(p) \not \in \alpha \) for every point p of \(\mathcal {S}\);

     
  2. (Q2)

    \(\langle \alpha ,e(p_1)\rangle \not = \langle \alpha ,e(p_2)\rangle \) for any two distinct points \(p_1\) and \(p_2\) of \(\mathcal {S}\).

     
We denote by \(\Sigma /\alpha \) the quotient projective space whose points are those subspaces of \(\Sigma \) that contain \(\alpha \) as a hyperplane. Since \(\alpha \) satisfies properties (Q1) and (Q2), it is easily verified that the map which associates with each point x of \(\mathcal {S}\) to the point \(\langle \alpha ,e(x)\rangle \) of \(\Sigma /\alpha \) defines a full projective embedding of \(\mathcal {S}\) into \(\Sigma /\alpha \). We call this embedding a quotient of e and denote it by \(e/\alpha \).

If \(\mathcal {S}\) is a fully embeddable slim partial linear space, then by Ronan [12], \(\mathcal {S}\) admits up to isomorphism a unique full projective embedding \(\widetilde{e}:\mathcal {S} \rightarrow \widetilde{\Sigma }\) such that every full projective embedding e of \(\mathcal {S}\) is isomorphic to a quotient of \(\widetilde{e}\). The full projective embedding \(\widetilde{e}\) is called the universal embedding of \(\mathcal {S}\). We have \(er(\mathcal {S}) = \dim (\widetilde{\Sigma })+1\). If \(\mathcal {S}\) admits a faithful full projective embedding, then the universal embedding \(\widetilde{e}\) of \(\mathcal {S}\) is also faithful.

2.2 Near polygons

A partial linear space \(\mathcal {S}=(\mathcal {P},\mathcal {L},\mathrm {I})\) is called a near polygon if for every point p and every line L, there exists a unique point on L nearest to p. If \(d \in \mathbb {N}\) is the maximal distance between two points of \(\mathcal {S}\) (= the diameter of \(\mathcal {S}\)), then the near polygon is also called a near 2d-gon. A near 0-gon is a point, and a near 2-gon is a line. Near quadrangles are usually called generalized quadrangles. A near polygon is called dense if every line is incident with at least three points and if every two points at distance 2 have at least two common neighbors.

2.3 Polar and dual polar spaces

A partial linear space \(\mathcal {S}=(\mathcal {P},\mathcal {L},\mathrm {I})\) is called a polar space if for every point p and every line L, either one or all points of L are collinear with p. The radical of a polar space is the set of all points x which are collinear with any other point. A polar space is called nondegenerate if its radical is empty. A subspace of a polar space is said to be singular if any two of its points are collinear. The rankr of a nondegenerate polar space is the maximal length r of a chain \(S_0 \subset S_1 \subset \cdots \subset S_r\) of singular subspaces where \(S_0 = \emptyset \) and \(S_i \not = S_{i+1}\) for all \(i \in \{ 0,\ldots ,r-1 \}\). A nondegenerate polar space of rank 2 is just a nondegenerate generalized quadrangle. The rank of a singular subspace S of a nondegenerate polar space is the maximal length k of a chain \(S_0 \subset S_1 \subset \cdots \subset S_k\) of singular subspaces such that \(S_0 = \emptyset \), \(S_k = S\) and \(S_i \not = S_{i+1}\) for all \(i \in \{ 0,\ldots ,k-1 \}\). Singular subspaces of rank r are also called maximal singular subspaces, and those of rank \(r-1\) are called next-to-maximal singular subspaces. A nondegenerate polar space is called thick if every line is incident with at least three points and if every next-to-maximal singular subspace is contained in at least three maximal singular subspaces.

With every (thick) polar space \(\mathcal {S}\) of rank \(r \ge 1\), there is associated a partial linear space \(\Delta \), which is called a (thick) dual polar space of rank r. The points of \(\Delta \) are the maximal singular subspaces of \(\mathcal {S}\), the lines of \(\Delta \) are the next-to-maximal singular subspaces of \(\mathcal {S}\), and incidence is reverse containment. Every thick dual polar space of rank r is a dense near 2r-gon.

2.4 Near-polar spaces

In [15], Shult introduced a class of point-line geometries. These point-line geometries were called near-polar spaces in [2]. Near-polar spaces of diameter n are inductively defined as follows.

A near-polar space of diameter 0 is just a point and a near-polar space of diameter 1 is a line having at least three points. A near-polar space of diameter \(n \ge 2\) is a point-line geometry \(\mathcal {S}\) satisfying the following five axioms:
  1. (E1)

    \(\mathcal {S}\) is connected and its diameter is equal to n;

     
  2. (E2)

    Every line of \(\mathcal {S}\) is incident with at least three points;

     
  3. (E3)

    Every geodesic \(x_0,x_1,\ldots ,x_k\) in \(\mathcal {S}\) can be completed to a geodesic \(x_0,x_1,\dots ,x_k,x_{k+1},\ldots ,x_n\) of length n;

     
  4. (E4)

    For every point x of \(\mathcal {S}\), the set \(H_x\) of points of \(\mathcal {S}\) at distance at most \(n-1\) from x is a hyperplane of \(\mathcal {S}\);

     
  5. (E5)

    If \(x_1\) and \(x_2\) are two points of \(\mathcal {S}\) with \(k := \mathop {\text{ d }}\nolimits (x_1,x_2) < n\), then the subgeometry of \(\mathcal {S}\) induced on the convex closure \(\langle x_1,x_2 \rangle \) is a near-polar space of diameter k.

     
The hyperplane \(H_x\) mentioned in Axiom (E4) is called the singular hyperplane of\(\mathcal {S}\)with deepest pointx.

The near-polar spaces of diameter 2 are precisely the nondegenerate polar spaces in which each line is incident with at least three points. Every near-polar space of diameter \(n \ge 2\) is a strong parapolar space in the sense of Cohen and Cooperstein [4]. The convex closures of the pairs of points at distance 2 from each other are also called symplecta.

Every thick dual polar space and more generally every dense near polygon is a near-polar space. The class of near-polar spaces also includes some half-spin geometries, some Grassmann spaces and some exceptional geometries, see Shult [15, Section 6].

We will now discuss full projective embeddings of near-polar spaces. Most of what we say here is based on De Bruyn [5].

Suppose \(e:\mathcal {S} \rightarrow \Sigma \) is a full projective embedding of a near-polar space \(\mathcal {S}=(\mathcal {P},\mathcal {L},\mathrm {I})\). By Shult [15, Lemma 6.1 (ii)], every singular hyperplane \(H_x\), \(x \in \mathcal {P}\), of \(\mathcal {S}\) is a maximal (proper) subspace. This implies that \(\Pi _x := \langle e(H_x)\rangle _\Sigma \) is either \(\Sigma \) or a hyperplane of \(\Sigma \). The embedding e is called polarized if \(\Pi _x\) is a hyperplane of \(\Sigma \) for every point x of \(\mathcal {S}\). If e is polarized, then the subspace \(\mathcal {N}_e := \underset{x \in \mathcal {P}}{\bigcap } \Pi _x\) is called the nucleus of e. By De Bruyn [5, Proposition 3.4], the nucleus \(\mathcal {N}_e\) satisfies the conditions (Q1) and (Q2) of Sect. 2.1 and the embedding \(\bar{e} := e/\mathcal {N}_e\) is polarized.

Suppose now that \(\mathcal {S}\) is a slim near-polar space. Then \(\mathcal {S}\) admits a faithful full polarized embedding, see Brouwer and Shpectorov [3] or De Bruyn [5, Proposition 3.11 (i)]. So, \(\mathcal {S}\) also has a universal embedding \(\widetilde{e}: \mathcal {S} \rightarrow \widetilde{\Sigma }\). This universal embedding is necessarily polarized and faithful. The embedding \(\widetilde{e}/\mathcal {N}_{\widetilde{e}}\) is called the minimal full polarized embedding of \(\mathcal {S}\). For every full polarized embedding e of \(\mathcal {S}\), the embedding \(\bar{e} = e/\mathcal {N}_e\) is isomorphic to \(\widetilde{e}/\mathcal {N}_{\widetilde{e}}\). Every full embedding of \(\mathcal {S}\) is isomorphic to \(\widetilde{e}/\alpha \) for some subspace \(\alpha \) of \(\widetilde{\Sigma }\) satisfying Properties (Q1) and (Q2). If \(\alpha _1\) and \(\alpha _2\) are two subspaces of \(\widetilde{\Sigma }\) satisfying (Q1) and (Q2), then \(e/\alpha _1 \cong e/\alpha _2\) if and only if \(\alpha _1=\alpha _2\).

Suppose again that \(\mathcal {S}\) is a slim near-polar space and that \(e:\mathcal {S} \rightarrow \Sigma \) is a full polarized embedding of \(\mathcal {S}\). This means that for every point x of \(\mathcal {S}\), the subspace \(\langle e(H_x) \rangle _\Sigma \) is a hyperplane \(\Pi _x\) of \(\Sigma \). By De Bruyn [5, Propositions 3.5 and 3.11 (ii)], the map \(x \mapsto \Pi _x\) defines a polarized full embedding \(e^*\) of \(\mathcal {S}\) into a subspace of the dual \(\Sigma ^*\) of \(\Sigma \). The embedding \(e^*\) is called the dual embedding of e. The nucleus of \(e^*\) is empty. So, the dual embedding \(e^*\) is isomorphic to the minimal full polarized embedding of \(\mathcal {S}\).

2.5 Extraspecial 2-groups

In the sequel, we will adopt the following conventions when dealing with groups. For elements ab of a group G, we write \([a,b]=a^{-1}b^{-1}ab\) and \(a^b=b^{-1}ab\). For elements xyz of G, we have \([xy,z]=[x,z]^y[y,z]\) and \([x,yz]=[x,z][x,y]^z\). We denote by \(C_n\) the cyclic group of order n.

A finite 2-group G is called extraspecial if its Frattini subgroup \(\Phi (G)\), commutator subgroup \(G'=[G,G]\) and center Z(G) coincide and have order 2. We refer to [7, Section 20, pp. 78–79] or [8, Chapter 5, Section 5] for the properties of finite extraspecial 2-groups which we will mention now.

An extraspecial 2-group is of order \(2^{1+2n}\) for some integer \(n \ge 1\). Let \(D_{8}\) and \(Q_{8}\), respectively, denote the dihedral and the quaternion groups of order 8. A non-abelian 2-group of order 8 is extraspecial and is isomorphic to either \(D_{8}\) or \(Q_{8}\). If G is an extraspecial 2-group of order \(2^{1+2n}\), \(n \ge 1\), then the exponent of G is 4 and G is either a central product of n copies of \(D_{8}\), or a central product of \(n-1\) copies of \(D_{8}\) and one copy of \(Q_{8}\). If the former (respectively, latter) case occurs, then the extraspecial 2-group is denoted by \(2_{+}^{1+2n}\) (respectively, \(2_{-}^{1+2n}\)).

Suppose G is an extraspecial 2-group of order \(2^{2n+1}\), \(n \ge 1\), and set \(G' = \{ 1,\lambda \}\). Then \(V = G/G'\) is an elementary abelian 2-group and hence can be regarded as a 2n-dimensional vector space over \(\mathbb {F}_2\). For all \(x,y \in G\), we define
$$\begin{aligned} f(xG',yG') = \left\{ \begin{array}{l@{\quad }l} 0 \in \mathbb {F}_2 &{} \text{ if } [x,y]=1, \\ 1 \in \mathbb {F}_2 &{} \text{ if } [x,y]=\lambda . \end{array} \right. \end{aligned}$$
Then f is a nondegenerate alternating bilinear form on V. For all \(x\in G\), \(x^2\in G'= \{ 1,\lambda \}\) as \(G/G'\) is elementary abelian. We define
$$\begin{aligned} q(x G') = \left\{ \begin{array}{l@{\quad }l} 0 \in \mathbb {F}_2 &{} \text{ if } x^2=1, \\ 1 \in \mathbb {F}_2 &{} \text{ if } x^2=\lambda . \end{array} \right. \end{aligned}$$
Then q is a nondegenerate quadratic form on V. The bilinear form associated with q is precisely f, that is,
$$\begin{aligned} q(x G' y G') = q(x G') + q(y G') + f(x G' , y G') \end{aligned}$$
for all \(x,y \in G\). The nondegenerate quadratic form q defines a nonsingular quadric of \(\mathrm {PG}(V)\), which is of hyperbolic type if \(G=2_{+}^{1+2n}\) or of elliptic type if \(G=2_{-}^{1+2n}\).

2.6 Representations of slim partial linear spaces

Let \(\mathcal {S}=(\mathcal {P},\mathcal {L},\mathrm {I})\) be a slim partial linear space. A representation [10, p. 525] of \(\mathcal {S}\) is a pair \((R,\psi )\), where R is a group and \(\psi \) is a mapping from \(\mathcal {P}\) to the set of involutions in R, satisfying:
  1. (i)

    R is generated by the image of \(\psi \);

     
  2. (ii)

    \(\psi (x)\psi (y) = \psi (z)\) for every line \(\{ x,y,z \}\) of \(\mathcal {S}\).

     
If \(\{ x,y,z \}\) is a line of \(\mathcal {S}\), then condition (ii) implies that \(\psi (x),\psi (y),\psi (z)\) are mutually distinct and \([\psi (x),\psi (y)]=[\psi (x),\psi (z)]=[\psi (y),\psi (z)]=1\). The group R is called a representation group of \(\mathcal {S}\). The representation \((R,\psi )\) of \(\mathcal {S}\) is called faithful if \(\psi \) is injective. Depending on whether R is abelian or not, the representation \((R,\psi )\) itself will be called abelian or non-abelian. For an abelian representation, the representation group is an elementary abelian 2-group and hence can be considered as a vector space over the field \(\mathbb {F}_2\) with two elements. In this case, the representation thus corresponds to a full projective embedding of \(\mathcal {S}\).

We refer to [9] and [13, Sections 1 and 2] for representations of partial linear spaces with \(p+1\) points per line, where p is a prime.

Suppose \(\mathcal {S}_1\) and \(\mathcal {S}_2\) are two slim partial linear spaces. Let \((R_i,\psi _i)\), \(i \in \{ 1,2 \}\), be a representation of \(\mathcal {S}_i\). The representations \((R_1,\psi _1)\) and \((R_2,\psi _2)\) are called equivalent if there exist an isomorphism \(\theta _1\) from \(\mathcal {S}_1\) to \(\mathcal {S}_2\) and a group isomorphism \(\theta _2\) from \(R_1\) to \(R_2\) such that \(\psi _2 \circ \theta _1(x) = \theta _2 \circ \psi _1(x)\) for every point x of \(\mathcal {S}_1\). If \(\mathcal {S}_1 = \mathcal {S}_2\), then \((R_1,\psi _1)\) and \((R_2,\psi _2)\) are called isomorphic if there exists a group isomorphism \(\theta \) from \(R_1\) to \(R_2\) such that \(\psi _2(x) = \theta \circ \psi _1(x)\) for every point x of \(\mathcal {S}_1\).

Suppose \((R,\psi )\) is a representation of a slim partial linear space \(\mathcal {S}\). Let N be a normal subgroup of R such that \(\psi (x) \not \in N\) for every point x of \(\mathcal {S}\). For every point x of \(\mathcal {S}\), let \(\psi _N(x)\) denote the element \(\psi (x) N\) of the quotient group R / N. Then \((R/N,\psi _N)\) is a representation of \(\mathcal {S}\) which is called a quotient of \((R,\psi )\). If \((R_1,\psi _1)\) and \((R_2,\psi _2)\) are two representations of \(\mathcal {S}\), then \((R_2,\psi _2)\) is isomorphic to a quotient of \((R_1,\psi _1)\) if and only if there exists a group epimorphism \(\theta \) from \(R_1\) to \(R_2\) such that \(\psi _2(x) = \theta \circ \psi _1(x)\). If this is the case, then \((R_2,\psi _2)\) is isomorphic to \((R_1/N,(\psi _1)_N)\), where \(N = ker(\theta )\).

2.7 Polarized and universal representations of slim near-polar spaces

Let \(\mathcal {S}= (\mathcal {P},\mathcal {L},\mathrm {I})\) be a slim near-polar space of diameter \(n \ge 2\).
  • A representation \((R,\psi )\) of \(\mathcal {S}\) is called quasi-polarized if \([\psi (x),\psi (y)]=1\) for every two points x and y of \(\mathcal {S}\) at distance at most \(n-1\) from each other.

  • An abelian representation \((R,\psi )\) of \(\mathcal {S}\) is called polarized if the corresponding full projective embedding (in the sense of Sect. 2.6) is polarized.

  • A non-abelian representation \((R,\psi )\) of \(\mathcal {S}\) is called polarized if \([\psi (x),\psi (y)]=1\) for every two points x and y of \(\mathcal {S}\) at distance at most \(n-1\) from each other, that is, if the representation is quasi-polarized.

We will later show that with every polarized non-abelian representation of \(\mathcal {S}\), there is an associated full polarized embedding of \(\mathcal {S}\) (which is obtained by taking a suitable quotient).
(1) Let \(\widetilde{R}_u\) be the group defined by the generators \(r_x\), \(x \in \mathcal {P}\), and the following relations:
  • \(r_x^2 = 1\), where \(x \in \mathcal {P}\);

  • \(r_x r_y r_z =1\), where \(x,y,z \in \mathcal {P}\) such that \(\{ x,y,z \} \in \mathcal {L}\).

For every point x of \(\mathcal {S}\), we define \(\widetilde{\psi }_u(x) := r_x \in \widetilde{R}_u\).
(2) Let \(\widetilde{R}_p\) be the group defined by the generators \(r_x\), \(x \in \mathcal {P}\), and the following relations:
  • \(r_x^2 = 1\), where \(x \in \mathcal {P}\);

  • \([r_x,r_y]=1\), where \(x,y \in \mathcal {P}\) such that \(\mathop {\text{ d }}\nolimits (x,y)<n\);

  • \(r_x r_y r_z =1\), where \(x,y,z \in \mathcal {P}\) such that \(\{ x,y,z \} \in \mathcal {L}\).

For every point x of \(\mathcal {S}\), we define \(\widetilde{\psi }_p(x) := r_x \in \widetilde{R}_p\).

(3) As mentioned before, \(\mathcal {S}\) has faithful full projective embeddings. The universal projective embedding of \(\mathcal {S}\) can be constructed as follows. Let V be a vector space over \(\mathbb {F}_2\) with a basis B whose elements are indexed by the points of \(\mathcal {P}\), say \(B = \{ \bar{v}_x \, | \, x \in \mathcal {P} \}\). Let W be the subspace of V generated by all vectors \(\bar{v}_{x_1} + \bar{v}_{x_2} + \bar{v}_{x_3}\) where \(\{ x_1,x_2,x_3 \}\) is some line of \(\mathcal {S}\). Let \(\widetilde{V}\) be the quotient vector space V / W and for every point x of \(\mathcal {S}\), let \(\widetilde{v_x}\) be the vector \(\bar{v}_x + W\) of \(\widetilde{V}\). The map \(x \mapsto \langle \widetilde{v_x} \rangle \) defines a full projective embedding \(\widetilde{e}\) of \(\mathcal {S}\) into \(\mathrm {PG}(\widetilde{V})\) which is isomorphic to the universal embedding of \(\mathcal {S}\).

Proposition 2.1

  1. (1)

    \((\widetilde{R}_u,\widetilde{\psi }_u)\) is a faithful representation of \(\mathcal {S}\).

     
  2. (2)

    \((\widetilde{R}_p,\widetilde{\psi }_p)\) is a faithful polarized representation of \(\mathcal {S}\).

     
  3. (3)

    If \((R,\psi )\) is a representation of \(\mathcal {S}\), then \((R,\psi )\) is isomorphic to a quotient of \((\widetilde{R}_u,\widetilde{\psi }_u)\).

     
  4. (4)

    If \((R,\psi )\) is a quasi-polarized representation of \(\mathcal {S}\), then \((R,\psi )\) is isomorphic to a quotient of \((\widetilde{R}_p,\widetilde{\psi }_p)\).

     

Proof

We show that \((\widetilde{R}_p,\widetilde{\psi }_p)\) is a faithful representation. Since \(\widetilde{v_x} + \widetilde{v_x} = W\) for every \(x \in \mathcal {P}\), \((-\widetilde{v_x}) + (-\widetilde{v_y}) + \widetilde{v_x} + \widetilde{v_y} = W\) for all \(x,y \in \mathcal {P}\) and \(\widetilde{v_x} + \widetilde{v_y} + \widetilde{v_z} = W\) for every line \(\{ x,y,z \}\) of \(\mathcal {S}\), we know from von Dyck’s theorem that there exists an epimorphism from \(\widetilde{R}_p\) to the additive group of \(\widetilde{V}\) mapping \(r_x\) to \(\widetilde{v_x}\) for every point x of \(\mathcal {S}\). Since \(\widetilde{e}\) is a full projective embedding, \(\widetilde{v_x} \not = W\) and hence \(r_x \ne _{\widetilde{R}_p} 1\) for every \(x \in \mathcal {P}\). The latter fact implies that \((\widetilde{R}_p,\widetilde{\psi }_p)\) is a representation. Since \(\widetilde{e}\) is a faithful projective embedding, we have \(\widetilde{v_x} \not = \widetilde{v_y}\) for any two distinct points \(x,y \in \mathcal {P}\). This implies that \(r_x \ne _{\widetilde{R}_p} r_y\). So, \((\widetilde{R}_p,\widetilde{\psi }_p)\) is a faithful representation.

In a completely similar way, one can show that \((\widetilde{R}_u,\widetilde{\psi }_u)\) is a faithful representation.

Claims (3) and (4) are straightforward consequences of von Dyck’s theorem.

By construction, the representation \((\widetilde{R}_p,\widetilde{\psi }_p)\) is quasi-polarized and hence polarized if \(\widetilde{R}_p\) is non-abelian. Suppose \(\widetilde{R}_p\) is abelian. Then let \(e_p\) denote the full projective embedding of \(\mathcal {S}\) corresponding to \((\widetilde{R}_p,\widetilde{\psi }_p)\). Let \((\widetilde{R},\widetilde{\psi })\) denote the abelian representation corresponding to the universal projective embedding \(\widetilde{e}\) of \(\mathcal {S}\). By Claim (4), \((\widetilde{R},\widetilde{\psi })\) is isomorphic to a quotient of \((\widetilde{R}_p,\widetilde{\psi }_p)\), and hence \(\widetilde{e}\) is isomorphic to a quotient of \(e_p\). As \(\widetilde{e}\) cannot be a proper quotient of some full embedding of \(\mathcal {S}\), the projective embeddings \(\widetilde{e}\) and \(e_p\) are isomorphic. So, \(e_p\) is polarized, or equivalently, \((\widetilde{R}_p,\widetilde{\psi }_p)\) is polarized. \(\square \)

The representation \((\widetilde{R}_u,\widetilde{\psi }_u)\) is called the universal representation of \(\mathcal {S}\). The representation \((\widetilde{R}_p,\widetilde{\psi }_p)\) is called the universal polarized representation of \(\mathcal {S}\).

From Sect. 5 (see Lemma 5.3) it will follow that there exists a \(\widetilde{\lambda } \in \widetilde{R}_p\) such that \([\widetilde{\psi }_p(x),\widetilde{\psi }_p(y)] = \widetilde{\lambda }\) for every two points x and y at distance n from each other. If \(\widetilde{\lambda } = 1\), then the universal polarized representation is abelian and hence corresponds to the universal projective embedding of \(\mathcal {S}\) (which is always polarized). If \(\widetilde{\lambda } \not = 1\), then the universal polarized representation of \(\mathcal {S}\) is non-abelian. Both instances can occur. Indeed, the slim dual polar space \(DW(2n-1,2)\) and the slim dense near hexagons \(Q(5,2) \times \mathbb {L}_3\), \(Q(5,2) \otimes Q(5,2)\) have non-abelian polarized representations [6, 11], while no finite slim nondegenerate polar space has non-abelian representations [13, Theorem 1.5 (i)]. Computer computations showed that other dense near polygons (like the dual polar space DH(5, 4)) also have non-abelian polarized representations (in extraspecial 2-groups), but the authors are still looking for computer free descriptions of these representations.

3 Main results

For a finite slim near-polar space \(\mathcal {S}\), we denote the embedding rank \(er(\mathcal {S})\) also by \(er^+(\mathcal {S})\). The vector space dimension of the minimal full polarized embedding of \(\mathcal {S}\) will be denoted by \(er^-(\mathcal {S})\). We will see in Proposition 4.2 that the number \(er^-(\mathcal {S})\) is even. By [14], every non-abelian representation of a slim dense near hexagon is polarized. The following theorem is the first main theorem of this paper. It generalizes some results regarding slim dense near hexagons obtained in [14]. We will prove it in Sect. 5.

Theorem 3.1

Suppose \(\mathcal {S}\) is a finite slim near-polar space of diameter \(n \ge 2\) having some polarized non-abelian representation \((R,\psi )\). Then \(n \ge 3\) and the universal polarized representation \((\widetilde{R}_p,\widetilde{\psi }_p)\) of \(\mathcal {S}\) is also non-abelian. Moreover,
  1. (i)

    \(\psi \) is faithful and \(\psi (x)\notin Z(R)\) for every point x of \(\mathcal {S}\).

     
  2. (ii)

    R is a 2-group of exponent 4, \(|R'|=2\) and \(R'=\Phi (R)\subseteq Z(R)\).

     
  3. (iii)

    If \(|Z(R)|=2^{l+1}\), then Z(R) is isomorphic1 to either \((C_2)^{l+1}\) or \((C_2)^{l-1} \times C_4\).

     
  4. (iv)

    R is of order \(2^{\beta }\) for some integer \(\beta \) satisfying \(1+er^-(\mathcal {S}) \le \beta \le 1+er^+(\mathcal {S})\). We have \(\beta = 1+er^-(\mathcal {S})\) if and only if R is an extraspecial 2-group. We have \(\beta = 1+er^+(\mathcal {S})\) if and only if \((R,\psi )\) is isomorphic to \((\widetilde{R}_p,\widetilde{\psi }_p)\).

     
  5. (v)

    If \(l = er^+(\mathcal {S}) - er^-(\mathcal {S})\), then \(Z(\widetilde{R}_p)\) has order \(2^{l+1}\) and so is isomorphic to either \((C_2)^{l+1}\) or \((C_2)^{l-1} \times C_4\).

     

In Sect. 6, we prove the following results.

Theorem 3.2

Suppose \(\mathcal {S}\) is a finite slim near-polar space of diameter \(n \ge 3\) having polarized non-abelian representations. Then the following holds:
  1. (i)

    The polarized representations of \(\mathcal {S}\) are precisely the representations of the form \((\widetilde{R}_p/N,(\widetilde{\psi }_p)_N)\), where N is a subgroup of \(\widetilde{R}_p\) contained in \(Z(\widetilde{R}_p)\).

     
  2. (ii)

    If \(N_1\) and \(N_2\) are two subgroups of \(Z(\widetilde{R}_p)\), then the representations \((\widetilde{R}_p/N_1,(\widetilde{\psi }_p)_{N_1})\) and \((\widetilde{R}_p/N_2,(\widetilde{\psi }_p)_{N_2})\) of \(\mathcal {S}\) are isomorphic if and only if \(N_1=N_2\).

     

Remark

If \(l= er^+(\mathcal {S}) - er^-(\mathcal {S})\), then we will see in Sect. 6 that Theorems 3.1 (v) and 3.2 imply that the number of nonisomorphic polarized non-abelian representations of \(\mathcal {S}\) is equal to the sum2\(\sum _{i=0}^{l+1}\Big [ \begin{array}{c} l+1 \\ i \end{array} \Big ]_2 - \sum _{i=0}^{l}\Big [ \begin{array}{c} l \\ i \end{array} \Big ]_2\) if \(Z(\widetilde{R}_p) \cong (C_2)^{l+1}\), and equal to \(\sum _{i=0}^{l}\Big [ \begin{array}{c} l \\ i \end{array} \Big ]_2 - \sum _{i=0}^{l-1}\Big [ \begin{array}{c} l-1 \\ i \end{array} \Big ]_2\) if \(Z(\widetilde{R}_p) \cong (C_2)^{l-1} \times C_4\).

Theorem 3.3

Suppose \(\mathcal {S}\) is a finite slim near-polar space of diameter \(n \ge 3\) having polarized non-abelian representations. Set \(l := er^+(\mathcal {S}) - er^-(\mathcal {S})\). Then \(\mathcal {S}\) has a polarized non-abelian representation \((R,\psi )\) with \(|R|=2^{1+er^-(\mathcal {S})}\) if and only if \(Z(\widetilde{R}_p) \cong (C_2)^{l+1}\). If this is the case then there are up to isomorphism \(2^l\) such representations. Moreover, the representation groups of any two of them are isomorphic (to either \(2^{1+er^-(\mathcal {S})}_+\) or \(2^{1+er^-(\mathcal {S})}_-\)).

Theorem 3.4

Suppose \(\mathcal {S}\) is a finite slim near-polar space of diameter \(n \ge 3\) having polarized non-abelian representations. Suppose \(Z(\widetilde{R}_p) \cong C_2^{l-1} \times C_4\), where \(l = er^+(\mathcal {S}) - er^-(\mathcal {S}) \ge 1\). Then \(|R| \ge 2^{2+er^-(\mathcal {S})}\) for every polarized non-abelian representation \((R,\psi )\) of \(\mathcal {S}\). Moreover, there are up to isomorphism \(2^{l-1}\) polarized non-abelian representations \((R,\psi )\) with \(|R| = 2^{2+er^-(\mathcal {S})}\). If \((R,\psi )\) is such a representation, then \(Z(R) \cong C_4\).

4 Some properties of near-polar spaces

Let \(\mathcal {S}\) be a near-polar space of diameter \(n \ge 1\). Two points x and y of \(\mathcal {S}\) are called opposite if they are at maximum distance from each other, that is, \(\mathop {\text{ d }}\nolimits (x,y)=n\). For two distinct points xy of \(\mathcal {S}\), we write \(x\sim y\) if they are collinear.

Proposition 4.1

Let \(\mathcal {S}\) be a near-polar space of diameter \(n \ge 1\). Let \(\Gamma \) be the graph whose vertices are the ordered pairs of opposite points of \(\mathcal {S}\), with two distinct vertices \((x_1,y_1)\) and \((x_2,y_2)\) being adjacent whenever either \(x_1 = x_2\) and \(y_1 \sim y_2\); or \(x_1 \sim x_2\) and \(y_1 = y_2\). Then \(\Gamma \) is connected.

Proof

Let \((x_1,y_1)\) and \((x_2,y_2)\) be two arbitrary vertices of \(\Gamma \). We prove that \((x_1,y_1)\) and \((x_2,y_2)\) are contained in the same connected component of \(\Gamma \).

For every point x of \(\mathcal {S}\), the subgraph of the collinearity graph of \(\mathcal {S}\) induced on the set of points at distance n from x is connected by Shult [15, Lemma 6.1 (ii)]. So, if \(x_1=x_2\) or \(y_1=y_2\), then \((x_1,y_1)\) and \((x_2,y_2)\) belong to the same connected component of \(\Gamma \).

Assume that \(x_1\ne x_2\) and \(y_1\ne y_2\). We prove that there exists a point \(y_3\) at distance n from \(x_1\) and \(x_2\). If \(y_3\) is such a point, then \((a_1,b_1)\) and \((a_2,b_2)\) belong to the same connected component of \(\Gamma \) for every \((a_1,b_1,a_2,b_2) \in \{ (x_1,y_1,x_1,y_3),(x_1,y_3,x_2,y_3),(x_2,y_3,x_2,y_2) \}\), proving that \((x_1,y_1)\) and \((x_2,y_2)\) also belong to the same connected component of \(\Gamma \).

The point \(y_3\) alluded to in the previous paragraph is defined as a point of \(\mathcal {S}\) at distance n from \(x_1\) which lies as far away from \(x_2\) as possible. Suppose \(\mathop {\text{ d }}\nolimits (y_3,x_2) \le n-1\) for such a point \(y_3\). Then by Axiom (E3), there exists a point \(y_4\) collinear with \(y_3\) which lies at distance \(k:=\mathop {\text{ d }}\nolimits (y_3,x_2)+1\) from \(x_2\). By Axiom (E5), a near-polar space of diameter k can be defined on the convex closure \(\langle x_2,y_4 \rangle \). By applying Axiom (E4) to this near-polar space of diameter k, we see that the points of the line \(y_3y_4\) distinct from \(y_3\) lie at distance \(k=d(y_3,x_2)+1\) from \(x_2\). By Axioms (E2) and (E4) applied to \(\mathcal {S}\), at least one of the points of \(y_3y_4 \setminus \{ y_3 \}\) lies at distance n from \(x_1\). This contradicts the maximality of \(\mathop {\text{ d }}\nolimits (y_3,x_2)\). So, \(\mathop {\text{ d }}\nolimits (x_1,y_3)=\mathop {\text{ d }}\nolimits (x_2,y_3)=n\) as we needed to prove. \(\square \)

Proposition 4.2

Let \(\mathcal {S}= (\mathcal {P},\mathcal {L},\mathrm {I})\) be a finite slim near-polar space of diameter \(n \ge 1\), let V be a finite-dimensional vector space over \(\mathbb {F}_2\) and let \(e:\mathcal {S} \rightarrow \mathrm {PG}(V)\) be a full polarized embedding of \(\mathcal {S}\) into \(\mathrm {PG}(V)\). Then there exists a unique alternating bilinear form f on V for which the following holds:

If x is a point of \(\mathcal {S}\) and \(\bar{v}\) is the unique vector of V for which \(e(x) = \langle \bar{v} \rangle \), then \(\mathrm {PG}(\bar{v}^{\perp _f})\) is a hyperplane of \(\mathrm {PG}(V)\) which contains all the points e(y), where \(y \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\), and none of the points e(z), where \(z \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,z)=n\).

If e is isomorphic to the minimal full polarized embedding of \(\mathcal {S}\), then the alternating bilinear form f is nondegenerate and hence \(er^-(\mathcal {S})=\dim (V)\) is even.

Proof

For every point x of \(\mathcal {S}\), let \(\Pi _x\) denote the unique hyperplane of \(\mathrm {PG}(V)\) which contains all the points e(y), where \(y \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\), and none of the points e(z), where \(z \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,z)=n\).

(1) We first prove the existence of the alternating bilinear form in the case e is isomorphic to the minimal full polarized embedding of \(\mathcal {S}\). Then \(\underset{x\in \mathcal {P}}{\bigcap } \Pi _x = \emptyset \).

Recall that the map \(x \mapsto \Pi _x\) defines a full projective embedding \(e^*\) of \(\mathcal {S}\) into the dual \(\mathrm {PG}(V)^*\) of \(\mathrm {PG}(V)\). This embedding \(e^*\) is called the dual embedding of e and is isomorphic to the minimal full polarized embedding of \(\mathcal {S}\). So, there exists an isomorphism \(\phi \) from \(\mathrm {PG}(V)\) to \(\mathrm {PG}(V)^*\) mapping e(x) to \(\Pi _x\) for every point x of \(\mathcal {S}\).

We prove that \(\phi \) is a polarity of \(\mathrm {PG}(V)\), or equivalently that \(\phi ^2 = 1\). Since \(\phi ^2\) defines a collineation of \(\mathrm {PG}(V)\), it suffices to prove that \(\phi ^2(p)=p\) for every point p belonging to a generating set of \(\mathrm {PG}(V)\). So, it suffices to prove that \(\phi (\Pi _x) = \phi ^2(e(x)) = e(x)\) for every point x of \(\mathcal {P}\). If y is a point at distance at most \(n-1\) from x, then \(e(y) \in \Pi _x\) implies that \(\phi (\Pi _x) \in \Pi _y\). Hence, \(\phi (\Pi _x)\) is contained in the intersection I of all hyperplanes \(\Pi _y\), where \(y \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\). Since \(e^*\) is polarized, the hyperplanes \(\Pi _y\), where \(y \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\), generate a hyperplane of \(\mathrm {PG}(V)^*\). So, I is a singleton. Since \(e(x) \in \Pi _y\) for every \(y \in \mathcal {P}\) satisfying \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\), we also have \(e(x) \in I\). Hence, \(\phi (\Pi _x) = e(x)\) as we needed to prove.

We now prove that \(\phi \) is a symplectic polarity of \(\mathrm {PG}(V)\). To that end, it suffices to prove that \(p \in p^\phi \) for every point p of \(\mathrm {PG}(V)\). Since \(\mathrm {PG}(V) = \langle Im(e)\rangle \), it suffices to prove the following:
  1. (a)

    \(e(x) \in e(x)^\phi \) for every \(x \in \mathcal {P}\);

     
  2. (b)

    if \(L=\{ p_1,p_2,p_3 \}\) is a line of \(\mathrm {PG}(V)\) such that \(p_1 \in p_1^\phi \) and \(p_2 \in p_2^\phi \), then also \(p_3 \in p_3^\phi \).

     
Since \(e(x)^\phi = \Pi _x\) and \(e(x) \in \Pi _x\), Property (a) clearly holds. If \(p_2 \in p_1^\phi \), then \(\{ p_3 \} \subseteq L \subseteq p_1^\phi \cap p_2^\phi = L^\phi \subseteq p_3^\phi \). If \(p_2 \not \in p_1^\phi \), then \(p_1^\phi = \langle L^\phi , p_1\rangle \), \(p_2^\phi = \langle L^\phi ,p_2\rangle \) and \(p_3^\phi \) is the unique hyperplane through \(L^\phi \) distinct from \(p_1^\phi \) and \(p_2^\phi \), implying that \(p_3^\phi = \langle L^\phi ,p_3\rangle \). So, Property (b) also holds in that case.

If f is the nondegenerate alternating bilinear form of V corresponding to the symplectic polarity \(\phi \) of \(\mathrm {PG}(V)\), then f satisfies the required conditions.

(2) Suppose e is not isomorphic to the minimal full polarized embedding of \(\mathcal {S}\). Let \(\alpha \) be the intersection of all subspaces \(\Pi _x\), \(x \in \mathcal {P}\), let U be the subspace of V corresponding to \(\alpha \) and let W be a subspace of V such that \(V = U \oplus W\). For every point x of \(\mathcal {S}\), let \(e'(x)\) denote the unique point of \(\mathrm {PG}(W)\) contained in \(\langle \alpha ,e(x)\rangle \). Then \(e'\) is isomorphic to the minimal full polarized embedding of \(\mathcal {S}\). By part (1) above, we know that there exists a nondegenerate alternating bilinear form \(f_W\) on W such that if x is a point of \(\mathcal {S}\) and \(\bar{w}\) is the unique vector of W for which \(e'(x) = \langle \bar{w}\rangle \), then the hyperplane \(\mathrm {PG}(\bar{w}^{\perp _{f_W}})\) of \(\mathrm {PG}(W)\) contains all points \(e'(y)\), where \(y \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\), and none of the points e(z), where \(z \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,z)=n\). Now, for all \(\bar{u}_1,\bar{u}_2 \in U\) and all \(\bar{w}_1,\bar{w}_2 \in W\), we define
$$\begin{aligned} f(\bar{u}_1+\bar{w}_1,\bar{u}_2+\bar{w}_2) := f_W(\bar{w}_1,\bar{w}_2). \end{aligned}$$
Then f is an alternating bilinear form on V.

Suppose x is a point of \(\mathcal {S}\). Let \(\bar{v}\) be the unique vector of V for which \(e(x) = \langle \bar{v}\rangle \) and let \(\bar{w}\) be the unique vector of W for which \(e'(x) = \langle \bar{w}\rangle \). Then \(\langle \bar{w} \rangle = \langle U,\bar{v}\rangle \cap W\). We also have \(\langle \bar{v}^{\perp _f}\rangle = \langle U,\bar{w}^{\perp _{f_W}}\rangle \). Since \(\mathrm {PG}(\bar{w}^{\perp _{f_W}})\) contains all points \(e'(y)\), where \(y \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\), and none of the points \(e'(z)\), where \(z \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,z)=n\), we have that \(\mathrm {PG}(\bar{v}^{\perp _f})\) contains all points e(y), where \(y \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,y) \le n-1\), and none of the points e(z), where \(z \in \mathcal {P}\) and \(\mathop {\text{ d }}\nolimits (x,z)=n\). So, the alternating bilinear form f satisfies the required conditions.

(3) We now prove the uniqueness of the alternating bilinear form. Suppose \(f_1\) and \(f_2\) are two alternating bilinear forms on V satisfying the required conditions. Then \(g := f_1 - f_2\) is also an alternating bilinear form on V.

Suppose \(x_1\) and \(x_2\) are two points of \(\mathcal {S}\) and let \(\bar{v}_i\), \(i \in \{ 1,2 \}\), be the unique vector of V for which \(e(x) = \langle \bar{v}_i\rangle \). If \(\mathop {\text{ d }}\nolimits (x_1,x_2) \le n-1\), then \(f_1(\bar{v}_1,\bar{v}_2) =0= f_2(\bar{v}_1,\bar{v}_2)\) and hence \(g(\bar{v}_1,\bar{v}_2)=0\). If \(\mathop {\text{ d }}\nolimits (x_1,x_2)=n\), then \(f_1(\bar{v}_1,\bar{v}_2) = 1=f_2(\bar{v}_1,\bar{v}_2)\) and hence \(g(\bar{v}_1,\bar{v}_2)=0\). Since \(\mathrm {PG}(V) = \langle e(x) \, | \, x \in \mathcal {P} \rangle \), we get \(g=0\). Hence \(f_1=f_2\). \(\square \)

5 Structure of the representation groups

Let \(\mathcal {S}=(\mathcal {P},\mathcal {L},\mathrm {I})\) be a finite slim near-polar space of diameter \(n \ge 2\) and suppose \((R,\psi )\) is a polarized non-abelian representation of \(\mathcal {S}\). In this section, we will prove all the claims mentioned in Theorem 3.1.

Lemma 5.1

We have \(n \ge 3\).

Proof

By [13, Theorem 1.5 (i)], every representation of a finite slim nondegenerate polar space is abelian. So, \(\mathcal {S}\) is not a polar space and hence \(n \ge 3\). \(\square \)

Lemma 5.2

The universal polarized representation \((\widetilde{R}_p,\widetilde{\psi }_p)\) is non-abelian. Moreover, \(|\widetilde{R}_p| \ge 2^{1+er^+(\mathcal {S})}\).

Proof

As \((R,\psi )\) is a quotient of \((\widetilde{R}_p,\widetilde{\psi }_p)\), the universal polarized representation \((\widetilde{R}_p,\widetilde{\psi }_p)\) itself should also be non-abelian. Since the abelian representation corresponding to the universal projective embedding of \(\mathcal {S}\) is quasi-polarized, it should be a quotient of \((\widetilde{R}_p,\widetilde{\psi }_p)\) by Proposition 2.1 (4). This implies that \(|\widetilde{R}_p| \ge 2^{1+er^+(\mathcal {S})}\). \(\square \)

Later (Lemma 5.12) we will show that \(|\widetilde{R}_p| = 2^{1+er^+(\mathcal {S})}\).

Lemma 5.3

Let \(\Gamma \) be the graph as defined in Proposition 4.1. Then there exists an involution \(\lambda \in R\) such that \(\lambda = [\psi (x),\psi (y)]\) for every vertex (xy) of \(\Gamma \).

Proof

We first show that \([\psi (x_1),\psi (y_1)] = [\psi (x_2),\psi (y_2)]\) for any two adjacent vertices \((x_1,y_1)\) and \((x_2,y_2)\) of \(\Gamma \). Suppose \(x_1=x_2\) and \(y_1 \sim y_2\). Let \(y_3\) be the unique third point of the line \(y_1y_2\). Then \(\mathop {\text{ d }}\nolimits (x_1,y_3)=n-1\). Since \(\psi (y_3)\) commutes with \(\psi (x_1)\) and \(\psi (y_2)\), we have \([\psi (x_1),\psi (y_1)] = [\psi (x_1),\psi (y_2) \psi (y_3)] = [\psi (x_1),\psi (y_2)]\). The case where \(x_1 \sim x_2\) and \(y_1=y_2\) is treated in a similar way.

Now let x and y be two opposite points of \(\mathcal {S}\) and set \(\lambda =[\psi (x),\psi (y)]\). By Proposition 4.1, \(\Gamma \) is connected. So, by the first paragraph, \(\lambda \) is independent of the opposite points x and y. Also \(\lambda \not = 1\) since \((R,\psi )\) is polarized and non-abelian. Since \(\lambda ^{-1} = [\psi (x),\psi (y)]^{-1} = [\psi (y),\psi (x)]=\lambda \), we get \(\lambda ^2=1\). \(\square \)

Corollary 5.4

\(\langle \psi (x),\psi (y)\rangle \cong D_8\) for every two opposite points x and y of \(\mathcal {S}\).

Proof

Since x and y are opposite points, \((\psi (x)\psi (y))^2=[\psi (x),\psi (y)]=\lambda \) by Lemma 5.3 and so \(\psi (x)\psi (y)\) is of order 4. Hence \(\langle \psi (x),\psi (y)\rangle \cong D_8\) [1, 45.1]. \(\square \)

Lemma 5.5

\(\psi \) is faithful and \(\psi (x) \notin Z(R)\) for every point x of \(\mathcal {S}\).

Proof

Let x and y be two distinct points of \(\mathcal {S}\) and let z be a point that is opposite to x, but not to y (such a point exists by Axiom (E3)). Then \([\psi (y),\psi (z)]=1\) and \([\psi (x),\psi (z)] = \lambda \ne 1\) by Lemma 5.3. Hence, \(\psi (x) \not = \psi (y)\).

For a given point x, choose a point w opposite to x. Then \([\psi (x),\psi (w)]=\lambda \ne 1\). So \(\psi (x) \notin Z(R)\). \(\square \)

Lemma 5.6

\(R' = \{ 1,\lambda \} \subseteq Z(R)\).

Proof

Set \(T=\langle \lambda \rangle =\{1,\lambda \}\). Then \(T \subseteq R'\) by Lemma 5.3. We first show that \(T\subseteq Z(R)\). Since \(R = \langle \psi (x) \mid x \in \mathcal {P}\rangle \), it is sufficient to prove that \([\psi (x),\lambda ]=1\) for every point x of \(\mathcal {S}\). Let y be a point of \(\mathcal {S}\) opposite to x. Since \(\psi (x),\psi (y),\lambda = [\psi (x),\psi (y)]\) all are involutions, a direct calculation shows that \([\psi (x),\lambda ]=1\).

Being a central subgroup, T is normal in R. In the quotient group R / T, the generators \(\psi (x)T\), \(x\in \mathcal {P}\), pairwise commute. So R / T is abelian and hence \(R'\subseteq T\).

\(\square \)

Corollary 5.7

For \(a,b,c \in R\), \([ab,c] = [a,c][b,c]\) and \([a,bc] = [a,b] [a,c]\).

Proof

By Lemma 5.6, we have \([ab,c] = [a,c]^b [b,c] = [a,c] \cdot [b,c]\) and \([a,bc] = [a,c] \cdot [a,b]^c = [a,b] \cdot [a,c]\). \(\square \)

Lemma 5.8

  1. (1)

    For every \(r \in R\), we have \(r^2 \in \{ 1,\lambda \}\).

     
  2. (2)

    R is a finite 2-group of exponent 4 and \(R'=\Phi (R)\).

     

Proof

We show that \(r^2 \in \{ 1,\lambda \}\) for every \(r \in R \setminus \{ 1 \}\). Set \(r = \psi (x_1) \psi (x_2) \ldots \psi (x_n)\), where \(x_1,x_2,\ldots ,x_n\) are points of \(\mathcal {S}\). Since \(\lambda ^2=1\), \(\psi (x_i)^2=1\) and \([\psi (x_i),\psi (x_j)] \in \{ 1,\lambda \} \subseteq Z(R)\) for all \(i,j \in \{ 1,\ldots ,n \}\), we have \(r^2 \in \{ 1,\lambda \}\). It follows that \(r^4=1\). Since R is non-abelian, the exponent of R cannot be 2 and hence equals 4.

Since \(R=\langle \psi (x) \, | \, x\in \mathcal {P}\rangle \) and \(\mathcal {S}\) is finite, the quotient group \(R/R'=\langle \psi (x)R'\, | \, x\in \mathcal {P}\rangle \) is a finite elementary abelian 2-group. Since \(|R'|=2\) by Lemma 5.6, we get that R is also a finite 2-group. Then the two facts that \(R'\) is the smallest normal subgroup K of R such that R / K is abelian and that \(\Phi (R)\) is the smallest normal subgroup H of R such that R / H is elementary abelian [1, 23.2, p. 105] imply \(R'=\Phi (R)\). \(\square \)

Since the quotient group \(R/R'\) is an elementary abelian 2-group, we can consider \(V=R/R'\) as a vector space over \(\mathbb {F}_2\). For every point x of \(\mathcal {S}\), let e(x) be the projective point \(\langle \psi (x) R'\rangle \) of \(\mathrm {PG}(V)\). Notice that, by Lemmas 5.5 and 5.6, \(\psi (x) R'\) is indeed a nonzero vector of V.

Lemma 5.9

The map e defines a faithful full projective embedding of \(\mathcal {S}\) into \(\mathrm {PG}(V)\).

Proof

Since \( R/R'=\langle \psi (x)R' \mid x \in \mathcal {P}\rangle \), the image of e generates \(\mathrm {PG}(V)\).

We prove that \(\psi (x_1) R' \not = \psi (x_2) R'\) for every two distinct points \(x_1\) and \(x_2\) of \(\mathcal {S}\). Suppose to the contrary that \(\psi (x_1) R' = \psi (x_2) R'\). Since \(\psi \) is faithful by Lemma 5.5, we have \(\psi (x_1) \not = \psi (x_2)\). So, \(\psi (x_1) = \psi (x_2)\lambda \). By Axiom (E3), there exists a point \(x_3\) opposite to \(x_1\), but not to \(x_2\). Then \(\lambda =[\psi (x_1),\psi (x_3)]=[\psi (x_2)\lambda ,\psi (x_3)]=[\psi (x_2),\psi (x_3)]=1\), a contradiction.

Let \(L = \{ x_1,x_2,x_3 \}\) be a line of \(\mathcal {S}\). We have \(e(x_i)=\langle \psi (x_i) R'\rangle \), for \(i \in \{ 1,2,3 \}\). Since \(\psi (x_1)\psi (x_2) = \psi (x_3)\), we have \(\psi (x_3) R'= \psi (x_1) R'\psi (x_2) R'\). Hence \(\{ e(x_1),e(x_2),e(x_3) \}\) is a line of \(\mathrm {PG}(V)\). \(\square \)

Definition

For all \(a,b\in R\), we define
$$\begin{aligned} f(aR',bR')=\left\{ \begin{array}{l@{\quad }l} 1 &{} \text { if }[a,b]=\lambda , \\ 0 &{} \text { if }[a,b]=1. \end{array}\right. \end{aligned}$$
Since \(R' = \{ 1,\lambda \} \subseteq Z(R)\), the map \(f:V \times V \rightarrow \mathbb {F}_2\) is well defined.

Lemma 5.10

The map \(f:V \times V \rightarrow \mathbb {F}_2\) is an alternating bilinear form of V.

Proof

The claim that f is an alternating bilinear form follows from the following facts.
  • Since \([a,a]=[1,a]=[a,1]=1\), we have \(f(aR',aR')=f(R',aR')=f(aR',R')=0\) for all \(a \in R\).

  • Let \(x_1,x_2,y_1 \in R\). Since \([x_1x_2,y_1] = [x_1,y_1][x_2,y_1]\), we have \(f(x_1R' x_2 R',y_1R') = f(x_1R',y_1R') + f(x_2R',y_1R')\).

  • Let \(x_1,y_1,y_2 \in R\). Since \([x_1,y_1y_2] = [x_1,y_1][x_1,y_2]\), we have \(f(x_1R',y_1R' y_2R') = f(x_1R',y_1R') + f(x_1R',y_2R')\). \(\square \)

Lemma 5.11

The embedding e of \(\mathcal {S}\) into \(\mathrm {PG}(V)\) is polarized.

Proof

For every point x of \(\mathcal {S}\), we define a certain subspace \(\Pi _x\) of \(\mathrm {PG}(V)\). Let \(\bar{v}\) be the unique vector of V for which \(e(x) = \langle \bar{v}\rangle \). Then \(\Pi _x\) is the subspace of \(\mathrm {PG}(V)\) corresponding3 to the subspace \(\bar{v}^{\perp _f}\) of V.

Let \(x_1\) and \(x_2\) be two points of \(\mathcal {S}\) and let \(\bar{v}_i=\psi (x_i)R'\), \(i \in \{ 1,2 \}\). So \(e(x_i) = \langle \bar{v}_i\rangle \). Then the following holds:
$$\begin{aligned} \mathop {\text{ d }}\nolimits (x_1,x_2) \le n-1\Leftrightarrow & {} [\psi (x_1),\psi (x_2)]=1 \nonumber \\\Leftrightarrow & {} f(\psi (x_1) R' , \psi (x_2) R')=0 \nonumber \\\Leftrightarrow & {} f(\bar{v}_1,\bar{v}_2)=0 \nonumber \\\Leftrightarrow & {} \bar{v}_2 \in \bar{v}_1^{\perp _f} \nonumber \\\Leftrightarrow & {} e(x_2) \in \Pi _{x_1}. \nonumber \end{aligned}$$
Now from the above it follows that \(\Pi _x= \langle e(H_x)\rangle _{\mathrm {PG}(V)}\) is a hyperplane of \(\mathrm {PG}(V)\) for every point x of \(\mathcal {S}\), where \(H_x\) is the singular hyperplane of \(\mathcal {S}\) with deepest point x. So e is polarized. \(\square \)

Definition

We call e the full polarized embedding of\(\mathcal {S}\)associated with the non-abelian representation\((R,\psi )\).

Lemma 5.12

  1. (1)

    R is of order \(2^{\beta }\) for some \(\beta \) satisfying \(1+er^-(\mathcal {S}) \le \beta \le 1+er^+(\mathcal {S})\).

     
  2. (2)
    The following are equivalent:
    • \((R,\psi )\) is isomorphic to \((\widetilde{R}_p,\widetilde{\psi }_p)\);

    • \(\beta = 1 + er^+(\mathcal {S})\);

    • e is isomorphic to the universal embedding of \(\mathcal {S}\).

     
  3. (3)
    The following are equivalent:
    • \(\beta = 1 + er^-(\mathcal {S})\);

    • R is an extraspecial 2-group;

    • e is isomorphic to the minimal full polarized embedding of \(\mathcal {S}\).

     

Proof

By Lemmas 5.9 and 5.11, e defines a full polarized embedding of \(\mathcal {S}\) into \(\mathrm {PG}(V)\). So, \(er^-(\mathcal {S})\le \dim (V)\le er^+(\mathcal {S})\). Since \(|R/R'|=2^{\beta -1}\), we have \(\dim (V)=\beta -1\) and hence \(1+er^-(\mathcal {S}) \le \beta \le 1+er^+(\mathcal {S})\). The lower bound occurs if and only if e is isomorphic to the minimal full polarized embedding of \(\mathcal {S}\). The upper bound occurs if and only if e is isomorphic to the universal embedding of \(\mathcal {S}\). From Lemma 5.2, the upper bound and the fact that \((R,\psi )\) is isomorphic to a quotient of \((\widetilde{R}_p,\widetilde{\psi }_p)\), it follows that \(\beta = 1 + er^+(\mathcal {S})\) if and only if \((R,\psi )\) is isomorphic to \((\widetilde{R}_p,\widetilde{\psi }_p)\).

Now, R is extraspecial if and only if \(R'=Z(R)\), that is, if and only if the alternating bilinear form f is nondegenerate. For every point x of \(\mathcal {S}\), let \(\bar{v}_{x}\) be the unique vector of V for which \(e(x) = \langle \bar{v}_x\rangle \). Then \(\langle e(H_x)\rangle = \mathrm {PG}(\langle \bar{v}_{x} \rangle ^{\perp _f})\) (see the proof of Lemma 5.11) is a hyperplane of \(\mathrm {PG}(V)\) for every point x of \(\mathcal {S}\). It follows that f is nondegenerate if and only if the nucleus \(\mathcal {N}_e\) of e is empty, that is, if and only if e is a minimal full polarized embedding of \(\mathcal {S}\). Thus R is extraspecial if and only if \(er^-(\mathcal {S})=\dim (V)=\beta -1\). \(\square \)

For every \(r \in R\), we set \(\theta (r) := r R' \in V\). Observe that if \(r_1,r_2 \in R\), then \(f(\theta (r_1),\theta (r_2))=0\) if \([r_1,r_2]=1\) and \(f(\theta (r_1),\theta (r_2))=1\) if \([r_1,r_2]=\lambda \). We denote by \(\mathcal {R}_f\) the radical of the alternating bilinear form f. The subspace of \(\mathrm {PG}(V)\) corresponding to \(\mathcal {R}_f\) is precisely \(\mathcal {N}_e\).

Lemma 5.13

If N is a subgroup of R contained in Z(R), then \(\theta (N) \subseteq \mathcal {R}_f\).

Proof

Let \(g \in N\) and \(h \in R\). Then \([g,h]=1\) implies that \(f(\theta (g),\theta (h))=0\). Since \(\theta (R)=V\), it follows that \(\theta (g) \in \mathcal {R}_f\). Hence, \(\theta (N) \subseteq \mathcal {R}_f\). \(\square \)

Lemma 5.14

If U is a subspace of \(\mathcal {R}_f\), then \(\theta ^{-1}(U)\) is a subgroup of R contained in Z(R). If \(\dim (U)=l\), then \(\theta ^{-1}(U)\) is an abelian subgroup isomorphic to either \(C_2^{l+1}\) or \(C_2^{l-1} \times C_4\).

Proof

Clearly, \(\theta ^{-1}(U)\) is a subgroup of R. If \(g \in \theta ^{-1}(U)\) and \(h \in R\), then we have \(f(\theta (g),\theta (h))=0\) since \(\theta (g) \in U \subseteq \mathcal {R}_f\). This implies that \([g,h]=1\). So, \(\theta ^{-1}(U) \subseteq Z(R)\). In particular, \(\theta ^{-1}(U)\) is abelian. By the classification of finite abelian groups, \(\theta ^{-1}(U)\) is isomorphic to the direct product of a number of cyclic groups. Since the exponent of R is equal to 4, each of these cyclic groups has order 2 or 4. Lemma 5.8 (1) then implies that there is at most one cyclic group of order 4 in this direct product. If \(\dim (U)=l\), then \(|\theta ^{-1}(U)| = 2^{l+1}\) and hence \(\theta ^{-1}(U)\) must be isomorphic to either \((C_2)^{l+1}\) or \((C_2)^{l-1} \times C_4\). \(\square \)

Corollary 5.15

  1. (1)

    We have \(\mathcal {R}_f = \theta (Z(R))\).

     
  2. (2)

    We have \(|Z(R)| = |R| \cdot 2^{- er^-(\mathcal {S})}\).

     
  3. (3)

    If \(l = er^+(\mathcal {S}) - er^-(\mathcal {S})\), then the center \(Z(\widetilde{R}_p)\) of \(\widetilde{R}_p\) is isomorphic to either \(C_2^{l+1}\) or \(C_2^{l-1} \times C_4\).

     

Proof

  1. (1)

    By Lemmas 5.13 and 5.14, we have \(\theta (Z(R)) \subseteq \mathcal {R}_f\) and \(\theta ^{-1}(\mathcal {R}_f) \subseteq Z(R)\), implying that \(\mathcal {R}_f = \theta (Z(R))\).

     
  2. (2)

    Since \(\lambda \in Z(R)\), we have \(|Z(R)| = 2 \cdot |\theta (Z(R))| = 2 \cdot |\mathcal {R}_f| = 2 \cdot |V| \cdot 2^{-er^-(\mathcal {S})} = |R| \cdot 2^{- er^-(\mathcal {S})}\).

     
  3. (3)

    This follows from Lemma 5.14 and Claim (2). \(\square \)

     

We will now study the quotient representations of \((R,\psi )\). For such quotient representations, we need normal subgroups N of R such that \(\psi (x) \not \in N\) for every point x of \(\mathcal {S}\).

Lemma 5.16

The normal subgroups of R are the following:
  1. (1)

    the subgroups of R containing \(\lambda \);

     
  2. (2)

    the subgroups of R not containing \(\lambda \) that are contained in Z(R).

     

Proof

Clearly, the subgroups in (1) and (2) above are normal in R. Suppose N is a normal subgroup of R not containing \(\lambda \). For all \(n \in N\) and all \(r \in R\), we then have \([n,r] \in N \cap R' = N \cap \{ 1,\lambda \} = \{ 1 \}\), implying that \(N \subseteq Z(R)\). \(\square \)

Remark

If N is a (normal) subgroup of R contained in Z(R), then the condition that \(\psi (x) \not \in N\) for every point x of \(\mathcal {S}\) is automatically satisfied by Lemma 5.5.

Lemma 5.17

Let N be a normal subgroup of R such that \(\psi (x) \not \in N\) for every point x of \(\mathcal {S}\). Then the quotient representation \((R/N,\psi _N)\) is abelian if and only if \(\lambda \in N\).

Proof

The representation \((R/N,\psi _N)\) is abelian if and only if \([\psi (x) N , \psi (y) N] = [\psi (x) , \psi (y)] \cdot N = N\) for every two points x and y of \(\mathcal {S}\), that is, if and only if \(\lambda \in N\).

\(\square \)

Lemma 5.18

Let N be a (necessarily normal) subgroup of R containing \(\lambda \) such that \(\psi (x) \not \in N\) for every point x of \(\mathcal {S}\). Set \(U:=\theta (N)\), and let \(\alpha \) denote the subspace of \(\mathrm {PG}(V)\) corresponding to U. Then \(e(x) \not \in \alpha \) for every point x of \(\mathcal {S}\), and the full projective embedding of \(\mathcal {S}\) corresponding to the abelian representation \((R/N,\psi _N)\) is isomorphic to \(e/\alpha \).

Proof

If \(x \in \mathcal {P}\), then the facts that \(\psi (x) \not \in N\), \(\lambda \in N\) and \(R' = \{ 1,\lambda \}\) imply that \(\psi (x) R'\) does not belong to the set \(\{ nR' \, | \, n \in N \}\), that is, \(e(x) \not \in \alpha \). So, \(e/\alpha \) is well defined as a projective embedding.

Consider the quotient vector space V / U and the associated projective space \(\mathrm {PG}(V/U)\). The map which sends each point x of \(\mathcal {S}\) to the point \(\langle (\psi (x) R') \cdot U \rangle \) of \(\mathrm {PG}(V/U)\) is then a full projective embedding isomorphic to \(e/\alpha \). The map
$$\begin{aligned} \phi : R/N \rightarrow V/U ;\; r N \mapsto \theta (r) \cdot U \qquad (r \in R), \end{aligned}$$
which is well defined as \(U = \theta (N)\), is an isomorphism of groups. (The injectivity of the map follows from the fact that \(\theta ^{-1}(U)=N\) which is a consequence of the fact that \(\lambda \in N\).) The fact that \(\phi \circ \psi _N(x) = \phi (\psi (x) \cdot N) = \theta (\psi (x)) \cdot U = (\psi (x) R') \cdot U\) for every point x of \(\mathcal {S}\) implies that the full projective embedding of \(\mathcal {S}\) corresponding to the abelian representation \((R/N,\psi _N)\) is isomorphic to \(e/\alpha \). \(\square \)

Lemma 5.19

Let N be a subgroup of R contained in Z(R). Set \(U:=\theta (N) \subseteq \mathcal {R}_f\), and let \(\alpha \subseteq \mathcal {N}_e\) denote the subspace of \(\mathrm {PG}(V)\) corresponding to U. Then:
  1. (1)

    If \(\lambda \not \in N\), then the projective embedding associated with the non-abelian representation \((R/N,\psi _N)\) is isomorphic to \(e/\alpha \).

     
  2. (2)

    The representation \((R/N,\psi _N)\) is polarized.

     

Proof

  1. (1)

    Consider the normal subgroup \(\overline{N} := \langle N,\lambda \rangle \subseteq Z(R)\) of R. By Lemma 5.5, this group does not contain any element \(\psi (x)\) where \(x \in \mathcal {P}\). We have \(\theta (\overline{N}) = \theta (N) = U\). So, by Lemma 5.18, the projective embedding corresponding to the abelian representation \((R/\overline{N},\psi _{\overline{N}})\) is isomorphic to \(e_\alpha \) where \(\alpha \) is the subspace of \(\mathrm {PG}(V)\) corresponding to U. It is straightforward to verify that the projective embedding associated with the non-abelian representation \((R/N,\psi _N)\) is isomorphic to the projective embedding corresponding to the abelian representation \((R/\overline{N},\psi _{\overline{N}})\). (Observe that \(R/\overline{N} \cong (R/N)/(\overline{N}/N)\) and \((R/N)' = \overline{N}/N\).)

     
  2. (2)

    If \((R/N,\psi _N)\) is non-abelian, then the fact that \([\psi (x) N , \psi (y) N] = [\psi (x),\psi (y)] \cdot N = N\) for any two non-opposite points x and y implies that \((R/N,\psi _N)\) is polarized. If \((R/N,\psi _N)\) is abelian, then the fact that \(\alpha \subseteq \mathcal {N}_e\) implies that \(e/\alpha \) is polarized and hence that \((R/N,\psi _N)\) is polarized by Lemma 5.18. \(\square \)

     

Lemma 5.20

Let N be a normal subgroup of R such that \(\psi (x) \not \in N\) for every point x of \(\mathcal {S}\). Then the representation \((R/N,\psi _N)\) is polarized if and only if \(N \subseteq Z(R)\).

Proof

If \(N \subseteq Z(R)\), then \((R/N,\psi _N)\) is polarized by Lemma 5.19. Conversely, suppose that \((R/N,\psi _N)\) is polarized. If \(\lambda \not \in N\), then \(N \subseteq Z(R)\) by Lemma 5.16. So, we may suppose that \(\{ 1,\lambda \} \subseteq N\). Then \((R/N,\psi _N)\) is an abelian representation of \(\mathcal {S}\). Now, \(R/N \cong (R/R')/(N/R')\), where \(R'=\{1,\lambda \}\). The embedding e has \(\mathrm {PG}(V)\) as target projective space, where \(V = R/R'\) is regarded as an \(\mathbb {F}_2\)-vector space. The full projective embedding \(e'\) corresponding to \((R/N,\psi _N)\) has \(\mathrm {PG}(R/N)\) as target projective space, where the elementary abelian 2-group R / N is again regarded as an \(\mathbb {F}_2\)-vector space. Since \(e'\) is polarized, we should have \(\theta (N) = N/R' \subseteq \mathcal {R}_f\), that is, \(N \subseteq Z(R)\). \(\square \)

6 Classification of the polarized non-abelian representations

In this section, we shall prove all the claims mentioned in Theorems 3.2, 3.3 and 3.4. Let \(\mathcal {S}=(\mathcal {P},\mathcal {L},\mathrm {I})\) be a finite slim near-polar space of diameter \(n \ge 3\) that has polarized non-abelian representations. We set \((\widetilde{R},\widetilde{\psi })\) equal to \((\widetilde{R}_p,\widetilde{\psi }_p)\), the universal polarized representation of \(\mathcal {S}\). There then exists an element \(\widetilde{\lambda } \in \widetilde{R} \setminus \{ 1 \}\) such that \([\widetilde{\psi }(x),\widetilde{\psi }(y)] = \widetilde{\lambda }\) for every two opposite points x and y of \(\mathcal {S}\). Recall also that \(\widetilde{R}' = \{ 1, \widetilde{\lambda } \}\) and that the quotient group \(\widetilde{R} / \widetilde{R}'\) is an elementary abelian 2-group which can be regarded as a vector space \(\widetilde{V}\) over \(\mathbb {F}_2\). Let \(\widetilde{f}\) denote the alternating bilinear form on \(\widetilde{V}\) associated with \((\widetilde{R},\widetilde{\psi })\) as described in Sect. 5 (see Lemma 5.10). The radical of \(\widetilde{f}\) is denoted by \(\mathcal {R}_{\widetilde{f}}\). For every \(r \in \widetilde{R}\), we put \(\widetilde{\theta }(r) := r \widetilde{R}' \in \widetilde{V}\) and for every point x of \(\mathcal {S}\), we put \(\widetilde{e}(x)\) equal to the point \(\langle \widetilde{\psi }(x) \widetilde{R}' \rangle \) of \(\mathrm {PG}(\widetilde{V})\). Then \(\widetilde{e}\) is isomorphic to the universal embedding of \(\mathcal {S}\). By Sect. 5, we also know the following.

Proposition 6.1

The polarized representations of \(\mathcal {S}\) are precisely the representations of the form \((\widetilde{R}/N,\widetilde{\psi }_N)\), where N is a subgroup contained in \(Z(\widetilde{R})\).

Recall that if N is a subgroup contained in \(Z(\widetilde{R})\), then N is necessarily normal and \(\widetilde{\psi }(x) \not \in Z(\widetilde{R})\) for every point x of \(\mathcal {S}\), implying that the quotient representation \((\widetilde{R}/N,\widetilde{\psi }_N)\) is well defined.

Proposition 6.2

If \(N_1\) and \(N_2\) are two subgroups of \(\widetilde{R}\) contained in \(Z(\widetilde{R})\), then the quotient representations \((\widetilde{R}/N_1,\widetilde{\psi }_{N_1})\) and \((\widetilde{R}/N_2,\widetilde{\psi }_{N_2})\) of \(\mathcal {S}\) are isomorphic if and only if \(N_1=N_2\).

Proof

We prove that if the representations \((\widetilde{R}/N_1,\widetilde{\psi }_{N_1})\) and \((\widetilde{R}/N_2,\widetilde{\psi }_{N_2})\) are isomorphic, then \(N_1 \subseteq N_2\). By symmetry, we then also have that \(N_2 \subseteq N_1\).

Let \(\phi \) be a group isomorphism from \(\widetilde{R}/N_1\) to \(\widetilde{R}/N_2\) such that \(\phi (\widetilde{\psi }(x) N_1) = \widetilde{\psi }(x) N_2\) for every point x of \(\mathcal {S}\).

Let \(g\in N_1\). Since \(\widetilde{R} = \langle \widetilde{\psi }(x) \, | \, x \in \mathcal {P} \rangle \), there exist (not necessarily distinct) points \(x_1,x_2,\ldots ,x_k\) such that \(g = \widetilde{\psi }(x_1) \widetilde{\psi }(x_2) \ldots \widetilde{\psi }(x_k)\). Then \(N_2 = \phi (N_1) = \phi (gN_1) = \phi ({\widetilde{\psi }(x_1)N_1} \ldots \widetilde{\psi }(x_k)N_1) = \phi (\widetilde{\psi }(x_1) N_1) \ldots \phi (\widetilde{\psi }(x_k) N_1) = \widetilde{\psi }(x_1) N_2 \ldots \widetilde{\psi }(x_k) N_2 = g N_2\). Hence, \(g \in N_2\). Since g is an arbitrary element of \(N_1\), we have \(N_1 \subseteq N_2\). \(\square \)

By Corollary 5.15 (3), we know that \(Z(\widetilde{R})\) is isomorphic to either \((C_2)^{l+1}\) or \((C_2)^{l-1} \times C_4\), where \(l := er^+(\mathcal {S}) - er^-(\mathcal {S})\).

Proposition 6.3

  1. (i)

    The number of nonisomorphic polarized representations of \(\mathcal {S}\) is equal to the sum \(\sum _{i=0}^{l+1}\Big [ \begin{array}{c} l+1 \\ i \end{array} \Big ]_2\) if \(Z(\widetilde{R}) \cong (C_2)^{l+1}\), and equal to \(2 \cdot \sum _{i=0}^{l}\Big [ \begin{array}{c} l \\ i \end{array} \Big ]_2 - \sum _{i=0}^{l-1}\Big [ \begin{array}{c} l-1 \\ i \end{array} \Big ]_2\) if \(l \ge 1\) and \(Z(\widetilde{R}) \cong (C_2)^{l-1} \times C_4\).

     
  2. (ii)

    The number of nonisomorphic polarized non-abelian representations of \(\mathcal {S}\) is equal to \(\sum _{i=0}^{l+1}\Big [ \begin{array}{c} l+1 \\ i\end{array} \Big ]_2 - \sum _{i=0}^{l} \Big [ \begin{array}{c} l \\ i \end{array} \Big ]_2\) if \(Z(\widetilde{R}) \cong (C_2)^{l+1}\), and equal to \(\sum _{i=0}^{l}\Big [ \begin{array}{c} l \\ i \end{array} \Big ]_2 - \sum _{i=0}^{l-1}\Big [ \begin{array}{c} l-1 \\ i \end{array} \Big ]_2\) if \(l \ge 1\) and \(Z(\widetilde{R}) \cong (C_2)^{l-1} \times C_4\).

     

Proof

By Lemma 5.17 and Propositions 6.1 and 6.2, the number of nonisomorphic polarized (non-abelian) representations of \(\mathcal {S}\) is equal to the number of subgroups of \(Z(\widetilde{R})\) (not containing \(\widetilde{\lambda }\)).

If \(Z(\widetilde{R}) \cong (C_2)^{l+1}\), then \(Z(\widetilde{R})\) is an elementary abelian 2-group and so the number of subgroups of \(Z(\widetilde{R})\) (containing \(\widetilde{\lambda }\)) is equal to \(\sum _{i=0}^{l+1}\Big [ \begin{array}{c}l+1 \\ i \end{array} \Big ]_2\)\(\left( \sum _{i=0}^{l}\Big [ \begin{array}{c} l \\ i \end{array} \Big ]_2\right) \).

If \(Z(\widetilde{R}) \cong (C_2)^{l-1} \times C_4\), then \(Z(\widetilde{R}) / \langle \widetilde{\lambda } \rangle \cong (C_2)^l\) and hence the total number of subgroups of \(Z(\widetilde{R})\) containing \(\widetilde{\lambda }\) is equal to \(\sum _{i=0}^{l}\Big [ \begin{array}{c}l \\ i \end{array} \Big ]_2\). If G is a subgroup of \(Z(\widetilde{R})\) not containing \(\widetilde{\lambda }\), then G only has elements of order 1 and 2. The subgroup of \(Z(\widetilde{R})\) consisting of all elements of order 1 and 2 is isomorphic to \((C_2)^l\) and hence the number of subgroups of \(Z(\widetilde{R})\) not containing \(\widetilde{\lambda }\) is equal to \(\sum _{i=0}^{l}\Big [ \begin{array}{c} l \\ i \end{array} \Big ]_2 -\sum _{i=0}^{l-1}\Big [ \begin{array}{c} l-1 \\ i \end{array} \Big ]_2\). \(\square \)

Lemma 6.4

The following are equivalent:
  1. (1)

    \(Z(\widetilde{R})\) is elementary abelian, that is, isomorphic to \(C_2^{l+1}\);

     
  2. (2)

    \(\mathcal {S}\) has a non-abelian representation \((R,\psi )\), where R is some extraspecial group;

     
  3. (3)

    \(\mathcal {S}\) has a non-abelian representation \((R,\psi )\), where \(|R|=2^{1+er^-(\mathcal {S})}\).

     
If one of these conditions holds, then the number of nonisomorphic polarized non-abelian representations \((R,\psi )\) with \(|R|=2^{1+er^-(\mathcal {S})}\) is equal to \(2^l\).

Proof

In Lemma 5.12 (3), we already showed that (2) and (3) are equivalent. By Lemma 5.17 and Proposition 6.1\(,\mathcal {S}\) has polarized non-abelian representations \((R,\psi )\) where \(|R| = 2^{1+er^-(\mathcal {S})}\) if and only if \(Z(\widetilde{R})\) has subgroups of order \(2^l\) not containing \(\widetilde{\lambda }\). Such subgroups do not exist if \(l \ge 1\) and \(Z(\widetilde{R}) \cong (C_2)^{l-1} \times C_4\). If \(Z(\widetilde{R}) \cong (C_2)^{l+1}\), then the number of such subgroups is equal to \(\Big [ \begin{array}{c} l+1 \\ l \end{array} \Big ]_2 - \Big [ \begin{array}{c} l \\ l-1\end{array} \Big ]_2 = 2^l\). \(\square \)

Lemma 6.5

If \(l \ge 1\) and \(Z(\widetilde{R}) \cong (C_2)^{l-1} \times C_4\), then \(|R| \ge 2^{2+er^-(\mathcal {S})}\) for every polarized non-abelian representation \((R,\psi )\) of \(\mathcal {S}\). The number of such polarized non-abelian representations (up to isomorphism) is equal to \(2^{l-1}\). If \((R,\psi )\) is a polarized non-abelian representation of \(\mathcal {S}\) for which \(|R| = 2^{2+er^-(\mathcal {S})}\), then \(Z(R) \cong C_4\).

Proof

By Lemmas 5.12 and 6.4, we know that \(|R| \ge 2^{2+er^-(\mathcal {S})}\) for every polarized non-abelian representation \((R,\psi )\) of \(\mathcal {S}\). The number of such polarized non-abelian representations (up to isomorphism) is equal to the number of subgroups of order \(2^{l-1}\) of \(Z(\widetilde{R})\) that do not contain \(\widetilde{\lambda }\), that is, equal to \(\Big [ \begin{array}{c} l \\ l-1 \end{array} \Big ]_2 - \Big [ \begin{array}{c} l-1 \\ l-2\end{array} \Big ]_2 = 2^{l-1}\). Suppose \((R,\psi )\) is a polarized non-abelian representation of \(\mathcal {S}\) with \(|R| = 2^{2+er^-(\mathcal {S})}\). Then Z(R) is isomorphic to either \(C_4\) or \(C_2 \times C_2\) by Lemma 5.14 and Corollary 5.15 (2). If \(Z(R) \cong C_2 \times C_2\), then Z(R) contains subgroups of order 2 not containing \(R'\) and so \((R,\psi )\) has a proper quotient which is a polarized non-abelian representation. This is impossible as the size of the representation group R is already as small as possible. \(\square \)

Lemma 6.6

If \(N_1\) and \(N_2\) are two subgroups of \(\widetilde{R}\) contained in \(Z(\widetilde{R})\) such that \(\widetilde{\lambda } \not \in N_1 \cup N_2\) and \(\widetilde{\theta }(N_1) = \widetilde{\theta }(N_2)\), then there exists an automorphism of \(\widetilde{R}\) mapping \(N_1\) to \(N_2\). As a consequence, the quotient groups \(\widetilde{R}/N_1\) and \(\widetilde{R}/N_2\) are isomorphic.

Proof

Set \(U := \widetilde{\theta }(N_1)=\widetilde{\theta }(N_2)=\langle \bar{v}_1,\bar{v}_2,\ldots ,\bar{v}_k \rangle \) for some vectors \(\bar{v}_1,\bar{v}_2,\ldots ,\bar{v}_k\) of \(\widetilde{V}\) where \(k = \dim (U)\). Put \(d := \dim (\widetilde{V})\) and extend \(\{ \bar{v}_1,\bar{v}_2,\ldots ,\bar{v}_k \}\) to a basis \(\{ \bar{v}_1,\bar{v}_2,\ldots ,\bar{v}_d \}\) of \(\widetilde{V}\). For every \(i \in \{ 1,2,\ldots ,d \}\), let \(g_i\) be an arbitrary element of \(\widetilde{\theta }^{-1}(\bar{v}_i)\). For all \(i,j \in \{ 1,2,\ldots ,d \}\), put \(a_{ij} := 1\) if \(\widetilde{f}(\bar{v}_i,\bar{v}_j)=1\) and \(a_{ij}:=0\) otherwise. The group \(\widetilde{R}\) has order \(2^{d+1}\) and consists of all elements of the form
$$\begin{aligned} \widetilde{\lambda }^{\epsilon _0} g_1^{\epsilon _1} g_2^{\epsilon _2} \ldots g_d^{\epsilon _d}, \end{aligned}$$
where \(\epsilon _0,\epsilon _1,\ldots ,\epsilon _d \in \{ 0,1 \}\). If \(i,j \in \{ 1,2,\ldots ,d \}\), we have \([g_i,g_j] = 1\) if \(\widetilde{f}(\bar{v}_i,\bar{v}_j)=0\) and \([g_i,g_j]=\widetilde{\lambda }\) if \(\widetilde{f}(\bar{v}_i,\bar{v}_j)=1\). So, the multiplication inside the group \(\widetilde{R}\) should be as follows. If \(\epsilon _0,\epsilon _1,\ldots ,\epsilon _d,\epsilon _0',\epsilon _1',\ldots ,\epsilon _d' \in \{ 0,1 \}\), then
$$\begin{aligned} (\widetilde{\lambda }^{\epsilon _0} g_1^{\epsilon _1} g_2^{\epsilon _2} \ldots g_d^{\epsilon _d}) \cdot (\widetilde{\lambda }^{\epsilon _0'} g_1^{\epsilon _1'} g_2^{\epsilon _2'} \ldots g_d^{\epsilon _d'}) = \widetilde{\lambda }^{\epsilon _0+\epsilon _0'+\epsilon _0''} g_1^{\epsilon _1 + \epsilon _1'} g_2^{\epsilon _2+\epsilon _2'} \ldots g_d^{\epsilon _d+\epsilon _d'}, \end{aligned}$$
where \(\epsilon _0'' := \sum _{i=1}^d \sum _{j=i+1}^d a_{ij} \epsilon _i' \epsilon _j\). Recall that \(\widetilde{\lambda } \not \in N_1 \cup N_2\). So, for every \(i \in \{ 1,2,\ldots ,k \}\), there exists a unique element \(g^{(1)}_i \in \{ g_i,g_i \widetilde{\lambda } \}\) belonging to \(N_1\) and a unique element \(g_i^{(2)} \in \{ g_i,g_i \widetilde{\lambda } \}\) belonging to \(N_2\). Then \(N_1 = \langle g^{(1)}_1,g^{(1)}_2,\ldots ,g^{(1)}_k \rangle \) and \(N_2 = \langle g^{(2)}_1,g^{(2)}_2,\ldots ,g^{(2)}_k \rangle \). Now, let I denote the subset of \(\{ 1,2,\ldots ,k \}\) consisting of all \(i \in \{ 1,2,\ldots ,k \}\) for which \(g^{(1)}_i \not = g^{(2)}_i\), or equivalently, for which \(g^{(2)}_i = g^{(1)}_i \widetilde{\lambda }\). Then the permutation of \(\widetilde{R}\) defined by
$$\begin{aligned} \widetilde{\lambda }^{\epsilon _0} g_1^{\epsilon _1} g_2^{\epsilon _2} \ldots g_d^{\epsilon _d} \mapsto \widetilde{\lambda }^{\epsilon _0 + \epsilon _0'} g_1^{\epsilon _1} g_2^{\epsilon _2} \ldots g_d^{\epsilon _d}, \end{aligned}$$
where \(\epsilon _0' := \sum _{i \in I} \epsilon _i\), is an automorphism \(\phi \) of R. Since \(\phi (g^{(1)}_i) = g^{(2)}_i\) for every \(i \in \{ 1,2,\ldots ,k \}\), we have \(\phi (N_1)=N_2\). \(\square \)

Corollary 6.7

If \((R_1,\psi _1)\) and \((R_2,\psi _2)\) are two polarized non-abelian representations of \(\mathcal {S}\) for which the associated full polarized embeddings are isomorphic, then also the representation groups \(R_1\) and \(R_2\) are isomorphic.

Proof

Let \(N_1\) and \(N_2\) be the subgroups of \(\widetilde{R}\) contained in \(Z(\widetilde{R})\) such that \((R_1,\psi _1) \cong (\widetilde{R}/N_1,\widetilde{\psi }_{N_1})\) and \((R_2,\psi _2) \cong (\widetilde{R}/N_2,\widetilde{\psi }_{N_2})\). Then \(\widetilde{\lambda } \not \in N_1 \cup N_2\). Let \(\alpha _1\) and \(\alpha _2\) be the subspaces of \(\mathcal {N}_{\widetilde{e}}\) corresponding to, respectively, \(U_1 := \widetilde{\theta }(N_1) \subseteq \mathcal {R}_{\widetilde{f}}\) and \(U_2 := \widetilde{\theta }(N_2) \subseteq \mathcal {R}_{\widetilde{f}}\). By Lemma 5.19 (1), the projective embeddings \(e/\alpha _1\) and \(e/\alpha _2\) are isomorphic. This implies that \(\alpha _1=\alpha _2\). Hence, \(\widetilde{\theta }(N_1)=\widetilde{\theta }(N_2)\). By Lemma 6.6, \(R_1 \cong R_2\). \(\square \)

Proposition 6.8

If \((R_1,\psi _1)\) and \((R_2,\psi _2)\) are two polarized non-abelian representations of \(\mathcal {S}\) such that \(|R_1|=|R_2|=2^\beta \), where \(\beta = 1 + er^-(\mathcal {S})\), then \(R_1\) and \(R_2\) are isomorphic (to either \(2^\beta _+\) or \(2^\beta _-\)).

Proof

Let \(N_1\) and \(N_2\) be the unique normal subgroups of \(\widetilde{R}\) contained in \(Z(\widetilde{R})\) such that \(\widetilde{\lambda } \not \in N_1 \cup N_2\) and \((\widetilde{R}/N_1,\widetilde{\psi }_{N_1}) \cong (R_1,\psi _1)\) and \((\widetilde{R}/N_2,\widetilde{\psi }_{N_2}) \cong (R_2,\psi _2)\). Then \(|N_1|=|N_2|=\frac{|\widetilde{R}|}{|R_1|} = 2^l\), where \(l=er^+(\mathcal {S}) - er^-(\mathcal {S})\). Since \(\widetilde{\lambda } \not \in N_1 \cup N_2\) and \(|Z(\widetilde{R})|=2^{l+1}\), we have \(Z(\widetilde{R}) = \langle N_1,\widetilde{\lambda } \rangle = \langle N_2,\widetilde{\lambda } \rangle \). Hence, \(\mathcal {R}_{\widetilde{f}} = \theta (Z(\widetilde{R})) = \theta (\langle N_1,\widetilde{\lambda } \rangle ) = \theta (N_1) = \theta (\langle N_2,\widetilde{\lambda } \rangle ) = \theta (N_2)\). By Lemma 6.6, \(R_1 \cong \widetilde{R}/N_1 \cong \widetilde{R}/N_2 \cong R_2\). \(\square \)

Footnotes

  1. 1.

    If \(l=0\), then \((C_2)^{-1}\) is not defined. In this case, this sentence should be understood as “Z(R) is isomorphic to \(C_2\)”.

  2. 2.

    The terms occurring in this sum are Gaussian binomial coefficients.

  3. 3.

    The map \(\phi _x:R\mapsto R'\) defined by \(\phi _x(r)=[\psi (x),r]\) is a homomorphism (see Corollary 5.7) which is surjective. The kernel of \(\phi _x\) is \(C_R(\psi (x))\) which has index 2 in R by the first isomorphism theorem. Then \(\bar{v}^{\perp _f}\) is precisely the image of \(C_R(\psi (x))\) in V under the canonical homomorphism \(R \rightarrow V; r \mapsto r R'\).

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGentBelgium
  2. 2.School of Mathematical SciencesNational Institute of Science Education and ResearchOdishaIndia

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