Linear representations of fundamental groups of Klein surfaces derived from spinor representations of Clifford algebras

Abstract

We study actions of multiplicative subgroups of Clifford algebras on Riemann surfaces. We show that every Klein surface of algebraic genus greater than 1 is isomorphic to the orbit space of such an action. We obtain linear representations of fundamental groups of Klein surfaces by using the spinor representations of Clifford algebras.

1 Introduction

Let $\textrm{Cl}_{n-t,t}$ be a Clifford algebra associated with a vector space $V=K^{n}$ for $K={\mathbb {R}}$ or $K={\mathbb {C}}$ and a quadratic form $Q_t$ defined by

$$\begin{aligned} Q(x_1,\ldots ,x_n)=(x_1^2+\ldots +x_t^2)-(x_{t+1}^2+\ldots +x_{n}^2)\;\textrm{for}\;(x_1,\ldots ,x_n)\in V. \end{aligned}$$

We will identify vectors of V with their images in $\textrm{Cl}_{n-t,t}$. A multiplicative subgroup $M_{n-t,t}$ of $\textrm{Cl}_{n-t,t}$ generated by an ortogonal basis of V is called a base group. Let $M_{n-t,t}^+=M_{n-t,n}\bigcap \textrm{Cl}_{n-t,t}^+$ for the subalgebra $\textrm{Cl}_{n-t,t}^+$ preserved by an automorphism of $\textrm{Cl}_{n-t,t}$ which maps v to $-v$ for all $v\in V$.

We prove that for any Klein surface Y of algebraic genus $d\ge 2$ there are actions of base groups $G_t=M_{d+1-t,t}$ for $t=0,1$ on a Riemann surface X of genus $g=1+2^{d+1}(d-1)$ such that the orbit space $X/G_t$ is isomorphic to Y. The surface Y has a double cover $Y^+$ being a Riemann surface. We show that for a proper Klein surface Y an action of $G_t$ can be defined in such a way that $Y^+$ is isomorphic to the orbit space $X/G_t^+$ for $G_t^+=M_{d+1-t,t}^+$. Let $\pi _Y$ and $\pi _{Y^+}$ be the fundamental groups of Y and $Y^+$. Using the spinor representation of a complex Clifford algebra $\textrm{Cl}_{d,1}$ for an odd d and spinor representation of complex algebra $\textrm{Cl}_{d+1,0}^+$ for an even d we obtain linear representations $\rho :\pi _Y\rightarrow \textrm{Gl}(2^m,{\mathbb {C}})$ and $\rho :\pi _{Y^+}\rightarrow \textrm{Gl}(2^m,{\mathbb {C}})$, respectively, for $m=\frac{d+d_{(2)}}{2}$ and $d_{(2)}\in \{0,1\}$ such that $d_{(2)}\equiv d\;\textrm{mod}\;2$.

2 Preliminaries

This chapter contains elementary information needed to understand the paper. Sections on tensor algebras, exterior algebras, Clifford algebras and their spinor representations are based on books [6, 7]; sections on NEC groups and Klein surfaces are based on the book [3].

2.1 Tensor algebras and external algebras

Let V and W be vector spaces over a field K. The tensor product of V and W is a K-vector space $V\otimes W$ with a bilinear map $j:V\times W\rightarrow V\otimes W$ such that for every K-vector space Z and every bilinear map $f:V\times W\rightarrow Z$ there is a unique linear map ${\bar{f}}:V\otimes W\rightarrow Z$ for which $f={\bar{f}}\circ j$.

Vectors $v\otimes w=j(v,w)$ for $v\in V$ and $w\in W$ are called elementary tensors. If V and W are finite dimensional and $\{v_i\}_1^n$ and $\{w_i\}_1^m$ are their basis, respectively, then the set

$$\begin{aligned} \{v_i\otimes w_j:1\le i\le n\;\textrm{and }\;1\le j\le m\} \end{aligned}$$

is a basis of $V\otimes W$. So $\textrm{dim}V\otimes W=\textrm{dim}V\cdot \textrm{dim}W$.

The tensor product of a finite family of K-vector spaces $\{V_i\}_1^n$ is defined as

$$\begin{aligned} \otimes _{i=1}^n V_i=(\ldots ((V_1\otimes V_2)\otimes V_3)\otimes \ldots )\otimes V_n. \end{aligned}$$

In the case when $V_1=\ldots =V_n=V$ the tensor product $\otimes _{i=1}^n V_i$ is denoted by $V^{\otimes n}$ and $V^{ \otimes 0}=K$.

A multiplication of tensors is a bilinear map $V^{\otimes n}\times V^{\otimes m}\rightarrow V^{\otimes n+m}$ which to each pair of tensors $t_1=v_1\otimes \ldots \otimes v_n\in V^{\otimes n}$ and $t_2=w_1\otimes \ldots \otimes w_m\in V^{\otimes m}$ assigns tensor

$$\begin{aligned} t_1\cdot t_2=v_1\otimes \ldots \otimes v_n\otimes w_1\otimes \ldots \otimes w_m\in V^{\otimes n+m}. \end{aligned}$$

The tensor algebra of a K-vector space V is a direct sum $T(V)=\oplus _{n\ge 0}V^{\oplus n}$ with the multiplication defined by

$$\begin{aligned} \left( \sum _p t_p^1\right) \cdot \left( \sum _q t_q^2\right) =\sum _s\left( \sum _{p+q=s}t_p^1\cdot t_q^2\right) \end{aligned}$$

for $\sum _p t_p^1, \;\sum _q t_q^2\in T(V)$.

An endomorphism $A\in \textrm{End}(V^{\otimes n})$ defined by the formula

$$\begin{aligned} A(v_1\otimes \ldots \otimes v_n)=\frac{1}{n !}\sum _{\sigma \in S_n}\textrm{sgn}\; \sigma v_{\sigma (1)}\otimes \ldots \otimes v_{\sigma (n)} \end{aligned}$$

for $t=v_1\otimes \ldots \otimes v_n\in V^{\otimes n}$ is called antisymmetrization. The vector space $\bigwedge ^{n} V=A(V^{\otimes n})$ is called the n-th exterior power of V and $\bigwedge ^{0} V$ denotes the field K.

Vector $t=A(v_1\otimes \ldots \otimes v_n)$ is denoted by $v_1\wedge \ldots \wedge v_n$. If $v_i=v_j$ for some $i\ne j$, then $t=0$. Thus $\textrm{dim} \bigwedge ^n V=0$ for $n> \textrm{dim}V$. Moreover, for any permutation $\tau \in S_n$,

$$\begin{aligned} v_{\tau (1)}\wedge \ldots \wedge v_{\tau (n)}=\mathrm{sgn \tau }v_1\wedge \ldots \wedge v_n. \end{aligned}$$

Consequently, if $\textrm{dim} V=k$ and $\{ v_i\}_1^k$ is a basis of V, then the set

$$\begin{aligned} \{1\}\cup \{v_{i_1}\wedge \ldots \wedge v_{i_n}:\; 1\le i_1<\ldots < i_n\le k\} \end{aligned}$$

is a basis of $\bigwedge ^n V$ and

$$\begin{aligned} \textrm{dim}\bigwedge ^n V=\frac{k!}{n!(k-n)!}. \end{aligned}$$

The exterior algebra of a vector space V of dimension k is the direct sum $\bigwedge V=\oplus _{n=0}^k\bigwedge ^n V$ with the multiplication defined by

$$\begin{aligned} t\wedge t'=A(t\otimes t')\;\textrm{for}\;t,t'\in \bigwedge V. \end{aligned}$$

This algebra is the quotient of the tensor algebra T(V) modulo the ideal generated by tensors $v\otimes v$ for all $v\in V$. Moreover, $\textrm{dim}\bigwedge V=\sum _{n=0}^k\left( {\begin{array}{c}k\\ n\end{array}}\right) =2^k$.

2.2 Clifford algebras

Let V be a finite-dimensional vector space over a field K. A quadratic form is a function $Q:V\rightarrow K$ such that

$$\begin{aligned} Q(\alpha v)=\alpha ^2 Q(v)\;\textrm{for}\;\textrm{all}\;v\in V\;\textrm{and}\;\alpha \in K \end{aligned}$$

and the mapping $B:V\times V\rightarrow K$ defined by the formula

$$\begin{aligned} B(v_1,v_2)=\frac{1}{2}[Q(v_1+v_2)-Q(v_1)-Q(v_2)] \end{aligned}$$

(1)

is a bilinear form. It is said that the quadratic form Q is nondegenerate, if for every $0\ne v\in V$ there exists $w\in V$ such that $B(v,w)\ne 0$.

The Clifford algebra associated with V and Q is an associative algebra Cl(V, Q) over K together with a linear map $j:V\rightarrow Cl(V,Q)$ such that

$$\begin{aligned} (j(v))^2=Q(v)\cdot 1\;\textrm{for}\;\textrm{all}\;v\in V \end{aligned}$$

(2)

and for every algebra A over K and every linear map $f:V\rightarrow A$ with

$$\begin{aligned} (f(v))^2=Q(v)\cdot 1_A\;\textrm{for}\;\textrm{all }\;v\in V, \end{aligned}$$

(3)

there is a unique algebra homomorphism ${\bar{f}}:Cl(V,Q)\rightarrow A$ for which $f={\bar{f}}\circ j$. Linear maps $f:V\rightarrow A$ satisfying the condition (3) are called Clifford maps. It is said that the Clifford algebra is universal for Clifford maps.

Any two Clifford algebras associated with V and Q are isomorphic. The Clifford algebra Cl(V, Q) can be seen as the quotient of the tensor algebra T(V) modulo the ideal generated by elements of the form $v\otimes v-Q(v)\cdot 1$ for all $v\in V$.

To simplify the notation, from now on we will write v instead of j(v) for $v\in V$. By Eqs. (1) and (2) we have

$$\begin{aligned} 2B(v,w)=Q(v+w)-Q(v)-Q(w)=(v+w)^2-v^2-w^2=vw+wv \end{aligned}$$

what implies that

$$\begin{aligned} vw+wv=2B(v,w). \end{aligned}$$

(4)

Let $\{v_i\}_1^n$ be an ortogonal basis of V with $B(v_i,v_j)=0$ for $i\ne j$. Then the set

$$\begin{aligned} \{1\}\cup \{v_{i_1}v_{i_2}\cdot \cdot \cdot v_{i_l}:1\le i_{1}<\ldots < i_{l}\le n\} \end{aligned}$$

(5)

is a basis of Cl(V, Q), where 1 denotes the multiplicative unit of $\textrm{Cl}(V,Q)$ being the image of $1\in K$ under the projection $TV\rightarrow Cl(V,Q)$.

The subalgebra of Cl(V, Q) generated by 1 and all elements $v_{i_1}v_{i_2}\cdot \cdot \cdot v_{i_l}$ of above basis with even l is denoted by $Cl(V,Q)^+$. This subalgebra is fixed by an automorphism of Cl(V, Q) which maps v to $-v$ for all $v\in V$. We have $\textrm{dim} Cl(V,Q)=2^{n}$ and $\textrm{dim} Cl(V,Q)^+=2^{n-1}$. If $Q\equiv 0$, then Cl(V, Q) is isomorphic to the exterior algebra $\bigwedge V$. Otherwise, Cl(V, Q) and $\bigwedge V$ are isomorphic only as vector spaces.

2.3 Spinor representation of a Clifford algebra

Let V be a finite-dimensional vector space over a field K and let Q be a non-degenerate quadratic form on V. A subspace W of V is called a totally isotropic subspace, if $Q(w)=0$ for all vectors $w\in W$. In the case when W does not contain any non-zero vector w with $Q(w)=0$, it is said that W is an anisotropic subspace. A subspace $W'$ of V is called ortogonal to W, if $B(w,w')=0$ for all $w\in W$ and $w'\in W'$, where B is a bilinear form associated with Q by the formula (1).

There exists a Witt decomposition of V into a direct sum of three subspaces $V=W\oplus U\oplus T$ such that W and U are maximal totally isotropic subspaces of the same dimension m, while T is anisotropic and orthogonal to $W\oplus U$. Moreover, for any basis $\{w_1,\ldots ,w_m\}$ of W, there exists a basis $\{u_1,\ldots ,u_m\}$ of U such that $B(w_i,u_j)=\delta _{ij}$, where $\delta _{ij}$ are the Kronecker symbols. If the field K is algebraically closed, then T has dimension 1 or 0 according to whether $n=\textrm{dim} V$ is odd or even, respectively.

First assume that n is even and $V=W\oplus U$. The set

$$\begin{aligned} \textrm{Cl}(V,Q)f=\{af:a\in \textrm{Cl}(V,Q)\}\;\textrm{for }\; f=w_1\cdot \cdot \cdot w_m \end{aligned}$$

is a subalgebra of $\textrm{Cl}(V,Q)$ generated by

$$\begin{aligned} \{f\}\bigcup \{u_{i_1}\cdot \cdot \cdot u_{i_k}f: 1\le i_1<\ldots <i_k\le m\}. \end{aligned}$$

Since $Q|_U\equiv 0$, it follows that the algebra $\textrm{Cl}(U,Q|_U)$ is isomorphic to the external algebra $\bigwedge U$ which is spanned by the set

$$\begin{aligned} {\mathcal {B}}=\{1\}\bigcup \{u_{i_1}\wedge \ldots \wedge u_{i_k}: 1\le i_1<\ldots <i_k\le m\}. \end{aligned}$$

Let $\varphi :\bigwedge U\rightarrow \textrm{Cl}(V,Q)f$ and $\psi : \textrm{Cl}(V,Q)\rightarrow \textrm{End}(\textrm{Cl}(V,Q)f)$ be given by

$$\begin{aligned} \varphi (u_{i_1}\wedge \ldots \wedge u_{i_k})=u_{i_1}\cdot \cdot \cdot u_{i_k}f\; \textrm{for}\;u_{i_1}\wedge \ldots \wedge u_{i_k}\in {\mathcal {B}} \end{aligned}$$

and

$$\begin{aligned} \psi (a)(bf)=abf, \;\textrm{for}\;a,b\in \textrm{Cl}(V,Q), \end{aligned}$$

respectively. Then the assignments

$$\begin{aligned} a\rightarrow \eta (a)=\varphi ^{-1}\circ \psi (a)\circ \varphi \;\textrm{for}\;a\in \textrm{Cl}(V,Q) \end{aligned}$$

(6)

define an isomorphism $\eta : \textrm{Cl}(V,Q)\rightarrow \textrm{End}(\bigwedge U)$ called the spinor representation of the Clifford algebra $\textrm{Cl}(V,Q)$.

In the case when K is algebraically closed and $n=\textrm{dim}V=2\,m+1$, the Witt decomposition of V has the form $V=W\oplus U\oplus \textrm{Lin}(v_0)$ for a subspace $\textrm{Lin}(v_0)$ spanned by a vector $v_0\in V$ ortogonal to $V'=W\oplus U$ with $Q(v_0)\ne 0$. The mapping $Q':V'\rightarrow K$ defined by the formula

$$\begin{aligned} Q'(v)=-Q(v_0)Q(v)\;\textrm{for }\;v\in V' \end{aligned}$$

is a nondegenerate quadratic form on a vector space $V'$ of even dimension 2m. Thus by the previus case there exists an isomorphism $\eta :\textrm{Cl}(V',Q')\rightarrow \textrm{End}(\bigwedge U)$. A linear map $f:V'\rightarrow \textrm{Cl}(V,Q)^+$ defined by

$$\begin{aligned} f(v)=v_0v\;\textrm{for}\;v\in V' \end{aligned}$$

is a Clifford map because $f(v)^2=-Q(v_0)Q(v)\cdot 1=Q'(v)\cdot 1$. Since algebra $\textrm{Cl}(V',Q')$ is universal for Clifford maps, there exists a unique algebra homomorphism $\bar{f}:\textrm{Cl}(V',Q')\rightarrow \textrm{Cl}(V,Q)^+$ such that $f=\bar{f}\circ j$ for $j:V'\rightarrow \textrm{Cl}(V',Q')$. The homomorphism ${\bar{f}}$ is an isomorphism because it is injective and

$$\begin{aligned} \text {dimCl}(V,Q)^+=2^{\text {dim}V-1}=2^{\text {dim}V'}=\text {dimCl}(V',Q'). \end{aligned}$$

An isomorphism $\mu =\eta \circ {\bar{f}}^{-1}: \textrm{Cl}(V,Q)^+\rightarrow \textrm{End}(\bigwedge U)$ is called the spinor representation of the algebra $\textrm{Cl}(V,Q)^+$.

2.4 Non-Euclidean-crystallographic groups (NEC groups)

An $\textrm{NEC}$ group is a discrete in the topology of ${\mathbb {R}}^4$ subgroup $\Lambda$ of the group $\textrm{Aut}({\mathcal {H}})$ of isometries of the hyperbolic plane ${\mathcal {H}}$ with compact orbit space ${{\mathcal {H}}}/\Lambda$. If an NEC group $\Lambda$ is contained in the group $\textrm{Aut}^+( {\mathcal {H}})$ of orientation preserving isometries, then it is called a Fuchsian group. Otherwise, it is said that $\Lambda$ is a proper NEC group and $\Lambda ^+=\Lambda \cap \textrm{Aut}^+( {\mathcal {H}})$ is called the canonical Fuchsian subgroup of $\Lambda$.

The basics of $\textrm{NEC}$ group theory were developed by Wilkie [15], Macbeath [8] and Natanzon [10, 11]. The algebraic structure of an $\textrm{NEC}$ group $\Lambda$ is given by the so-called signature which has the form

$$\begin{aligned} \sigma (\Lambda )=(g;\pm ;[m_1,\ldots ,m_r];\{(n_{1,1},\ldots ,n_{1,s_1}),\ldots ,(n_{k,1},\ldots ,n_{k,s_k})\}). \end{aligned}$$

(7)

The number g is called the orbit genus, the integers $m_i$ are said to be proper periods, the brackets $(n_{i,1},\ldots ,n_{i,s_i})$ are called period cycles and the integers $n_{i,j}$ are the link periods of $\Lambda$. The set of proper periods may be empty as well as the set of period cycles. In addition, an individual period-cycle may be empty too. For example, the signature $(g;+;[-];\{-\})$ has no proper periods and no period cycle; the signature $(g;-;[m];\{(-), (-)\})$ has one proper period m and two empty period cycles. A Fuchsian group can be regarded as an NEC group with the signature $(h;+;[m_{1},\ldots \, \!,m_{t}];\{-\})$, usually shortened to $(h;m_{1}, \ldots \, \!, m_{t})$.

If there is a sign $+$ in the signature $\sigma (\Lambda )$, then the presentation of $\Lambda$ consists of generators $a_i,b_i\;(i=1,\ldots ,g),$ $x_i\;(i=1,\ldots ,r),$ $c_{ij},e_i\;(i=1,\ldots ,k,j=1,\ldots ,s_i)$ and the relations

$$\begin{aligned} \begin{array}{ll} x_i^{m_i}=1,&{} i=1,\ldots ,r,\\ c_{ij-1}^2=c_{ij}^2= (c_{ij-1}c_{ij})^{n_{ij}}=1, &{} i=1,\ldots ,k,j=1,\ldots ,s_i,\\ e_ic_{i0}e_i^{-1}=c_{is_i}, &{} i=1,\ldots ,k\\ x_1\cdot \cdot \cdot x_re_1\cdot \cdot \cdot e_k [a_1,b_1]\cdot \cdot \cdot [a_{g},b_{g}]=1.&{}\\ \end{array} \end{aligned}$$

If there is a sign − in the signature $\sigma (\Lambda )$, then we just replace the generators $a_i,b_i$ by $d_i\;(i=1,\ldots ,g)$ and the last relation by

$$\begin{aligned} x_1\cdot \cdot \cdot x_re_1\cdot \cdot \cdot e_k d_1^2\cdot \cdot \cdot d_{g}^2=1. \end{aligned}$$

The last relation in the presentation of $\Lambda$ will be called long relation.

The generators $a_i,b_i$ are hyperbolic, $d_i$ are glide reflections, $x_i$ are elliptic, $e_i$ are hyperbolic or elliptic, and $c_{ij}$ are reflections.

Any generators of an NEC group satisfying the above relations are called canonical generators.

In [8] Macbeath proved the following

Theorem 2.1

Let $\Lambda$ and $\Lambda '$ be $\textrm{NEC}$ groups with signatures (7) and

$$\begin{aligned} \sigma (\Lambda ')=(g';\pm ;[m_1',\ldots ,m_{r'}'];\{(n_{1,1}',\ldots ,n_{1,s_1'}'),\ldots ,(n_{k',1}',\ldots ,n_{k',s_{k'}'}')\}), \end{aligned}$$

respectively. Let $P_i=(n_{i,1},\ldots ,n_{i,s_i})$ and $P_i'=(n_{i,1}',\ldots ,n_{i,s_i'}')$ be the period cycles in $\sigma (\Lambda )$ and $\sigma (\Lambda ')$, respectively. Then $\Lambda$ and $\Lambda '$ are isomorphic if and only if the following conditions are satisfied:

• (i) the sign in $\sigma (\Lambda )$ is the same as in $\sigma (\Lambda ')$,

• (ii) $g'=g$, $r=r'$ and $k=k'$,

• (iii) $(m_1',\ldots ,m_{r'}')$ is a permutation of $(m_1,\ldots ,m_r)$,

• (iv) there is a permutation $\varphi$ of $\{1,\ldots ,k\}$ such that $s_i=s_{\varphi (i)}'$ for $i=1,\ldots ,k$.

• (v) if the sign is $"+"$ then either for each i, $P'_{i}$ is a cyclic permutation of $P_{\varphi (i)}$ or for each i, $P'_{i}$ is a cyclic permutation of the inverse of $P_{\varphi (i)}$; if the sign is $"-"$, then for each i either $P'_{i}$ is a cyclic permutation of $P_{\varphi (i)}$ or is a cyclic permutation of the inverse of $P_{\varphi (i)}$.

There is a closed subset $E\subset {\mathcal {H}}$ associated with an NEC group $\Lambda$, called a fundamental region of $\Lambda$. It has the property that for every $z\in {\mathcal {H}}$ there exists $\lambda \in \Lambda$ such that $\lambda (z)\in E$ and this $\lambda$ is unique if $\lambda (z)\in \textrm{Int} E$. The hyperbolic area $\mu (\Lambda )$ of E depends only on the signature of $\Lambda$. If $\sigma (\Lambda )$ is given by (7), then

$$\begin{aligned} \mu (\Lambda )=2\pi \left[ \alpha g+k-2+\sum _{i=1}^r\left( 1- \frac{1}{m_i}\right) +\frac{1}{2}\sum _{i=1}^k\sum _{j=1}^{s_i}\left( 1- \frac{1}{n_{i,j}}\right) \right] , \end{aligned}$$

(8)

where $\alpha =1$ or $\alpha =2$ according to whether the sign in $\sigma (\Lambda )$ is − or $+$, respectively (e.g. [5, 14]). An abstract group with an algebraic structure determined by a signature (7) is an $\textrm{NEC}$ group if and only if the right hand of (8) is positive.

If $\Gamma$ is a finite index subgroup of an NEC group $\Lambda$, then it is an NEC group itself and there is a Hurwitz-Riemann formula, which says that

$$\begin{aligned}{}[\Lambda :\Gamma ]=\frac{\mu (\Gamma )}{\mu (\Lambda )}. \end{aligned}$$

(9)

2.5 Klein surfaces

An NEC group with a signature

$$\begin{aligned} \sigma =(\gamma ;\pm ;[- ];\{(- ),{\mathop {\ldots }\limits ^{k}},( - )\}) \end{aligned}$$

(10)

is called a surface NEC group. The orbit space of the hyperbolic plane ${\mathcal {H}}$ under the action of this group is a surface of topological genus $\gamma$ with k boundary components which is orientable or non-orientable according to whether the sign $+$ or − occurs in the signature. The integer $\alpha \gamma +k-1$ is called the algebraic genus of the surface, where $\alpha =2$ in the orientable case and $\alpha =1$ in the non-orientable case.

A Klein surface is a compact topological surface equipped with a dianalitic structure. A Riemann surface can be seen as an orientable Klein surface without boundary. Preston proved in [12] that any Klein surface Y of algebraic genus $d\ge 2$ is an orbit space ${\mathcal {H}}/\Gamma$ for some surface NEC group $\Gamma$ with the signature (10).

Alling and Greenleaf [1] constructed certain double cover $Y^+$ of Y being a Riemann surface. If $\Gamma$ is a proper NEC group, then $Y^+\simeq {\mathcal {H}}/\Gamma ^+$ for the canonical Fuchsian subgroup $\Gamma ^+< \Gamma$ with the signature $(d;+;[-];\{-\})$ which consists of all preserving automorphisms of $\Lambda$. If Y is an orientable surface without boundary, then $Y^+$ consists of two connected components $Y_1$ and $Y_2$ with different analytic structures each one homeomorphic to Y and there is an anticonformal isomorphism from $Y_1$ to $Y_2$.

By Proposition 3 of May in [9], an automorphism group of $Y={\mathcal {H}}/\Gamma$ is isomorphic to the quotient group $\Lambda /\Gamma$ for some NEC group $\Lambda$ containing $\Gamma$ as a normal subgroup. So an action of a finite group G on a Klein surface of algebraic genus $d\ge 2$ is associated with a short exact sequence of homomorphisms

$$\begin{aligned} 1\rightarrow \Gamma \rightarrow \Lambda {\mathop {\rightarrow }\limits ^{\theta }}G\rightarrow 1,\end{aligned}$$

(11)

in which $\Lambda$ is an NEC group and $\Gamma$ is a surface NEC group isomorphic to the fundamental group of the surface. This action is denoted by $(\Lambda ,\theta ,G)$.

If there does not exist another NEC group containing $\Lambda$ properly, then $\Lambda$ is called a maximal NEC group and $G=\Lambda /\Gamma$ is the full automorphism group of the Klein surface. The detailed exposition of maximality can be found in [4].

A signature $\sigma$ is called maximal, if for every $\textrm{NEC}$ group $\Lambda '$ with a signature $\sigma '$ containing an NEC group $\Lambda$ with the signature $\sigma$ and having the same Teichmüller dimension, the equality $\Lambda =\Lambda '$ holds. For any maximal signature $\sigma$, there exists a maximal NEC group with the signature $\sigma$. The lists of non-maximal signatures are given in [2, 4].

3 Clifford actions on Riemann surfaces

3.1 The maximal base subgroups of Clifford algebras

We start this chapter by introducing a notation that will be valid throughout the entire paper. Let t and n be two integers such that $0\le t< n$. By $\textrm{Cl}(n-t,t)$ we denote the Clifford algebra $\textrm{Cl}(V,Q)$ associated with a vector space $V=K^n$ for $K={\mathbb {R}}$ or $K={\mathbb {C}}$ and a quadratic form $Q_{t}$ defined by

$$\begin{aligned} Q_{t}(v)=(x_1^2+\ldots x_t^2)-(x_{t+1}^2\ldots +x_{n}^2)\;\textrm{for}\;v=(x_1,\ldots ,x_n)\in V.\end{aligned}$$

(12)

We use the same symbols for vectors of V and their images in the algebra $\textrm{Cl}(n-t,t)$. There is an automorphism of $\textrm{Cl}(n-t,t)$ which maps v to $-v$ for all $v\in V$. The subalgebra of $\textrm{Cl}(n-t,t)$ fixed by this automorphism will be denoted by $\textrm{Cl}(n-t,t)^+$.

Let $\mathcal {B=}\{v_p\}_1^n$ be the canonical basis of V such that $v_p=(x_1,\ldots ,x_n)$, where $x_p=1$ and $x_i=0$ for $i\ne p$. By the base group of the algebra $\textrm{Cl}(n-t,t)$ we mean its multiplicative subgroup $M_{n-t,t}$ generated by $v_1,\ldots ,v_n$.

Lemma 3.1

For $n>1$, the base group $M_{n-t,t}$ of the algebra $\textrm{Cl}(n-t,t)$ has order $2^{n+1}$.

Proof

In $\textrm{Cl}(n-t,t)$ an element $v_p$ satisfies the relation $v_p^2=Q_{t}(v_p)\cdot 1$ what implies that $v_p^2=1$ for $p=1,\ldots ,t$ and $v_p^2=-1$ for $p>t$. Moreover, for $p\ne r$, we have $v_pv_r+v_rv_p=2B_t(v_p,v_r)=0$, where $B_t$ is the bilinear form associated with $Q_t$. Thus $v_pv_rv_p^{-1}=-v_r$. The element $-1$ is central in the group $M_{n-t,t}$ and the quotient $M_{n-t,t}/\langle -1\rangle$ is an abelian group generated by n elements of order 2 what implies that $|M_{n-t,t}|=2^{n+1}$.$\square$

By the proof of Lemma 3.1 we get the relations in the group $M_{n-t,t}$ which will be used later in the paper.

Corollary 3.2

The generators $v_1,\ldots ,v_n$ of the group $M_{n-t,t}$ for $t=0,1$ satisfy the following relations:

$t=1:$ $v_1^2=1$, $v_p^4=1,\; v_pv_1v_p^{-1}=v_1v_p^2$ for $p>1,$

$v_q^2=v_p^2$ for $1<p,q\le n$ and $\;v_qv_pv_q^{-1}=v_p^{-1}$ for $p\ne q$,

$t=0:$ $v_p^4=1$, $v_p^2=v_q^2$ for $1\le p,q\le n$ and $v_qv_pv_q^{-1}=v_p^{-1}$ for $p\ne q$.

3.2 Clifford actions defining Klein surfaces

An action of the base group $M_{n-t,t}$ on a Riemann surface X of genus $g\ge 2$ will be called a (n-t,t,g)-Clifford action. This action is full if $M_{n-t,t}$ is the group of all automorphisms of X. We restrict our attention to Clifford actions for which the orbit space $X/M_{n-t,t}$ is isomorphic to a Klein surface Y of algebraic genus $d>1$. In this case we will say that Y is definable by the $(n-t,t,g)$-Clifford action.

Let $M_{n-t,t}^+=\textrm{Cl}(n-t,t)^+\bigcap M_{n-t,t}$. The orbit space $Y'=X/M_{n-t,t}^+$ is a double cover of Y which will be called the Clifford double cover defined by the (n-t,t,g)-Clifford action.

Theorem 3.3

Every Klein surface Y of algebraic genus $d\ge 2$ is definable by a $(n-t,t,g)$-Clifford action for $g=1+2^{d+1}(d-1)$, $n=d+1$ and $t=0,1$. If Y is a proper Klein surface except a sphere with three boundary components, then the Clifford double cover defined by this Clifford action is isomorphic to the canonical double cover of Y and in the exceptional case this is true only for $t=0$. Moreover, for any Klein surface of genus $d>3$, there exists a full $(n-t,t,g)$-Clifford action defining a Klein surface homeomorphic to Y.

Proof

Assume that Y is a Klein surface of algebraic genus $d\ge 2$ with k boundary components. Then Y is isomorphic to the orbit space of the hyperbolic plane ${\mathcal {H}}$ under the action of a surface NEC group $\Lambda$ with a signature $\sigma$ given by (10) in which the sign is $+$ or − according to whether Y is orientable or not, respectively. To simplify the notation we will write $"{ sign}(\sigma )=+"$ or $"{ sign}(\sigma )=-"$. In the first case $\Lambda$ is generated by hyperbolic elements $a_1,b_1,\ldots ,a_\gamma ,b_\gamma$ and reflections $c_{1,0},\ldots ,c_{k,0}$ and connecting generators $e_{1},\ldots ,e_k$ which satisfy the relations $[e_{i},c_{i,0}]=1$ for $i=1,\ldots ,k$ and

$$\begin{aligned}{}[a_1,b_1]\cdot \cdot \cdot [a_\gamma ,b_\gamma ]e_{1}\cdot \cdot \cdot e_k=1. \end{aligned}$$

In the second case there are generating glide reflections $d_1,\ldots ,d_\gamma$ instead of hyperbolic generators and the long relation has the form

$$\begin{aligned} d_1^2\cdot \cdot \cdot d_\gamma ^2e_1\cdot \cdot \cdot e_k=1. \end{aligned}$$

Let $M_{n-t,t}$ be the base group of the algebra $\textrm{Cl}_{n-t,t}$ for $t=0,1$ and $n=d+1=\alpha \gamma +k$, where $\alpha =2$ or $\alpha =1$ according to whether $"{ sign}(\sigma )=+"$ or $"{ sign}(\sigma )=-"$, respectively. In order to find an action of the group $M_{n-t,t}$ on a Riemann surface for which the orbit space is isomorphic to Y we need to find a smooth epimorphism $\theta :\Lambda \rightarrow M_{n-t,t}$ for which $\Gamma =\textrm{ker}\theta$ is a torsion free Fuchsian group. Then $M_{n-t,t}\simeq \Lambda /\Gamma$ is an automorphism group of the Riemann surface $X={\mathcal {H}}/\Gamma$. By the Hurwitz–Riemann formula, X has genus $g=1+2^{d+1}(d-1)$ and

$$\begin{aligned} X/M_{n-t,t}\simeq ({\mathcal {H}}/\Gamma )/(\Lambda /\Gamma )\simeq {\mathcal {H}}/\Lambda \simeq Y. \end{aligned}$$

The preimage $\Lambda '=\theta ^{-1}(M_{n-t,t}^+)$ is a subgroup of $\Lambda$ with index 2 and the orbit space $X/M_{n-t,t}^+\simeq {\mathcal {H}}/\Lambda '$ is a double cover of Y.

Using the relations listed in Corollary 3.2, we will define an epimorphism $\theta :\Lambda \rightarrow M_{n-t,t}$ for which the long relation is preserved and all generating reflections of $\Lambda$ are mapped to elements of order 2 and none product of generators of the group $\Lambda$ containing an odd number of anti-conformal elements is mapped to 1. These conditions guarantee that kernel of $\theta$ is a surface Fuchsian group.

We start with the case when Y is a Riemann surface uniformized by a surface Fuchsian group $\Lambda$ with a signature $\sigma (\Lambda )=(\gamma ;+;[-];\{-\})$ for some $\gamma \ge 2$. For the base group $M_{n-t,t}$ with $n=2\gamma$, let $\theta :\Lambda \rightarrow M_{n-t,t}$ be induced by:

$$\begin{aligned} \theta (a_i)=v_nv_{2i-1},\;\;\theta (b_i)=v_nv_{2i}\;\textrm{for}\;1\le i\le \gamma -1\end{aligned}$$

(13)

and $\theta (a_\gamma )=v_{n-1},\theta (b_{\gamma })=v_n$ if $\gamma$ is even or $\theta (a_{\gamma })=v_{n-2}v_{n-1}$ and $\theta (b_{\gamma })=v_n$ if $\gamma$ is odd. Thanks the relations listed in Corollary 3.2 we have $\prod _{i=1}^\gamma [\theta (a_i),\theta (b_i)]=1$. So $\Gamma =\mathrm{ker \theta }$ is a surface Fuchsian group and the Riemann surface $X={\mathcal {H}} /\Gamma$ has an automorphism group $\Lambda /\Gamma =M_{n-t,t}$ such that $X/M_{n-t,t}\simeq {\mathcal {H}}/\Lambda \simeq Y$.

The pre-image $\Lambda '=\theta ^{-1}(M_{n-t,t}^+)$ is a surface Fuchsian group which by the Hurwitz–Riemann formula has signature $(2\gamma -1;+;[-];\{-\})$. We can choose elements $A_1,B_1,\ldots A_{2\gamma -1},B_{2\gamma -1}$ as the canonical generators of $\Lambda '$, where for even $\gamma$:

$$\begin{aligned} \begin{array}{lll}A_i=a_i, &{} B_i=b_i &{} \textrm{for}\; 1\le i\le \gamma -1\\ A_{\gamma -1+i}=A_{2\gamma -1} b_{\gamma -i}A_{2\gamma -1}^{-1},&{} B_{\gamma -1+i}=A_{2\gamma -1} a_{\gamma -i}A_{2\gamma -1}^{-1} &{} \textrm{for}\; 1\le i\le \gamma -1\\ A_{2\gamma -1}=a_\gamma b_\gamma ,&{} B_{2\gamma -1}=b_\gamma a_\gamma . &{}\\ \end{array} \end{aligned}$$

and for odd $\gamma$:

$$\begin{aligned} \begin{array}{lll}A_i=b_\gamma a_i b_\gamma ^{-1}, &{} B_i=b_\gamma b_i b_\gamma ^{-1} &{} \textrm{for}\; 1\le i\le \gamma -1\\ A_{\gamma -1+i}=a_i,&{} B_{\gamma -1+i}=b_i &{} \textrm{for}\; 1\le i\le \gamma -1\\ A_{2\gamma -1}=a_{\gamma },&{} B_{2\gamma -1}=b_{\gamma }^2. &{}\\ \end{array} \end{aligned}$$

We leave to a reader checking that $\prod _{i=1}^{2\gamma -1}[A_i,B_i]=1$ and that $\theta$-images of $A_1,B_1,\ldots A_{2\gamma -1},B_{2\gamma -1}$ generate the group $M_{n-t,t}^+$. The orbit space $Y'=X/M_{n-t,t}^+$ is isomorphic to Riemann surface ${\mathcal {H}}/\Lambda '$. The quotient group ${\mathbb {Z}}_2\simeq \Lambda /\Lambda '$ acts in natural way on $Y'$ and the orbit space is isomorphic to Y.

Next, we assume that $\Lambda$ is a proper NEC group with a signature (10) and $\Lambda ^+$ is its canonical Fuchsian subgroup consisting of all conformal elements in $\Lambda$. If there exists a smooth epimorphism $\theta :\Lambda \rightarrow M_{n-t,t}$ which maps all conformal generators of $\Lambda$ to elements of the group $M_{n-t,t}^+$ and maps all anti-conformal generators to elements outside $M_{n-t,t}^+$, then $\theta (\Lambda ^+)\subseteq M_{n-t,t}^+$ because any element of $\Lambda ^+$ is a product of the canonical generators of $\Lambda$ containing an even number of anti-conformal elements. If additionally, the generators $v_1,\ldots ,v_n$ of $M_{n-t,t}$ are $\theta$-images of anti-conformal elements of the group $\Lambda$ then they are themselves anti-conformal and since any element of $M_{n-t,t}^+$ is a product of even number of these generators, it follows that $M_{n-t,t}^+\subseteq \textrm{Aut}^+(X)=\theta (\Lambda ^+)$. Consequently, $M_{n-t,t}^+=\theta (\Lambda ^+)\simeq \Lambda ^+/\Gamma$ for $\Gamma =\mathrm{ker \theta }$ which means that the group $M_{n-t,t}^+$ acts on the Riemann space $X\simeq {\mathcal {H}}/\Gamma$ and the orbit space is isomorphic to the canonical double cover $Y^+={\mathcal {H}}/\Lambda ^+$ of $Y={\mathcal {H}}/\Lambda$. So in order to prove that Y is definable by a $(n-t,t,g)$-Clifford action for which the Clifford cover is the canonical double cover of Y, we need to define a smooth epimorphism $\theta :\Lambda \rightarrow M_{n-t,t}$ such that the image $\theta (\lambda )$ of any generator $\lambda$ of $\Lambda$ belongs to $M_{n-t,t}^+$ if and only if $\lambda$ is conformal, and all generators $v_p$ of $M_{n-t,t}$ are $\theta$-images of anti-conformal elements of $\Lambda$. Using the relations given in Corollary 3.2, it is easy to check that $\theta$ defined below is a smooth epimorphism which satisfies the above conditions. The definition is divided into a few cases depending on parameters $\gamma$, k and $\varepsilon$, where $\varepsilon =1$ is $\gamma$ is odd and $\varepsilon =0$ if $\gamma$ is even.

A smooth epimorphism $\theta :\Lambda \rightarrow M_{n,0}$ can be defined as follows:

• (1) $\textrm{sign}(\sigma )=-$, $\gamma =1$ and $k\ge 2$ $\theta (d_1)=v_1, \theta (e_1)=\theta (e_2)=v_1v_{k+1}, \theta (e_j)=1\;\textrm{for}\;3\le j\le k$ and $\theta (c_{i0})=v_1v_{k+1}v_{i+1}\;\textrm{for}\;i=1,\ldots ,k-1,\;\theta (c_{k0})=\theta (c_{k-10}).$

• (2) $\textrm{sign}(\sigma )=-$, $\gamma >1$, $\gamma +k\ge 3$ $\theta (c_{j0})=v_1v_2v_{\gamma +j}$ for $1\le j\le k$, $\theta (d_i)=v_i\;\textrm{for}\;1\le i\le \gamma -1\;\textrm{and}\;\theta (e_j)=1\;\textrm{for}\;2\le j\le k.$ If $k>0$, then $\theta (d_\gamma )=v_\gamma$ and $\theta (e_1)=v_\gamma ^{2\varepsilon }$. If $k=0$, then $\theta (d_\gamma )=(v_1v_2)^\varepsilon v_\gamma$.

• (3) $\textrm{sign}(\sigma )=+$, $\gamma >0$, $k>0$ and $2\gamma +k\ge 3$ $\theta (a_i)=v_{2i-1}v_{2\gamma +k},\;\theta (b_i)=v_{2i}v_{2\gamma +k},\;\textrm{for }\;1\le i\le \gamma$, $\theta (c_{j0})=v_1v_2v_{2\gamma +j}\;\textrm{for}\;j=1,\ldots ,k,\;\theta (e_1)=v_1^{2\varepsilon }\;\textrm{and}\;e_j=1\;\textrm{for}\;j>1.$

• (4) $\textrm{sign}(\sigma )=+$, $\gamma =0$ and $k\ge 3$ $\theta (e_1)=v_1v_k, \theta (e_2)=v_2v_k,\theta (e_3)=v_2v_1\;\textrm{and }\;\theta (e_j)=1\;\textrm{for}\;j>3,$ $\theta (c_{k-10})=\theta (c_{k0})=v_1v_2v_k,\;\textrm{and }\;\theta (c_{j0})=v_{1}v_2v_{j+2}\;\textrm{for}\;j=1,\ldots ,k-2.$

Next, we define a smooth epimorphism $\theta :\Lambda \rightarrow M_{n-1,1}$ as follows:

• (1) $\gamma >0$, $k>0$ and $\alpha \gamma +k\ge 3$ $\theta (c_{i0})=v_1\;\textrm{for }\;1\le i\le k$, $\theta (e_i)=v_{k+1}v_{i+1}\;\textrm{for }\;\;1\le i< k$, $\theta (e_k)=(\sqcap _{i=1}^{k-1}v_{k+1}v_{i+1})^{-1}v_{k+1}^{2\varepsilon }$, if $\textrm{sign}=-$, then $\theta (d_j)=v_{k+j}\;\textrm{for }\;1\le j\le \gamma$, if $\textrm{sign}=+$, then $\theta (a_i)=v_{k+2i-1}v_{1}$, $\theta (b_i)=v_{k+2i}v_{1}$ for $1\le i\le \gamma$,

• (2) $\gamma =0$ and $k\ge 3$: if $k=3$ then $\theta (c_{10})=v_1$, $\theta (c_{20})=v_1v_2$, $\theta (c_{30})=v_1v_3$ and $\theta (e_1)=\theta (e_2)=\theta (e_3)=1$( the exceptional case in which Y is not double definable), if $k>3$ then $\theta (c_{10})=v_2v_3v_4$, $\theta (c_{i0})=v_1$ for $2\le i\le k$, $\theta (e_1)=v_3v_4$, $\theta (e_i)=v_2v_{i+1}$ for $2\le i\le k-1$, $\theta (e_k)=(\theta (e_1)\cdot \cdot \cdot \theta ( e_{k-1}))^{-1}$.

• (3) $\gamma \ge 3$, $k=0$ and $\textrm{sign}(\sigma )=-$ $\theta (d_j)=v_{j}$ for $j=1,\ldots ,\gamma -1$ and $\theta (d_\gamma )=(v_2v_3)^{1-\varepsilon } v_{\gamma }$,

By browsing the lists of non-maximal NEC signatures we check that the signature (10) is non-maximal only in few cases listed in Table below.

non-maximal surface signatures	d
$(2;+;[-];\{-\})$	3
$(1;+;[-];\{(-)\})$	2
$(0;+;[-];\{(-),(-),(-)\})$	2
$(1;-;[-];\{(-),(-)\})$	2
$(2;-;[-];\{(-)\})$	2
$(3;-;[-];\{-\})$	2

For $d>3$, the signature (10) is maximal. There exists a maximal NEC group with any given maximal signature which is not contained properly in any other NEC group. Let $\Lambda '$ be a maximal NEC group with a signature (10) for $d>3$. By Theorem 2.1, NEC groups with the same signatures are isomorphic. Thus there is an isomorphism $\tau :\Lambda '\rightarrow \Lambda$. Composing $\tau$ with $\theta$ we get an epimorphism $\theta ':\Lambda '\rightarrow M_{n-t,t}$ with kernel $\Gamma$ which defines a full action $(\Lambda ',\theta ',M_{n-t,t})$ on the Riemann surface $X={\mathcal {H}}/\Gamma$. It means that $M_{n-t,t}$ is the group of all automorphisms of X. Otherwise, there would be an NEC group $\Lambda ''$ containing $\Lambda '$ as a proper subgroup, against the assumption that $\Lambda '$ is maximal.

Corollary 3.4

For a given integer $n\ge 3$, let $g=1+2^{n}(n-2)$ and let $n_{(2)}\in \{0,1\}$ such that $n_{(2)}\equiv n\;\textrm{mod}\; 2$. Then for any $t\in \{0,1\}$ there are at least n $(n-t,t,g)$-Clifford actions defining non-homeomorphic non-orientable Klein surfaces of algebraic genus $n-1$ and there are at least $1+\frac{n-n_{(2)}}{2}$ such actions defining non-homeomorphic orientable Klein surfaces of algebraic genus $n-1$.

Proof

Let $(\gamma ,k,\alpha )$ be a triple of nonnegative integers such that $n=\alpha \gamma +k$, $\alpha \in \{1,2\}$, and $\gamma \ne 0$ if $\alpha =1$. For $\alpha =1$, there are n such triples because $\gamma$ can be any integer in the range $1\le \gamma \le n$ and k is uniquely determined for a fixed $\gamma$. For $\alpha =2$, $\gamma$ can be any integer in the range $0\le \gamma \le \frac{n-n_{(2)}}{2}$ and we get $1+\frac{n-n_{(2)}}{2}$ different triples. Let $\sigma$ be a signature (10) corresponding to a given triple $(\gamma ,k,\alpha )$, where $\textrm{sign}(\sigma )=-$ for $\alpha =1$ and $\textrm{sign}(\sigma )=+$ for $\alpha =2$. There exists an NEC group $\Lambda$ with the signature $\sigma$ and the orbit space $Y={\mathcal {H}} /\Lambda$ is a Klein surface of algebraic genus $d=n-1\ge 2$ which is non-orientable or orientable according to whether $\alpha =1$ or $\alpha =2$, respectively. According to Theorem 3.3, the surface Y is definable by a $(n-t,t,g)$-Clifford action for $g=1+2^{n}(n-2)$ and $t\in \{1,0\}$. $\square$

4 Linear representations of surface NEC groups

In the previous chapter we proved that every Klein surface Y of algebraic genus $d\ge 2$ is the orbit space of a Riemann surface under the action of a base group of some Clifford algebra. Using the spinor representation of this algebra, described in section 2.3, we will get a linear representation of the fundamental group of Y or linear representation of the fundamental group of the double Clifford cover of Y depending on whether d is odd or even. For this purpose we need Clifford algebras under an algebraically closed field. Therefore in this chapter we assume that $\textrm{Cl}(n-t,t)$ for $t=1,0$ is the Clifford algebra associated with a complex vector space $V={\mathbb {C}}^n$ and a quadratic form $Q_t$ defined by (12) and $M_{n-t,t}$ is the base group of $\textrm{Cl}(n-t,t)$ generated by images of vectors of the canonical basis $\{v_i\}_{i=1}^n$ of V. Vectors of this basis and their images in $\textrm{Cl}(n-t,t)$ will be denoted with the same symbols. By $\sigma _1,\sigma _2$ and $\sigma _3$ we denote the following Pauli matrices:

$$\begin{aligned} \sigma _1=\left[ \begin{array}{cc} 0&{} \quad 1\\ 1&{}\quad 0\\ \end{array}\right] ,\; \sigma _2=\left[ \begin{array}{cc} 0&{} \quad -i\\ i&{}\quad 0\\ \end{array}\right] \;\textrm{and}\;\sigma _3=\left[ \begin{array}{cc} 1&{} \quad 0\\ 0&{}\quad -1\\ \end{array}\right] . \end{aligned}$$

Theorem 4.1

Let $\eta _m:\textrm{Cl}(2m-1,1)\rightarrow \textrm{End}(Z)$ be the spinor representation of the Clifford algebra $\textrm{Cl}(2m-1,1)$ for a vector space Z of dimension $2^m$. Then in some basis of Z the endomorphism $\eta _m(v_p)$ is represented by a matrix $A_{m,p}$, where $A_{1,1}=\sigma _1,\;A_{1,2}=-i\sigma _2$ and for $m\ge 2$ the matrix $A_{m,p}$ is defined as follows:

$$\begin{aligned} A_{m,p}=\left[ \begin{array}{cc} A_{m-1,p}&{} \quad 0\\ 0&{}\quad A_{m-1,p}\\ \end{array}\right] ,\;\;p=1,\ldots ,m-1,\end{aligned}$$

(14)

$$\begin{aligned} A_{m,p}=\left[ \begin{array}{cc} A_{m-1,p-2}&{} \quad 0\\ 0&{}\quad A_{m-1,p-2}\\ \end{array}\right] ,\;\;p=m+2,\ldots ,2m,\end{aligned}$$

(15)

$$\begin{aligned} A_{m,m}=\left[ \begin{array}{cc} 0&{} \quad -D_m\\ D_m&{}\quad 0\\ \end{array}\right] \;\textrm{and}\; A_{m,m+1}=\left[ \begin{array}{cc} 0&{} \quad D_m(-i)\\ D_m(-i)&{}\quad 0\\ \end{array}\right] ,\end{aligned}$$

(16)

with

$$\begin{aligned} D_2=\sigma _3\;\textrm{and }\;D_{m}=\left[ \begin{array}{cc} D_{m-1}&{} \quad 0\\ 0&{}\quad -D_{m-1}\\ \end{array}\right] \;\textrm{for}\;m>2. \end{aligned}$$

Proof

Let $\{v_p\}_{p=1}^{n}$ be the canonical basis of a vector space $V={\mathbb {C}}^{n}$ for $n=2\,m$. Then the maximal totally isotropic subspaces W and U of dimension m in Witt decomposition $V=W\oplus U$ can be spanned by sets $\{w_p\}_1^m$ and $\{u_p\}_1^m$, respectively, where

$$\begin{aligned}{} & {} w_1=\frac{1}{2}(v_1-v_n),\;\;u_1=\frac{1}{2}(v_1+v_n), \\{} & {} w_p=\frac{1}{2}(iv_{n+1-p}-v_p)\;\textrm{and }\;u_p=\frac{1}{2}(iv_{n+1-p}+v_p)\;\textrm{for }\;2\le p\le m. \end{aligned}$$

The images of these vectors in $\textrm{Cl}(n-1,1)$ satisfy the relations:

$$\begin{aligned} w_p^2=0,u_p^2=0\;\textrm{and}\;w_pu_j+u_jw_p=\delta _{p,j}\;\textrm{for}\;\;\;p,j=1,\ldots ,m.\end{aligned}$$

(17)

For $f=w_1\cdot \cdot \cdot w_m$, the subalgebra $\textrm{Cl}(n-1,1)f=\{af:a\in \textrm{Cl}(n-1,1)\}$ of $\textrm{Cl}(n-1,1)$ is generated by

$$\begin{aligned} {\mathcal {B}}=\{f\}\cup \{u_{j_1}\cdot \cdot \cdot u_{j_k}f: 1\le j_1< \ldots < j_k\le m\}. \end{aligned}$$

The external algebra $Z=\bigwedge U$ of dimension $2^m$ with basis

$$\begin{aligned} \{1\}\cup \{u_{j_1}\wedge \ldots \wedge u_{j_k}: 1\le j_1< \ldots < j_k\le m\}\end{aligned}$$

(18)

is isomorphic to the Clifford algebra associated with vector space U and the zero quadratic form $Q_1|_U$. Let $\varphi :Z\rightarrow \textrm{Cl}(n-1,1)f$ be given by

$$\begin{aligned} \varphi (1)=f,\;\varphi (u_{j_1}\wedge u_{j_2}\wedge \ldots \wedge u_{j_k})=u_{j_1} u_{j_2}\cdot \cdot \cdot u_{j_k}f. \end{aligned}$$

The spinor representation $\eta _m:\textrm{Cl}(n-1,1)\rightarrow \textrm{End}(Z)$ of the algebra $\textrm{Cl}(n-1,1)$ is defined by

$$\begin{aligned} \eta _m(a)(u)= \varphi ^{-1}(a\varphi (u))\;\textrm{for}\;a\in \textrm{Cl}(n-1,1)\;\textrm{and}\;u\in Z. \end{aligned}$$

For an ordered subset $S=\{z_1,\ldots ,z_k\}\subset Z$ and $z\in Z$, let $Z\wedge z$ denote an ordered set $\{z_1\wedge z,\ldots ,z_k\wedge z\}$. There is a sequence

$$\begin{aligned} {\mathcal {B}}_1\subset {\mathcal {B}}_2\subset \ldots \subset {\mathcal {B}}_m \end{aligned}$$

of ordered subsets of $\bigwedge U$ such that ${\mathcal {B}}_1=\{1,u_1\}$ and ${\mathcal {B}}_{j+1}={\mathcal {B}}_j\cup ({\mathcal {B}}_j\wedge u_{j+1})$ for $j=1,\ldots ,m-1$. So ${\mathcal {B}}_2=\{1,u_1,u_2,u_1\wedge u_2\}$, ${\mathcal {B}}_3=\{1,u_1,u_2,u_1\wedge u_2,u_3,u_1\wedge u_3,u_2\wedge u_3,u_1\wedge u_2\wedge u_3\}$ and so on. In particular, the set ${\mathcal {B}}_m$ is the basis of Z given by (18). Let $A_{m,p}\in M_{2^m\times 2^m}({\mathbb {C}})$ denote the matrix of $\eta _m(v_p)$ in this basis.

For $m=1$, the canonical basis of V consists of two vectors

$$\begin{aligned} v_1=u_1+w_1\;\textrm{and }\;v_2=u_1-w_1.\end{aligned}$$

(19)

Thus by the relations (17) we have

$$\begin{aligned} \eta _1(v_1)(1)=\varphi ^{-1}((u_1+w_1)w_1)=u_1 \end{aligned}$$

and

$$\begin{aligned} \eta _1(v_1)(u_1)=\varphi ^{-1}((u_1+w_1)u_1w_1)=\varphi ^{-1}(w_1u_1w_1)=\varphi ^{-1}((1-u_1w_1)w_1)=1 \end{aligned}$$

what implies that $A_{1,1}=\left[ \begin{array}{cc} 0&{} \quad 1\\ 1&{} \quad 0\\ \end{array}\right] =\sigma _1$. By similar calculation we get $A_{1,2}=-i\sigma _2$.

Now let $P=\{1,\ldots ,m-1\}$ and $R=\{m+2,\ldots n\}$ for $m>1$. Vectors $v_1$ and $v_2$ are given by (19) while for $p=2,\ldots ,m$ we have

$$\begin{aligned} v_p=(u_p-w_p),\;\;v_{n+1-p}=(u_p+w_p)(-i).\end{aligned}$$

(20)

Since for all $p\in P\cup R$ vector $v_p$ is a linear combination of vectors $u_i$ and $w_i$ with $1\le i\le m-1$, it follows that $\eta _m(v_p)(u)\in \textrm{Lin}({\mathcal {B}}_{m-1})$ and $\eta _m(v_p)(u\wedge u_{m})\in \textrm{Lin}({\mathcal {B}}_{m-1}\wedge u_m)$ for any $u\in {\mathcal {B}}_{m-1}$. It means that there are zero square matrices of dimension $2^{m-1}$ in the right upper corner and in the left lower corner of the matrix $M_{m,p}$ for all $p\in P\cup R$. Moreover, if $p\in P$, then $\eta _m(v_p)(u)=\eta _{m-1}(v_p)(u)$ and $\eta _m(v_p)(u\wedge u_{m})=(\eta _{m-1}(v_p)(u))\wedge u_m$ what implies that there is the matrix $A_{m-1,p}$ in the left upper corner and in the right lower corner of the matrix $A_{m,p}$. Thus the endomorphism $\eta _m(v_p)$ in basis ${\mathcal {B}}_m={\mathcal {B}}_{m-1}\cup ({\mathcal {B}}_{m-1}\wedge u_{m})$ has the matrix

$$\begin{aligned} A_{m,p}=\left[ \begin{array}{cc} A_{m-1,p}&{} \quad 0\\ 0&{} \quad A_{m-1,p}\\ \end{array}\right] \;\textrm{for }\;p\in P. \end{aligned}$$

Since $\eta _m(v_{n+1-j})(u)=\eta _{m-1}(v_{n-1-j})(u)$ for $j=1,\ldots ,m-1$ and $u\in {\mathcal {B}}_{m-1}$, it follows that $\eta _m(v_{p})$ has the matrix

$$\begin{aligned} A_{m,p}=\left[ \begin{array}{cc} A_{m-1,p-2}&{} \quad 0\\ 0&{} \quad A_{m-1,p-2}\\ \end{array}\right] \;\textrm{for }\;p\in R. \end{aligned}$$

It remains to determine the matrices $M_{m,m}$ and $M_{m,m+1}$. If $u=u_{i_1}\wedge \ldots \wedge u_{ik}\in {\mathcal {B}}_{m-1}$, then by the relations (17), we have $\eta _m(w_m)(u)=0$ and $\eta _m(u_m)(u\wedge u_m)=0$. Thus

$$\begin{aligned} \eta _m(v_m)(u)=\eta _m(u_m-w_m)(u)=\eta _m(u_m)(u)=(-1)^ku\wedge u_m\end{aligned}$$

(21)

and

$$\begin{aligned} \eta _m(v_m)(u\wedge u_m)=\eta (-w_m)(u\wedge u_m)=(-1)^{k+1}u\end{aligned}$$

(22)

what implies that $A_{m,m}=\left[ \begin{array}{cc} 0&{} \quad -D_m\\ D_m&{} \quad 0\\ \end{array}\right]$ for some diagonal matrix $D_m\in M_{2^{m-1}\times 2^{m-1}}({\mathbb {C}})$. Similarly, for $v_{m+1}=(u_m+w_m)(-i)$ we have

$$\begin{aligned} \eta _m(v_{m+1})(u)=(-1)^k(-i) ( u\wedge u_m)\;\textrm{and}\;\eta (v_{m+1})(u\wedge u_m)=(-1)^k(-i) u \end{aligned}$$

what implies that $A_{m,m+1}=\left[ \begin{array}{cc} 0&{} \quad D_m(-i)\\ D_m(-i)&{} \quad 0\\ \end{array}\right] .$ By simple calculation for $m=2$ we get

$$\begin{aligned} A_{m,m}=A_{2,2}=\left[ \begin{array}{cc} 0&{} \quad -\sigma _3\\ \sigma _3&{} \quad 0\\ \end{array}\right] \;\textrm{and}\; A_{m,m+1}=A_{2,3}=\left[ \begin{array}{cc} 0&{} \quad \sigma _3(-i)\\ \sigma _3(-i)&{} \quad 0\\ \end{array}\right] . \end{aligned}$$

For $m=3$ we have

$$\begin{aligned} A_{m,m}=\left[ \begin{array}{cccc} 0&{} \quad 0&{} \quad -\sigma _3&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad \sigma _3\\ \sigma _3&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad -\sigma _3&{} \quad 0&{} \quad 0\\ \end{array}\right] \;\textrm{and}\; A_{m,m+1}=\left[ \begin{array}{cccc} 0&{} \quad 0&{} \quad -\sigma _3i&{} \quad 0\\ 0&{} \quad 0&{} \quad 0&{} \quad \sigma _3i\\ -\sigma _3i&{} \quad 0&{} \quad 0&{} \quad 0\\ 0&{} \quad \sigma _3i&{} \quad 0&{} \quad 0\\ \end{array}\right] . \end{aligned}$$

These two examples help us to notice that

$$\begin{aligned} D_{m}=\left[ \begin{array}{cc} D_{m-1}&{} \quad 0\\ 0&{} \quad -D_{m-1}\\ \end{array}\right] \end{aligned}$$

for any $m>2$. This is a consequence of relations (21) and (22) and the fact that ${\mathcal {B}}_m={\mathcal {B}}_{m-1}\cup ({\mathcal {B}}_{m-1}\wedge z_m)$.

Theorem 4.2

Let $\mu _m:\textrm{Cl}(n,0)^+\rightarrow \textrm{End}(Z)$ be the spinor representation of the Clifford algebra $\textrm{Cl}(n,0)^+$ for $n=2m+1$. Then the homomorphisms $\mu _m(v_nv_p)$ with $1\le p\le 2\,m$ are represented in some basis of Z by $A_{1,1}=(-i)\sigma _2,\;A_{1,2}=(-i)\sigma _1\;\textrm{for}\;m=1$ and by matrices $A_{m,p}$ given by (14), (15) and (16) for $m>1$.

Proof

Let $\{v_i\}_{1}^n$ be the canonical basis of a vector space $V={\mathbb {C}}^n$ for $n=2\,m+1$. There exists Witt decomposition $V=W\oplus U\oplus \textrm{Lin}(v_n)$ for W and U generated by

$$\begin{aligned} \left\{ w_p=\frac{1}{2}(iv_{n-p}-v_p)\right\} _{p=1}^m\;\textrm{and}\;\left\{ u_p=\frac{1}{2}(iv_{n-p}+v_p)\right\} _{p=1}^m, \end{aligned}$$

respectively. Here W and U are two maximal totally isotropic subspaces of V and $v_n$ is a non-isotropic vector ortogonal to $V'=W\oplus U$ with respect to the bilinear form $B_0$ associated with $Q_0$. The subspace $V'$ is spanned by the set $\{v_i\}_1^{n-1}$ and it has a non-degenerate quadratic form $Q'$ defined by

$$\begin{aligned} Q'(v')=-Q_0(v_n)Q(v')\;\textrm{for}\;v'\in V'. \end{aligned}$$

Let us notice that $Q'=Q_0|_{V'}$ because $Q_0(v_n)=-1$. A linear map $f:V'\rightarrow \textrm{Cl}(n,0)^+$ given by $f(v')=v_nv'$ satisfies the condition $f(v')^2=Q'(v')\cdot 1$ for all $v'\in V'$. Thus by universality of the Clifford algebra $\textrm{Cl}(V ',Q ')$, there is an isomorphism ${\bar{f}}:\textrm{Cl}(V',Q')\rightarrow \textrm{Cl}(n,0)^+$ such that $\bar{f}\circ j=f$ for the canonical map $j:V'\rightarrow \textrm{Cl}(V',Q')$. It is induced by

$$\begin{aligned} {\bar{f}}(v_p)=v_nv_p\;\textrm{for}\;p=1,\ldots ,n-1. \end{aligned}$$

Let $\eta _m:\textrm{Cl}(V',Q')\rightarrow \textrm{End}(\bigwedge U)$ be the spinor representation of the algebra $\textrm{Cl}(V',Q')$. Then $\mu _m=\eta _m\circ {\bar{f}}^{-1}:\textrm{Cl}(V,Q)^+\rightarrow \textrm{End}(\bigwedge U)$ is an isomorphism such that

$$\begin{aligned} \mu _m(v_nv_p)=\eta _m(v_p)\;\textrm{for}\;p=1,\ldots ,n-1. \end{aligned}$$

In order to determine the matrices $A_{m,p}$ of endomorphisms $\mu _m(v_nv_p)$ in basis ${\mathcal {B}}_m$ of $\bigwedge U$ we can use formulas

$$\begin{aligned} v_p=u_p-w_p\;\textrm{and}\; v_{n-p}=(-i)(u_p+w_p)\;\textrm{for}\;p=1,\ldots ,m \end{aligned}$$

and the relations (17) which are satisfied in the algebra $\textrm{Cl}(V',Q')$ by vectors $w_p$ and $u_p$. By repeating the argumentation from the proof of Theorem 4.1 we get that matrices $A_{m,p}$ are defined for $m>1$ by (14), (15) and (16); and for $m=1$ we have $A_{1,1}=(-i)\sigma _2,\;\textrm{and}\;A_{1,2}=(-i)\sigma _1.$

Theorem 4.3

Let $\pi _Y$ be the fundamental group of a proper Klein surface Y of algebraic genus $d\ge 2$, $\pi _{Y^+}$-the fundamental group of the Riemann surface $Y^+$ being a double cover of Y and let $m=(d+d_{(2)})/2$ for $d_{(2)}\in \{0,1\}$ such that $d_{(2)}\equiv d\;\textrm{mod}\;2$. If d is odd, then there is a linear representation $\rho :\pi _Y\rightarrow \textrm{Gl}(2^m,{\mathbb {C}})$ with image generated by the matrices $A_{m,1}\ldots A_{m,2m}$ defined in Theorem 4.1. If d is even, then there is a linear representation $\rho :\pi _{Y^+}\rightarrow \textrm{Gl}(2^m,{\mathbb {C}})$ with image generated by the matrices $A_{m,1}\ldots A_{m,2m}$ defined in Theorem 4.2.

Proof

According to Theorem 3.3, any proper Klein surface Y of algebraic genus $d\ge 2$ is definable by a $(n-t,t,g)$-Clifford action $(\Lambda ,\theta , M_{n-t,t})$ for $n=d+1$, $g=1+ 2^{d+1}(d-1)$ and $t\in \{0,1\}$ and the Clifford cover defined by this action is isomorphic to the canonical double cover $Y^+$ of Y. Here $\Lambda$ is a surface NEC group isomorphic to the fundamental group of Y and $\theta :\Lambda \rightarrow M_{n-t,t}$ is a smooth epimorphism. By composing $\theta$ with the spinor representation of the algebra $\textrm{Cl}(n-t,t)$ for even n or with the spinor representation of the algebra $\textrm{Cl}(n-t,t)^+$ for odd n, we get linear representations of the fundamental groups of Y or $Y^+$, respectively.

Let $m=\frac{d+d_{(2)}}{2}$ and let $t=d\;\textrm{mod}\;2$. For an odd d, the spinor representation $\eta _m$ of the algebra $\textrm{Cl}_{d,1}$ associates with every generator $v_p$ of the group $M_{d,1}$ an isomorphism of a vector space Z of dimension $2^m$. By Theorem 4.1, there is a basis ${\mathcal {B}}$ of Z in which endomorphisms $\eta _m(v_p)$ are represented by matrices $A_{m,p}\in \textrm{Gl}(2^m,{\mathbb {C}})$, where $A_{1,1}=\sigma _1\;\textrm{and}\; A_{1,2}=-i\sigma _2$ for $m=1$ and $A_{m,p}$ are given by (14), (15) and (16) for $m>1$. Thus we get an epimorphism $\rho =\eta \circ \theta :\Lambda \rightarrow G\subset \textrm{Gl}(2^m,{\mathbb {C}})$ onto the group generated by matrices $A_{m,1},\ldots A_{m,2m}$.

If d is even, then there is a spinor representation $\eta _m:\textrm{Cl}_{n,0}^+\rightarrow \textrm{Gl}(2^m,{\mathbb {C}})$ such that the generators $v_1v_n,\ldots ,v_{n-1}v_n$ of the group $M_{n,0}^+$ are represented by matrices given in Theorem 4.2. Thus composing $\theta |_{\Lambda ^+}$ with $\eta _m$ we get an epimorphism $\rho :\Lambda ^+\rightarrow G$ onto the group generated by these matrices.

Corollary 4.4

For any odd $d\ge 3$ and $m=\frac{d+1}{2}$, there exist a Klein surface $Y\simeq {\mathcal {H}}/\Lambda$ of algebraic genus d and a linear representation $\rho :\Lambda \rightarrow \textrm{Gl}(2^{m},{\mathbb {C}})$ which maps bijectively canonical generators of a canonical presentation of $\Lambda$ to matrices $A_{m,1},\ldots$, $A_{m,2\,m}$ defined in Theorem 4.1.

Proof

Let $\Lambda$ be an NEC group with the signature $(\gamma ;-;[-];\{(-)\})$ for $\gamma =d\ge 3$. Then $Y={\mathcal {H}}/\Lambda$ is a Klein surface of algebraic genus d. Let $d_1,\ldots ,d_\gamma$ be generating glide reflections of $\Lambda$ and let $c_{10}$ and $e_1$ be generators of $\Lambda$ associated with the only period cycle. Then $e_1=(d_1^2\cdot \ldots \cdot d_\gamma ^2)^{-1}$, $c_{10}^2=1$ and $e_1c_{10}e_1^{-1}=c_{10}$. There is a homomorphism $\theta :\Lambda \rightarrow M_{d,1}$ induced by $\theta (d_i)=v_{1+i}$ for $i=1,\ldots ,\gamma$, $\theta (c_{10})=v_1$ and $\theta (e_1)=v_{d+1}^2$. The generator $e_i$ is redundant because it can be expressed by $d_1,\ldots ,d_\gamma$. So generators of $\Lambda$ correspond bijectively to generators $v_1,\ldots ,v_{d+1}$ which according to Theorem 4.1 are represented by matrices $M_{m,1},\ldots ,A_{m,d+1}$ by the spinor representation of $\textrm{Cl}_{d,1}$.

Corollary 4.5

For any even $d\ge 2$ and $m=\frac{d}{2}$, there exist a Klein surface $Y\simeq {\mathcal {H}}/\Lambda$ of algebraic genus d and a linear representation $\rho :\Lambda \rightarrow \textrm{Gl}(2^{m},{\mathbb {C}})$ which maps bijectively canonical conformal generators of a canonical presentation of $\Lambda$ to matrices $A_{m,1},\ldots$, $A_{m,2\,m}$ defined in Theorem 4.2.

Proof

An NEC group $\Lambda$ with the signature $(\gamma ;+;[-];\{(-)\})$ for $\gamma =\frac{d}{2}$ is generated by elements $a_1,b_1,\ldots ,a_\gamma ,b_\gamma$ and $c_{10}$ such that $c_{10}^2=1$ and $e_1c_{10}e_1^{-1}=c_{10}$ for $e_1=([a_1,b_1]\cdot \cdot \cdot [a_\gamma ,b_\gamma ])^{-1}$. There is a smooth epimorphism $\theta :\Lambda \rightarrow M_{n,0}$ for $n=d+1$ induced by $\theta (a_i)=v_{2i-1}v_n$, $\theta (b_n)=v_{2i}v_n$ for $i=1,\ldots ,\gamma$, $\theta (c_{10})=v_1v_2v_n$ and $\theta (e_1)=v_n^{\varepsilon }$, where $\varepsilon =\gamma \;\textrm{mod}\;(2)$. The generator $e_1$ is redundant and the other conformal generators of $\Lambda$ are mapped to generators $v_1v_n,\ldots ,v_{n-1}v_n$ of the group $M_{n,0}^+$ which according to Theorem 4.2 are represented by matrices $A_{m,1},\ldots ,A_{m,2\,m+1}$ for $m=\frac{d}{2}$ by the spinor representation of the algebra $\textrm{Cl}_{n,0}^+$