The Weyl group of type $A_{1}$ root systems extended by an abelian group¹¹1 The research for this article was supported by a postdoctoral fellowship at the Department of Mathematics and Statistics at Dalhousie University.

We consider a root system $R$ extended by an abelian group $G$, a notion that is introduced in [Yos04]. It generalizes the concepts of extended affine root systems (see [AAB⁺97], for instance) and affine root systems in the sense of [Sai85], both of which are generalizations of root systems of affine Kac-Moody algebras (see [MP95], for instance). The Weyl group $\mathcal{W}$ of $R$ is not necessarily a Coxeter group, so a more general presentation is needed to capture the algebraic structure of $\mathcal{W}$. The group $\mathcal{U}$ is given by the so-called presentation by conjugation with respect to $R$:

There is a natural group homomorphism from $\mathcal{U}$ onto $\mathcal{W}$.

In this note we investigate the case that $R$ is of type $A_{1}$, i.e. the underlying finite root system consists of two roots. This type of root system $R$ allows for less rigidity then other types and is therefor of special interest as a prototype. We prove the following result that allows an answer to the question above using an algorithmic approach.

Suppose $R$ is a type $A_{1}$ root system extended by a free abelian group $G$. Then a subset $T^{\mathrm{ab}}$ of $G_{2}=G/2G$ can be associated to it in a natural way. This subset is called 2-independent, if its image under $G_{2}\to G_{2}\otimes G_{2},g\mapsto g\otimes g$ is a linearly independent set.

This result provides an attractive alternative to a characterization proved in [AS08] using so-called integral collections. Our answer to the question above is more general than that in [AS08] as $G$ is not required to be finitely generated. We expect that the idea of 2-independence that we have introduced will play an important role in understanding the question for root systems of the types $B_{n}$ and $C_{n}$.

In this section we provide the basic terminology for the following sections. The notion of a discrete symmetric space is a special case of the symmetric spaces introduced in [Loo69]. The associated category of reflection groups is introduced in [Hof08] and more details can be found there.

For the remainder of this section, let $T$ be a discrete symmetric space.

Let the group $\mathcal{U}$ be given by the presentation

There is map $\hskip 1.42262pt\begin{picture}(1.0,5.0)(-0.5,-3.0)\circle*{2.0}\end{picture}\hskip 1.42262pt^{\mathcal{U}}:~T\to\mathcal{U},~t\mapsto t^{\mathcal{U}}$ associated to the presentation. An action of $\mathcal{U}$ on $T$ can be defined satisfying $t^{\mathcal{U}}.s=t{\cdot}s$ for all $s$ and $t\in T$. With respect to this action the pair $(\mathcal{U},\hskip 1.42262pt\begin{picture}(1.0,5.0)(-0.5,-3.0)\circle*{2.0}\end{picture}\hskip 1.42262pt^{\mathcal{U}})$ is a $T$-reflection group. There is a unique $T$-morphism from $\mathcal{U}$ into any other $T$-reflection group.

In this section we introduce the concept of a type $A_{1}$ root system extended by an abelian group $G$ in an ad hoc manner. Thus we avoid presenting the details of the definition for more general types.

Let $(G,+)$ be an abelian group. Define the multiplication

Now let $T$ be a generating subset of $G$ such that $0\in T$ and $G{\cdot}T\subseteq T$. It is straightforward to verify that $T$ with the restriction of the multiplication above is a discrete symmetric space. The set $R:=T\times\{1,-1\}$ is a type $A_{1}$ root system extended by the abelian group $G$ in the sense of [Yos04] or [Hof08].

Consider the two-element group $\mathcal{V}:=\{1,-1\}$ with its action on $G$ characterized by $-1g=-g$ for all $g\in G$. Set $\mathcal{A}:=G\rtimes\mathcal{V}$. Then $\mathcal{A}$ acts on $T$ via

The map

turns $\mathcal{A}$ into a $T$-reflection group.

In general, if $B$ is a group, $A$ is an abelian group, and $f:B\times B\to A$ is a cocycle, then the set $A\times B$ with the multiplication given by

defines a group denoted by $A\times_{f}B$ which is a central extension of $B$.

The set $(G\wedge G)\times G\times\mathcal{V}$ with the multiplication

is a group. We denote it by $(G\wedge G)\times_{\wedge}G\rtimes\mathcal{V}$. It can equally be interpreted as the semidirect product of the Heisenberg group $(G\wedge G)\times_{\wedge}G$ with $\mathcal{V}$ or a central extension of $\mathcal{A}$ by $G\wedge G$ with cocycle $f:~\mathcal{A}^{2}\to G\wedge G,~\big{(}(g,v),(g^{\prime},v^{\prime})\big{)}\mapsto g\wedge(vg^{\prime}).$

Set

Let $\mathcal{W}$ be the subgroup of $(G\wedge G)\times_{\wedge}G\rtimes\mathcal{V}$ generated by $T^{\mathcal{W}}$. Then $(\mathcal{W},\hskip 1.42262pt\begin{picture}(1.0,5.0)(-0.5,-3.0)\circle*{2.0}\end{picture}\hskip 1.42262pt^{\mathcal{W}})$ is a $T$-reflection group with the action of $\mathcal{W}$ on $T$ induced by the action of $\mathcal{A}$ on $T$.

This definition of the Weyl group coincides with the definition of Weyl groups given in [Hof08] if $G$ is free abelian and the one given in [Aza99] if $G$ is finitely generated free abelian.

In this section we investigate the case where $G$ is an elementary abelian 2-group. So we may think of $G$ as a vector space over the Galois field $\mathbb{F}_{2}$ with two elements. From (1) it immediately follows that $T$ has the trivial multiplication.

Denote by $G\otimes_{\mathrm{sym}}G$ the subgroup of $G\otimes G$ generated by the elements of the set $\{v\otimes v~|~v\in G\}.$ The group homomorphism

factors through $G\wedge G$ giving a group homomorphism

If $B$ is an ordered basis of $G$ then $\{b_{1}\wedge b_{2}~|~b_{1}<b_{2}\in B\}$ is a basis of $G\wedge G$. Its image under $\pi$ is linearly independent, so $\pi$ is injective.

Define the map

In this section the action of $\mathcal{V}$ on $G$ is trivial, so the reflection group $\mathcal{A}$ is given by the direct product $\mathcal{A}=G\times\mathcal{V}$. The Weyl group $\mathcal{W}$ is given as the subgroup of $\big{(}(G\wedge G)\times_{\wedge}G\big{)}\times\mathcal{V}$ generated by the image of

Due to the preceding theorem, the Weyl group can also be given as the subgroup of $\big{(}G\otimes_{\mathrm{sym}}G)\times\mathcal{V}$ generated by the image of

Let $F:=F(T\setminus\{0\})$ be the free vector space on the set $T\setminus\{0\}$ with the embedding $\iota:T\setminus\{0\}\to F$. The initial reflection group is given by $\mathcal{U}=F\times\mathcal{V}$ with the map

Denote the reflection morphism $\mathcal{U}\to\mathcal{W}$ above by $\varphi$. Then Example 4.3 yields

In this section let $G$ be a free abelian group. We will reduce the situation to that of the former section. More details can be found in [Hof08] Section 2, in particular in Construction 2.10.

Let $\mathcal{U}$ be the initial $T$-reflection group and let $\mathcal{W}$ be the Weyl group. The abelianizations $\mathcal{U}^{\mathrm{ab}}$ and $\mathcal{W}^{\mathrm{ab}}$ are $T^{\mathrm{ab}}$-reflection groups, where $T^{\mathrm{ab}}$ is the image of $T$ under the quotient homomorphism $G\to G_{2}=G/2G$. This is a discrete symmetric space with the trivial multiplication. More precisely $\mathcal{U}^{\mathrm{ab}}$ is the initial $T^{\mathrm{ab}}$-reflection group and $\mathcal{W}^{\mathrm{ab}}=\big{(}G_{2}\wedge G_{2}\big{)}\times_{\wedge}G_{2}\times\mathcal{V}$ is the Weyl group for the discrete symmetric space $T^{\mathrm{ab}}$.

The $T$-reflection morphism $\mathcal{U}\to\mathcal{W}$ yields a $T^{\mathrm{ab}}$-morphism $\mathcal{U}^{\mathrm{ab}}\to\mathcal{W}^{\mathrm{ab}}$ and there is a group homomorphism $\psi$ making the following diagram commute:

According to [Hof08] Theorem 4.16 the map $\psi$ is an isomorphism. With Theorem 4.4 we have obtained the main result of this article:

Corollary 4.5 gives more information in some specific cases. In particular, it confirms the observation made in [Hof07] and [AS08] that $\mathcal{U}\to\mathcal{W}$ is not always injective. If $n$ is the rank of $G$ then testing for 2-dependence involves testing for linear dependence of $|T\setminus\{0\}|$ vectors in an $\frac{n(n+1)}{2}$-dimensional vector space over the Galois field $\mathbb{F}_{2}$. This is more practical than testing for the existence of a so-called non-trivial integral collection according to [AS08] Theorem 5.16. This theorem also requires $G$ to be finitely generated, a hypothesis that we don’t require for our Theorem 5.1.

The hypotheses “free” for $G$ is only used to apply Theorem 4.16 of [Hof08]. We would be interested in understanding if it could be weakened to “torsion free”, “involution free” or even omitted completely.