Structure of coincidence isometry groups

: Let L be a lattice of rank n in an n - dimensional Euclidean space. We show that the coincidence isometry group of L is generated by coincidence re ﬂ ections if and only if L contains an orthogonal subset of order n .


Introduction
The theory of coincidence site lattice (CSL) gives partial answers to some questions that appeared in the physics of interfaces and grain boundaries, see [1,2]. The CSL theory mainly studies the coincidence problem between two lattices in a finite-dimensional Euclidean space. Several mathematical approaches were used in the study of this problem, including matrix theory, number theory, and geometric algebra, see [3][4][5][6][7][8][9][10][11][12]. In this paper, we focus on the structure of the coincidence isometry group of a lattice in a finitedimensional Euclidean space.
Let V be an n-dimensional Euclidean space and let … α α , , n L is a sublattice of L with finite index. The coincidence isometry group of L was introduced by Baake [1], which consists of all coincidence isometries of L. It is clear that an isometry is a coincidence isometry of L if and only if the matrix of with respect to … α α , , n 1 is a rational matrix. It follows that the set of all coincidence isometries of L is a group which we denote by ( ) OC L . Aragón et al. [3] and Rodríguez et al. [13] used geometric algebra as a tool in the study of the coincidence isometry group. In particular, the coincidence problem was completely solved in the planar case. They also found that coincidence reflections play an important role and conjectured that if = L n is the lattice spanned by the canonical basis in n , then any coincidence isometry is a product of coincidence reflections. Zou in [14] showed that if the reflection defined by an arbitrary nonzero vector of L is a coincidence isometry of L, then any coincidence isometry of L is a product of coincidence reflections defined by the vectors of L. This result includes the conjecture as a special case and an algorithm to decompose a coincidence isometry into coincidence reflections was obtained. Zou also gave example to show that this result is not valid for any lattice. Huck [15] generalized this result to free modules over some subrings of .
In this paper, we show that for any lattice L of rank n in n , any coincidence isometry of L is a product of coincidence reflections defined by the vectors of L if and only if L contains an orthogonal subset of order n.
In Section 2, we recall some relevant definitions and known results. In Section 3, we obtain many interesting properties of the coincidence reflections of a lattice. The main result is proved in Section 4.
We also give a rough classification of coincidence isometry groups in dimension two of three. Section 5 is a concluding section.

Preliminaries
The set of integers, rational numbers and real numbers are denoted by , and , respectively. For a ring R, the set of all × m n matrices over R is denoted by We recall some notations in linear algebra and the definition of lattice.
Since any n-dimensional Euclidean space is isomorphic to n with the standard inner product, we only consider lattices in n and we write vectors in n as column vectors. Here we do not require that [ ] ′ L L : is finite for a sublattice ′ L of L. This is because we will study the relationship between the group of coincidence isometries of a lattice L and the group of coincidence isometries of a sublattice ′ L of L having lower rank. Now we introduce the coincidence isometry group of a lattice. . The set of all coincidence isometries of L is a subgroup of ( ) O V and is denoted by ( ) OC L .
There exists a canonical way to extend an isometry of V to an isometry of n : , .
Hence, the coincidence isometry group ( ) OC L of a lattice in V can be viewed as a subgroup of ( ) O n . and P is a nonsingular rational matrix, then Proof. It follows immediately from Theorem 2.1 and Lemma 7.2 of [5]. □ By Grimmer's theorem, we see that the coincidence isometry group of lattice with basis … α α , , n 1 consists precisely of the isometries whose matrices with respect to the basis … α α , , n 1 are rational matrices. be the respective -subspaces of n spanned by the basis vectors of L 1 and L 2 . It is routine to check that the following map  Moreover, in this case any coincidence isometry of L is a product of at most n reflections defined by the vectors in L.

Reflections in the coincidence isometry group
The well-known Cartan-Dieudonné theorem states that any isometry in ( ) O n is the product of at most n reflections. It was found that the coincidence reflections of a lattice L play an important role in the structure of ( ) OC L , see [10] and [14]. In this section, we obtain many interesting properties of a coincidence reflection in the coincidence isometry group. For any nonzero vector ∈ v V, let v denote the reflection defined by v, Proof. We put the proof in the appendix. □ Proof. By Lemma 3.1, there exists a linearly independent subset . Since is an isometry, one has ( ) = γ γ , 0 Proof. We put the proof in the appendix. □ Proof. We put the proof in the appendix. □ The following theorem is part of the main result.
be a lattice of n and let r be dimension of the -space spanned by 1 w r is a basis of V 1 . By Lemmas 3.3 and 3.1, we may assume that , which is a contradiction. This proves (1).
Let ( ) ∈ OC L . By Corollary 3.1, the reflection defined by ( ) w j is also a coincidence isometry of L and is an isometry, V 2 is also an -invariant subspace of V . It follows that the group homomorphism defined in Lemma 2.2 is an isomorphism in this case. If , and is also for any coincidence isometry which is a product of reflections. Hence, −I is not contained in the subgroup of ( ) OC L generated by reflections. □ Proof. Suppose to the contrary that be a maximal linearly independent subset of X, | | ≤ < r X n. Since ( ) OC L is generated by reflections, by Theorem 3.1 there exist vectors … ∈

Main results
In this section, we present our main result. We first give a criterion so that a reflection is a coincidence isometry.
Lemma 4.1. Let ∈ α n be a nonzero vector. Then ⊥ α has a basis consisting of vectors in n if and only if α is parallel to a nonzero vector of n .
Proof. We put the proof in the appendix. □ Proof. Let . By the property of Gram matrix, we have for > i 1, . By Lemma 4.1, GB is parallel to a nonzero vector in n . Now suppose that = GB rC and ∈ C . n By Lemma 4.1, there exist − n 1 linearly independent row vectors   We define a relation ~on the set of nonzero vectors in n such that α β if . This is clearly an equivalence relation. Now we state and prove our main result. be the partition corresponding to the equivalence relation S tructure of coincidence isometry groups  1521 be the sublattice of L, the -subspace of V and the -subspace of n generated by the elements of X i , respectively. Then = ⊕ = L L i r i be two nonzero vectors. We claim that:  (2) Every nonzero vector of L defines a coincidence reflection of L; (3) Any coincidence isometry of L is a product of at most n reflections defined by the vectors in L; (4) L contains an orthogonal subset of order n.

Then Theorem 2.2 states that
and Theorem 4.2 states that . Zou [14] showed that one of the following is satisfied: , 2 , then every non-zero vector of L defines a coincidence reflection of L and ( ) OC L is generated by these reflections; This example also shows that ( ) ( ) ⇒ 3 ̸ 2 in Remark 4.1. It should be mentioned that the coincidence problem was completely solved in the planar case by using Clifford Algebra, see [13]. We give a rough classification of the coincidence isometry groups in two and three dimensional based on our main result. We also put the proof of the following two examples in the appendix.  be a lattice in 3 . Suppose that X is a maximal orthogonal subset of . Then exactly one of the following is satisfied. 1 2 3 and one of the following holds: is rational, and ( ) are rational, and every nonzero vector of L defines a coincidence reflection of L.

Concluding remark
In this work, we focus on the structure of coincidence isometry groups of lattices in n . We obtain a necessary and sufficient condition for an arbitrary coincidence isometry of certain lattices L to be a product of at most n coincidence reflections defined by the vectors of L. In particular, we show that if L has an orthogonal basis, then there exists an orthogonal decomposition of = ⊕ = L L Without loss of generality, we may assume that ( , .      This finishes the proof. 1 2 . It is well known that each linear isometry of 2 is either a reflection or a rotation, see [17]. Let θ denote the rotation through the angle θ.