In three dimensions there are three different symmetry groups that are algebraically the Klein four-group V:.
The three elements of order two in the Klein four-group are interchangeable: the automorphism group of V is the group of permutations of these three elements. The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on four points:. In this representation, V is a normal subgroup of the alternating group A 4 and also the symmetric group S 4 on four letters.
In fact, it is the kernel of a surjective group homomorphism from S 4 to S 3. In the construction of finite rings , eight of the eleven rings with four elements have the Klein four-group as their additive substructure. In a similar fashion, the group of units of the split-complex number ring , when divided by its identity component, also results in the Klein four-group.
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The simplest simple connected graph that admits the Klein four-group as its automorphism group is the diamond graph shown below. It is also the automorphism group of some other graphs that are simpler in the sense of having fewer entities. These include the graph with four vertices and one edge, which remains simple but loses connectivity, and the graph with two vertices connected to each other by two edges, which remains connected but loses simplicity. In music composition the four-group is the basic group of permutations in the twelve-tone technique. In that instance the Cayley table is written; .
From Wikipedia, the free encyclopedia. This article is about the mathematical concept. Basic notions. It is not clear a priori that this is an embedding, although it will turn out that this is so, or that a shortest expression of an element in the image by elements of T will also be one by elements of S. It is possible to develop the subject of Coxeter groups entirely in combinatorial terms this is done - well, at least thoroughly attempted - in the book by Bourbaki , but certain geometric representations of Coxeter groups, in which the group acts discretely on a certain domain, and in which the generators are represented by reflections, allow one to visualize nicely what is going on.
Recall that a linear reflection is a linear transformation fixing a hyperplane passing through the origin and acting as multiplication by -1 on a line through the origin transverse to this hyperplane. There are other kinds of reflections, too, but they reduce to linear ones. They are discussed at the end of this section. In this section we shall look at some Coxeter groups defined geometrically. A polyhedral realization of a Coxeter group is a linear representation in which The group possesses a fundamental domain C which is a polyhedral cone; the generators in S are represented by reflections in the walls of this cone.
I do not exclude the case where the cone is invariant under translation. This for a group with a single generator a single hyperplane is admissible in any dimension. In these notes, we shall use only simplicial realizations where the polyhedral cone is simplicial, its walls parametrized by S.
The dimension of a simplicial realization must be at least the rank of the group. Every Coxeter group possesses at least one realization, as we shall see in a moment. Geometric properties of realizations translate naturally to combinatorial properties of the group.
From the geometry of the simplices neighbouring a fundamental domain, for example, you can read off the Coxeter matrix. This is because if L the intersection of two walls, then the configuration in the neighbourhood of L is essentially that in a realization of the group generated by the two reflections in those walls. This is a special case of a very general result proven in most generality by MacBeath around Given a realization, make a choice for each s in S of a pair a s , a s v defining reflection in the wall of the fundamental domain parametrized by s.
Such a linear function a s is determined up to a positive scalar multiple, and its equivalence class under such multiplications will be called a basic root of the realization and implicitly also of the choice of fundamental domain. This terminology is not common, but for general Coxeter groups this notion of root is entirely natural, since the particular choice of a s has no intrinsic significance. This is not the case if the Coxeter group is the Weyl group of a Kac-Moody Lie algebra, since in that case the roots themselves are part of the structure of the Lie algebra.
This gives rise to a new matrix DCD -1 where D is a diagonal matrix with positive entries. The next result is a consequence of observations made earlier about conjugacy classes of pairs of reflections: Theorem This theorem is apparently due originally to Vinberg. A real matrix C satisfying these conditions for a given Coxeter matrix is called an abstract Cartan matrix. It determines a Coxeter matrix in a natural way. The numbers n s, t will always be non-negative; the finite values of m s, t are determined by conditions 2 and 4 for the values of n s, t lying in [0, 4 ; and m s, t is infinite for n which are 4 or larger.
Any Cartan matrix clearly gives rise to a representation of the associated Coxeter group.
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In fact: Theorem The representation of a Coxeter group determined by any abstract Cartan matrix is a realization of the associated Coxeter group. This will be proven in the next section. Cartan matrices with integral matrices determine Kac-Moody Lie algebras. In this case the representation of its Weyl group on the lattice of roots is the one associated to this Cartan matrix. Coxeter groups which occur as the Weyl groups of Kac-Moody algebras are called crystallographic , and are distinguished by the property that for them the numbers m s, t are either 2 , 3 , 4 , 6 or infinite.
In particular, those Cartan matrices giving rise to realizations equivalent to the standard one are symmetrizable. Proposition If each m s, t is finite, then the isomorphism classes of Cartan realizations are parametrized by H 1 , R where is the Coxeter diagram. Conclude by applying the logarithm. If W is finite, then the Coxeter diagram has no circuits.
Because if the diagram were not a tree, the group would possess a continuous family of non-isomorphic representations of dimension r. Distinct classes can give rise to realizations with very different geometric properties.
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We have seen this already in the case of the infinite dihedral group, and here are the pictures for two different realizations of the Coxeter group whose Coxeter diagram is :. For the moment, fix a Cartan matrix. It gives rise to a representation of its associated Coxeter group, in which the elements of s act by reflections.
This will be called for the moment a Cartan representation.
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The principal result connecting combinatorics and the geometry of a Coxeter group is this: Theorem Suppose that a s is a basic root in a Cartan representation of W. This generalizes what we have already seen for groups of rank two. The proof is somewhat intricate. It begins with Lemma. In particular, we will see that the group of components is arithmetic and hence finitely presented under this assumption. I plan to sketch an algorithmic procedure which allows to build projective wonderful models for the complement of a toric arrangement in a n-dimensional algebraic torus T in analogy with the case of subspaces in a linear or projective space.
The main step of the construction is a combinatorical algorithm that produces a projective toric variety in which the closure of each layer of the arrangement is smooth. The explicit procedure of our consturction allows us to describe the integer cohomology rings of such models by generators and relations. Understanding differential Azumaya algebras cohomologically requires locally solving systems of linear differential equations which is done by introducing the Delta-finite Grothendieck topology.
We compare this topology with Zariski and etale topologies for appropriate coefficients and then use it to relate the differential and usual Brauer groups. Finally we look at the situation for smooth, projective varieties. This gives a precise mathematical explanation of the physicists' "affine E8 global symmetry of the half K3 surface".