Whenever we invoke a representation for a group --using a matrix group, or group of differential operators -- are we converting a problem in group variables to one in arithmetical variables (field variables, actually)?
Group is defined independently on its reprezentation space. for instance. Let G={g1,g2,g3} be three elements group. For the three elements group its structure is unique: g1*g1=g1,g1*g2=g2,g1*g3=g3,g2*g1=g2,g3*g1=g3,
g2*g2=g3,g2*g3=g1,g3*g2=g1,g3*g3=g2. If these elements are abstract ones then G is abstract group.
If one defines a space V={1} and an action of G elements
in V: g1*1=1,g2*1=1,g3*1=1, g2*2=3,g2*3=1,g3*2=1,g3*3=2 then each group element gets the following representation: g1=1,g2=1,g3=1.
One can generate trivial less example if assumes V={g1,g2,g3} which means that the representation space consists of the all group elements. The action G on V is defined by the group action in G (see above). What is the group representation in V? Let as
take into account VT ={g1,g2,g3}T and perform the following actions:
g1*VT={g1,g2,g3}T, g2*VT={g2,g3,g1}T, g3*VT={g3,g1,g2}T. Now we read for example the representtion of g2:
|0 1 0|
[ g2]=|0 0 1|
|1 0 1|.
Therefore one can work with an abstract group, number group, matrix group and others. The choice of representation depends on a problem to be solve.
The main feature of group theory which makes it as a strong universal tool in mathematics, physics, chemistry etc. is its concept in definition of some "group property" which governs the relation between members of a set . There is no restriction on the members of set, so they may consist of points in some topology or operators of some transformation etc. this gives a universality to this concept and wide ranging application from solution of polynomial equation to internal symmetries of elementary particles. Sometimes it is more easier to replace the matrix representation with character, a complex number which corresponds to trace of matrices with applications in group symmetry in chemistry. Therefore the group concept stands for some specific relation between members of a set but the members of set (i.e. representation of group) can be justified respect to the problem conditions.It have to be mentioned that "continuous group" ( pertaining to continuous symmetry) has more directed application in physics.
Group representations in the broad sense are just group homomorphisms. So the target doesn't need any structure beyond that of a group. The most familiar representations, however, are afforded by vector spaces. But you can also have representations on modules, whose underlying scalars do not form a field, but only a ring. An example is SL(n,Z), viewed as a submodule of M(n,Z), the nxn matrices with integer coefficients. Of course you can also view SL(n,Z) as a vector subspace of M(n,Q), M(n,R), or M(n,C). You can cook up more quixotic examples where the scalar ring is not an integral domain, in which case it cannot be embedded in a field.
There is a unique universal (till today unpopular) representation of a finite group $F$ through matrix $M$ of a $n$-orbit of its permutation representation G_n(V) of the least possible degree $n$. The symmetries of $M$ shows all possible properties of $G_n$ and hence $F$.
It depends on what you want. To perform calculations with group elements it is easier, indeed, to use either a matrix representation or a presentation by generators and relations.
Sam, I agree, but Steve - what do you mean by "group variables" in your question? What is the background of your question? After Klein's "Erlanger Programm" and the subsequent Bourbaki philosophy - what else do you expect? ;-))) How do you want to represent this approach else? ;-)
Thank you Rolf for your question. My interest is in transformation groups in physics. I should have said "group elememts" rather than group variables. When it comes to writing concrete examples of groups, they always seem t me to be matricies whose elelments are scalars, or differential operators, which are again compinations of scalars.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Theoretical Physics