# LOG#087. Group theory(VII).

Representation theory is the part of Group Theory which is used in the main applications. Matrices acting on the members of a vector space are assigned to every element of a group. The connections between particle physics and representation theory is “natural”. It was noted by Eugene Wigner firstly, and  the properties of elementary particles and the representation theory are due to some special groups,  the so-called Lie groups and Lie algebras. The deep connection between Lie groups and Lie algebras with particle physics is observed in the different quantum states of an elementary particle, as a consequence of the irreducible representations of the Poincaré group. We are going to study this a little bit.

Definition (29). Representation. Group representation is every linear map D in G, $D: G\longrightarrow GL(V)$. That is, a linear representation of a group is a group homomorphism onto the general linear group due to the elements of the group. Informally speaking, it is a rule that assings to every group element a square matrix.

Here V is certain vector space, sometimes called “the space of the representation”, and $GL(V)$ is the “general linear group” of non-singular matrices defined over V.

Definition (30). Equivalent representations. Two representations $D^\mu$ and $D^\nu$ are said to be equivalent representations of G, if there exists some matrix $A\in GL(V)$, an isomorphism, such as

$D^\mu(g)=AD^\nu (g)A^{-1}$ $\forall g\in G$

Definition (31). Character. The character of an element $g\in G$, given the representation

$D:G\longrightarrow G''$

is the number

$\chi^{G'}(g)=\mbox{Tr} D(g)$

Of course, the character is related to the trace of the matrix, so in the case of infinite groups we have to be more careful and precise with the definition of “trace”. But this point is not relevant or important in the present discussion. Clearly, the character could be interpreted as an homomorphism between the group and the group of real (complex) numbers.

Definition (32). Character of a representation. The character of a representation $G$ is the $n-plet$ of numbers

$\chi^{G}=\left(\chi^G(g_i)\right)_{i=1,2,\ldots,\mbox{ord}(G)}$

Property: if $G^\mu$ and $G^\nu$ are two equivalent representations of G, then their characters are the same

$\chi^{G^\mu}=\chi^{G^\nu}$

Remark (I): The space a representation IS the vectorial space V where the matrices (belonging to the general linear group) act.

Remark(II): The dimension of a representation IS the dimension of the vectorial space V.

Remark (III): Do you know some examples of group representations we have seen in fact already? Remember that the trace of a matrix verifies that $\mbox{Tr}(AB)=\mbox{Tr}(BA)$ and $\mbox{Tr}(ABC)=\mbox{Tr}(BCA)=\mbox{Tr}(CAB)$ and so on. This is the cyclic property of the trace.

Given two group representations, $D_1, D_2$ of $G$, we can build other representations from them in at least two simple ways:

Definition (33) . Direct sum representation. The direct sum is defined over the space $V_1\oplus V_2$. The dimension of the direct sum:

$\mbox{dim}(D)=\mbox{dim}(D_1)+\mbox{dim}(D_2)$

so

$D_1\oplus D_2\longrightarrow GL(V_1)\oplus GL(V_2)$

where in block for we have

$(D_1\oplus D_2)(g)=\begin{pmatrix} D_1(g) & 0\\ 0 & D_2(g)\end{pmatrix}$

Definition (34). Tensor product representation. Given two representations $D_1, D_2$ of G, we can build the tensorial product representation

$D_1\otimes D_2$ $\longrightarrow GL(V_1)\otimes GL(V_2)$

so

$(D_1\otimes D_2)(g)=D_1(g)\otimes D_2(g)$

as a tensor product of matrices.

The dimension of the tensor product representation is equal to:

$\mbox{dim}(D)=\mbox{dim}(D_1)\times \mbox{dim}D_2$

Question: Do you know the basic properties of tensorial products in vector spaces? Review their properties and work out some examples yourself.

Definition (35). Invariant subspace of a representation $D$ is a susbspace (or even the whole space itself) such as

$D(g)W\subseteq W$ $\forall g\in G$

We distinguish two types of representations according to the existence or not of non-trivial invariant subspaces with respect to them:

1st. Irreductible representations. These class of representations are those such as V does not contain any invariant subspace, except V itself. That is, irreducible representations are “single” pieces/boxes from which group representations are made of. They are the “atoms” of group representations.

2nd. Reducible representations. In this case, V contains some invariant subspaces different to itself.

3rd. Fully reducible representation is any representation, if exists, such as there is some invariant subspace $W$, such that there exists another $W^\perp$ invariant subspace that satisfies

$V=W\otimes W^\perp$

The study of the group representations is based on the study of the irreducible representations (irreps.), since reducible representations can be contructed from irreducible representations (reducible representations would be “molecules” made of irreducible representations).

Some properties of group representations are interesting:

1) Let G be a finite group. Then, all representation on an inner product space are equivalent to an  unitary representation.

2) Let G be a group and let D be an unitary representation. Then, if the representation is reducible, then it is fully reducible.

3) Every reducible representation of a finite group is fully reducible.

In summary: a finite group has always an unitary representation and if if is reducible, it is fully reducible.

See you in other group theory blog post!