Team 5 (Norman, Mark, Melissa, Shaun)

Representation Theory

Representation Theory is one aspect of abstract algebra and is concerned with representing groups in terms of matrices. This can include forms of linear algebra with matrices, transformations, and vector spaces.

Representing elements as linear transformations of vector spaces can be shown and defined in multiple ways. We will introduce the relevant concepts via notes, definitions, examples, pictures, etc. Note that an $n$ dimension vector space is isomorphic to $\mathbb{R}^n$.

Multiplication of matrices through representation theory:
As stated on Wikipedia, “The set of all invertible $n \times n$ matrices is a group under matrix multiplication and the representation theory of groups analyses a group by describing ("representing") its elements in terms of invertible matrices.” [1]

There are five classes of groups in Representation Theory. These are: permutation groups, matrix, transformation, abstract, and topological and algebraic. For example:
Permutation: Starts with a set denoted as $X$ and there is a collection of $G$ that separates $X$ into bijections of itself. These bijections are closed under inverses and compositions, where $G$ is based on $X$. According to Wikipedia, “If $X$ consists of $n$ elements and $G$ consists of all permutations, $G$ is the symmetric group $Sn$; in general, $G$ is a subgroup of the symmetric group of $X$.”[2]

Matrix: Sometimes called a linear group; this group $G$ is a set of matrices over some field (like $\mathbb{R}$) that is closed under products and inverses.

Group Representation takes abstract groups and describes them as matrices such that the group operation is represented by the multiplication of matrices. We can learn about finite groups, compact or locally compact groups, Lie groups, linear algebraic groups, and non-compact topological groups.

Recall that an isomorphism is a structure-preserving map that is 1-1 and onto. Representation theory is concerned with isomorphisms of the form:

\begin{align} \phi: G\rightarrow M \end{align}

where $G$ is an abstract group and $M \leq M_n_\times_n \mathbb{R}$.

Take the following example: in $\mathbb{R}^2$

recall that the dihedral group $D_4 = \{ r,s|r^4=s^2=1, rs=sr^-^1\}$
If $R$ and $S$ are any matrices satisfying the relations $R^4=S^2=I$and$RS=SR^-^1$ then the map $r \rightarrow R$and$s \rightarrow S$ extends uniquely to a homomorphism from $D_4$ to the matrix group generated by $R$ and $S$, hence gives a representation of $D_4$. An explicit example of matrices $R, S \epsilon M_2 (\mathbb{R})$ may be obtained as follows. If a regular N-gon is drawn on an x,y plane centered in the origin with $x=y$ as one of the lines of symmetry then the matrix $R$ that rotates the plane through $2\pi/n$ radians and the matrix $S$ that reflects the plane about the line $y=x$ both send this $n$-gon onto itself. It follows that the matrices act as symemetries of the $n$-gon and so satisfy the above relations. The matrices are readily computed and so the maps

\begin{align} r \rightarrow R= \[ \left( \begin{array}{ccc} cos(2\pi/n) & -sin(2\pi/n) \\ sin(2\pi/n) & cos(2\pi/n) \end{array} \right)\] \end{align}


\begin{align} s\rightarrow S= \[ \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right)\] \end{align}

extend uniquely to a degree 2 representation of $D_n$ into $GL_2(\mathbb{R})$. Since matrices $R$ and $S$ have orders n and 2 respectively, it follows that they generate a subgroup of $GL_2(\mathbb{R})$ of order $2n$ and hence the representation is faithful.
for example:

\begin{align} r \rightarrow R= \[ \left( \begin{array}{ccc} cos(\pi/2) & -sin(\pi/2) \\ sin(\pi/2) & cos(\pi/2) \end{array} \right)\] \end{align}
\begin{align} r \rightarrow R= \[ \left( \begin{array}{ccc} 0 & -1 \\ 1 & 0 \end{array} \right)\] \end{align}
\begin{align} s\rightarrow S= \[ \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right)\] \end{align}

One of the more important theorems in Representation theory is Maschke’s theorem which is as follows: “Let $G$ be a finite group and let $F$ be a field whose charictarisitc does not divide $|G|$. If $V$ is any $FG$ module and $U$ is any submodule of $V$ then $V$ has a submodule $W$ such that $U \bigoplus W=V$.” [3] (Page 815)
Take the case of the field $F$ being the Real Numbers, and $V= \mathbb{R}^3$, the submodule $U$ is any line or plane through the origin, and $W$ is the plane or line(s), orthogonal to $U$.

Works Cited
[3] Dummit, David S. Abstract Algebra (Second Edition). 1999.

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