Subrepresentations

6. Subrepresentations#

6.1. Subrepresentations#

A subrepresentation is a subvector space that is also a representation of \(G\) under the same action.

Definition (Subrepresentation)

Let \(V\) be a linear representation of a group \(G\). A subspace \(W\) of \(V\) is a subrepresentation if \(W\) is invariant under \(G\) - that is, if \(g.w\in W\) for all \(g\in G\) and all \(w\in W\).

Examples (Subrepresentation)

  • Every subspace is a subrepresentation of the trivial representation on any vector space since the trivial \(G\) action takes every subspace back to itself

  • The tautological representation of \(D_4\) on \(\mathbb{R}^2\) has no proper non-zero subrepresentations because there is no line taken back to itself under every symmetry of the square, that is, there is no line left invariant by \(D_4\).

  • The vertex permutation representation of \(D_4\) on \(\mathbb{R}^4\), induced by the action of \(D_4\) on a set of basis elements \(\{e_1,e_2,e_3,e_4\}\) indexed by the vertices of a square does have a proper non-trivial subrepresentation. For example, the one dimensional subspace spanned by \(e_1+e_2+e_3+e_4\) is fixed by \(D_4\) – when \(D_4\) acts, it permutes \(e_i\) so their sum remains unchanged. Therefore, for all \(g\in D_4\), we have

\[ g.(\lambda,\lambda,\lambda,\lambda)=(\lambda,\lambda,\lambda,\lambda) \]

for all vectors in the one-dimensional subspace of \(\mathbb{R}^4\). Then, \(D_4\) acts trivially on this one-dimensional subrepresentation.

  • Another representation of the vertex permutation representation of \(D_4\) on \(\mathbb{R}^4\) is the subspace \(W\subset \mathbb{R}^4\) of vectors whose coordinates sum to 0. When \(D_4\) acts by permutinf the coordinates, it leaves their sum unchanged, For example, \(B\) sends \((1,2,3,-6)\) to \((-6,3,2,1)\) if vectors are labelled counterclockwise from the upper right. The space \(W\) is a three-dimensional representation of \(D_4\) on \(\mathbb{R}^4\). Note that \(W\) is a non-trivial subrepresentation since the elements of \(G\) do move the vectors around in the space \(W\).

  • The standard representaion of \(S_n\): Let \(S_n\) acts on a vector space of dimension \(n\), say \(\mathbb{R}^n\), by permuting the \(n\) vectors of a fixed basis. Note that the subspace spanned by the sum of the basis vectors is fixed by the action of \(S_n\) – it is a subrepresentation on which \(S_n\) acts trivially. More interestingly, the \((n-1)\)-dimensional subspace

\[\begin{split} W=\left\{\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}:\sum_{i=1}^n x_i=0\right\} \subset\mathbb{R}^n \end{split}\]

is also invariant under the permutation action. This is called the standard representation of \(S_n\).

6.2. Direct sum of representations#

For representations that have non-zero subrepresentations, it is always possible to simplify the matrix representatives in such a way that all subrepresentations are clearly visible, as blocks in these matrices. Let us take the permutation representation of \(S_4\), \(\rho:S_3\to GL(\mathbb{R}^3)\):

\[\begin{split} \begin{align*} e\mapsto \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, \qquad (12)\mapsto \begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}, \qquad (13)\mapsto \begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}\\ (23)\mapsto \begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}, \qquad (123)\mapsto \begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}, \qquad (132)\mapsto \begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix} \end{align*} \end{split}\]

We know that it does have two subrepresentations:

  • a one-dimensional trivial representation

  • a two-dimensional standard representation

Let us define

\[\begin{split} M=\begin{pmatrix}1&-\frac{1}{2}&-\frac{1}{2}\\0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\1&1&1\end{pmatrix} \end{split}\]

Then by direct calculation:

\[\begin{split} \begin{align*} M \rho(e) M^{-1} =\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}\qquad M \rho((12)) M^{-1} =\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}&0\\\frac{\sqrt{3}}{2}&\frac{1}{2}&0\\0&0&1\end{pmatrix}\\ M \rho((13)) M^{-1} =\begin{pmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}&0\\-\frac{\sqrt{3}}{2}&\frac{1}{2}&0\\0&0&1\end{pmatrix}\qquad M \rho((23)) M^{-1} =\begin{pmatrix}1&0&0\\0&-1&0\\0&0&1\end{pmatrix}\\ M \rho((123)) M^{-1} =\begin{pmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}&0\\\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\0&0&1\end{pmatrix}\qquad M \rho((132)) M^{-1} =\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}&0\\-\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\0&0&1\end{pmatrix}\\ \end{align*} \end{split}\]

All these matrices are block-diagonal with a two-dimensional block and a one-dimensional block. By multiplying by \(M\) from the left and \(M^{-1}\) from the right we just changed the basis of the three-dimensional space. This change of basis exposed the subrepresentations of the permutation representation of \(S_3\).

Definition (Direct sum of representations)

Suppose that \(G\) acts on vector spaces \(V\) and \(W\). We can define an action of \(G\) coordinate-wise on their direct sum as:

\[ g.(v,w)=(g.v,g.w)\in V\oplus W \]

The matrix associated to every \(g\) acting on \(V\oplus W\) will be a block diagonal matrix

\[\begin{split} \begin{pmatrix} \rho_1(g)&0\\0&\rho_2(g) \end{pmatrix} \end{split}\]

obtained from the \(n\times n\) matrix \(\rho_1(g)\) describing the action of \(g\) on \(V\), and the \(m\times m\) matrix \(\rho_2(g)\) describing the action of \(g\) on \(W\). We call \(V\oplus W\) the direct sum of representations of \(G\).

We are interested in decomposing representations in direct susm of their subrepresentations. To do it we need one more notion:

Definition (Isomorphic representations)

Two representations \(V\) and \(W\) of \(G\) are isomorphic if there is a vector space isomorphism between them that preserves the action of \(G\), that is, if there exists an isomorphism \(\phi:V\to W\) such that

\[ R_V(g)=\phi^{-1}\circ R_W(g)\circ \phi \]

for all \(g\in G\), where \(R_V:G\to GL(V)\) and \(R_W:G\to GL(W)\) are two representations.

Example (Isomorphic representations)

We have shown that there exists a matrix \(M\) such that for all \(g\in S_3\) we have

\[\begin{split} M R(g) M^{-1}=\begin{pmatrix}R_1(g)&0\\0&R_2(g)\end{pmatrix} \end{split}\]

where \(R\) is the permutation representation of \(S_3\), \(R_1\) is the standard representation of \(S_3\), and \(R_2\) is the trivial representation of \(S_3\). This implies that

\[ R\cong R_1\oplus R_2 \]

6.3. Searching for subrepresentations#

Let us consider the following four-dimensional representation \(V\) of \(D_4\) (called the vertex permutation representation):

  • for the rotation: \(A \mapsto \begin{pmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix} \)

  • for the horizontal reflection: \(B \mapsto \begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix} \)

The remaining matrices can be found using the fact that we have a group representation, i.e. it is a group homomorphism. For example

\[\begin{split} A^2=A\cdot A\mapsto \phi(A)\cdot \phi(A)=\begin{pmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix} ^2=\begin{pmatrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{pmatrix} \end{split}\]

We want to find all subrepresentations of \(V\). To find interesting subrepresentations of \(V\), we can look for non-zero vectors in \(\mathbb{R}^4\) whose orbits under the action of \(D_4\) span proper subspaces of \(V\). One way is to find vectors with small orbits. For example, the vector \((1,1,1,1)\) is fixed by \(D_4\). It spans a one-dimensional subrepresentation of \(V\) where \(D_4\) acts trivially.

Another vector with a small orbit is \(w=(-1,1,-1,1)\). Note that \(A\) acts on \(w\) to produce \((1,-1,1,-1)=-w\). Also the reflection \(B\) acts the same way and produces \(-w\). All other elements of \(D_4\) act by superpositions of these two elements, and therefore the orbit of \(w\) is just \(\{w,-w\}\). Thus the one-dimensional subspace spanned by \(w\) is a subrepresentation of \(V\). This is not a trivial representation since some elements act by the multiplication by \(-1\).

There is also a two-dimensional subrepresentation of \(V\) that can be found by considering the orbit of \(u=(1,1,-1,-1)\). Its orbit is:

\[ \mathrm(Orb)(u)=\{(1,1,-1,-1),(-1,1,1,-1),(-1,-1,1,1),(1-1,-1,1)\} \]

that are points on a two-dimensional subspace \(T\) of \(\mathbb{R}^4\) that has a basis \((1,1,-1,-1)\) and \((-1,1,1,-1)\). It is easy to check that the vertex representation action of \(D_4\) on \(T\) is identical with the tautological representation of \(D_4\) on this two-plane.

Finally, since the three subrepresentations described above span \(V\), there is a direct sum decomposition of representations:

\[ V\cong T\oplus \mathbb{R}(1,1,1,1)\oplus \mathbb{R} (1,-1,1,-1) \]

Importantly, none of these subrepresentations can be further decomposed.

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