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Spin representations, Clifford action

2 Dimensional Representations

{$V_j=\langle x_j,y_j \rangle$}

{$C(V_j)=\langle 1,x_j,y_j,x_jy_j \rangle$}

{$a_j=\frac{1}{2}(x_j + iy_j)$}

{$\bar{a}_j=\frac{1}{2}(x_j - iy_j)$}

{$x_j=a_j+\bar{a}_j$}

{$y_j=i(\bar{a}_j-a_j)$}

{$W_j\oplus W_j^*=\langle a_j, \bar{a}_j \rangle$}

Representation given by action on {$\bar{a}_j$}

{$W^*_j=\langle \bar{a}_j \rangle $}

{$S'_j = \langle 1, \bar{a}_j \rangle $}

{$C(V_j)=\langle 1,x_j,y_j,x_jy_j \rangle\cong \textrm{End}(\langle 1, \bar{a}_j \rangle)$}

{$1\rightarrow \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$}

{$a_j\rightarrow \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ \end{pmatrix}$}

{$\bar{a}_j\rightarrow \begin{pmatrix} 0 & 0 \\ 1 & 0 \\ \end{pmatrix}$}

{$a_j\bar{a}_j\rightarrow \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix}$}

{$\bar{a}_ja_j\rightarrow \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix}$}

{$a_j^2=\bar{a}_j^2=0$}

{$x_j\rightarrow \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$}

{$y_j\rightarrow \begin{pmatrix} 0 & -i \\ i & 0 \\ \end{pmatrix}$}

{$x_jy_j\rightarrow \begin{pmatrix} i & 0 \\ 0 & -i \\ \end{pmatrix}$}

{$y_jx_j\rightarrow \begin{pmatrix} -i & 0 \\ 0 & i \\ \end{pmatrix}$}

Representation given by action on {$a_j$}

{$W_j=\langle a_j \rangle $}

{$S_j = \langle 1, a_j \rangle $}

{$C(V_j)=\langle 1,x_j,y_j,x_jy_j \rangle\cong \textrm{End}(\langle 1, a_j \rangle)$}

{$1\rightarrow \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$}

{$a_j\rightarrow \begin{pmatrix} 0 & 0 \\ 1 & 0 \\ \end{pmatrix}$}

{$\bar{a}_j\rightarrow \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ \end{pmatrix}$}

{$a_j\bar{a}_j\rightarrow \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix}$}

{$\bar{a}_ja_j\rightarrow \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix}$}

{$a_j^2=\bar{a}_j^2=0$}

{$x_j\rightarrow \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$}

{$y_j\rightarrow \begin{pmatrix} 0 & i \\ -i & 0 \\ \end{pmatrix}$}

{$x_jy_j\rightarrow \begin{pmatrix} -i & 0 \\ 0 & i \\ \end{pmatrix}$}

{$y_jx_j\rightarrow \begin{pmatrix} i & 0 \\ 0 & -i \\ \end{pmatrix}$}

The Cartan subalgebra of the Lie algebra for the spin group is generated by {$\bar{a}_j\wedge a_j$} which is mapped to {$\frac{1}{4}[\bar{a}_j,a_j]=\frac{1}{4}\bar{a}_ja_j - \frac{1}{4}a_j\bar{a}_j$}. Acting on an arbitrary element {$c_j+d_ja_j\in V$}, and noting the factor {$\sqrt{2}$} in applying a vector, we have {$(\frac{1}{4}\bar{a}_ja_j - \frac{1}{4}a_j\bar{a}_j)(c_j+d_ja_j)=(\frac{1}{4}c_j - \frac{1}{4}d_j)2$}. This means the action is given by

{$\begin{pmatrix} \frac{1}{2} & 0 \\ 0 & -\frac{1}{2} \\ \end{pmatrix}\begin{pmatrix} c_j \\ d_j \end{pmatrix} = \begin{pmatrix} \frac{1}{2}c_j \\ -\frac{1}{2}d_j \end{pmatrix}$}

The weight spaces are {$1$} and {$a_j$}. The Cartan subalgebra element {$\bar{a}_j\wedge a_j$} has eigenvalue {$-\frac{1}{2}$} on {$a_j$}, which contains {$a_j$}, and eigenvalue {$\frac{1}{2}$} on {$1$}, which does not contain {$a_j$}.

Note that the weight {$\lambda = \frac{1}{2}$} of {$S_+=\langle 1 \rangle$} is the negative of the weight {$-\lambda = -\frac{1}{2}$} of {$S_-=\langle a_j \rangle$}. Thus there is an isomorphism from {$S_+$} to {$S^*_-$}, unique up to scale.

Relating the representations

Note that the Lie algebra for {$\textrm{Spin}(2)$} has a single generator, the bivector {$x_jy_j$}.

The representation {$\bar{\phi}$} based on {$\bar{a}_j$} sends

{$x_jy_j\rightarrow \begin{pmatrix} i & 0 \\ 0 & -i \\ \end{pmatrix}$}

The representation {$\phi$} based on {$a_j$} sends

{$x_jy_j\rightarrow \begin{pmatrix} -i & 0 \\ 0 & i \\ \end{pmatrix}$}

Thus these are dual representations. {$\phi(x_jy_j)=-\bar{\phi}(x_jy_j)^T$} and {$\bar{\phi}(x_jy_j)^T=-\phi(x_jy_j)$}.

Note also that the values of the dual representation is given by the value on {$y_jx_j$}, which is to say, the value in the opposite Clifford algebra.