# Uniqueness of spinor representation

$$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$$I asked a similar question on math stack exchange here, but I wonder if it may be better received here.

Let $$n$$ be even, then the standard complex Clifford algebra admits an isomorphism: $$\mathbb{C}\text{l}(n)\cong \text{End}(\mathbb{C}^N)$$ where $$N=2^{n/2}$$. We call $$\mathbb{C}^N$$ the space of Dirac spinors. Let $$\gamma:\mathbb{R}^n\rightarrow\text{Cl}(n)$$ be the canonical injection, and $$f:\text{Cl}(n)\rightarrow \mathbb{C}\text{l}(n)\cong \text{Cl}(n)\otimes_\mathbb{R}\mathbb{C}$$ be the inclusion map. The above isomorphism then induces a blinear map called spinor multiplication with a vector: \begin{align*} \mathbb{R}^n\times \mathbb{C}^N&\longrightarrow \mathbb{C}^N\\ (x,\psi)&\longmapsto x\cdot \psi =(f\circ \gamma(x))\cdot \psi \end{align*} where we identify $$f\circ \gamma(x)$$ with it's unique endomorphism under the aforementioned isomorphism. Identifying $$\text{Spin}(n)$$ as a subset of the real Clifford algebra then yields a faithful representation of $$\text{Spin}(n)$$ on $$\mathbb{C}^N$$, called the spinor representation. We denote this representation by $$\kappa$$, and one can easily show that: $$\kappa(g)\cdot(x\cdot \psi)=(\lambda(g)\cdot x)\cdot(\kappa(g)\cdot \psi)$$ where $$\lambda:\text{Spin}(n)\rightarrow \SO(n)$$ is the double covering homomorphism.

My question is then this, does the above property uniquely determine the spinor representation up to isomorphism? I.e. if we have two faithful representations of $$\text{Spin}(n)$$ on $$\mathbb{C}^N$$ satisfying the property above, denoted by $$\kappa$$ and $$\rho$$, does there exist an isomorphism $$F:\mathbb{C}^N\rightarrow \mathbb{C}^N$$ satisfying: \begin{align*} F(\kappa(g)\cdot \psi)=\rho(g)\cdot F(\psi) \end{align*} for all $$\psi\in \mathbb{C}^N$$.

My motivation for this question is mostly due to the case where $$n=4$$. When $$n=4$$, we have that the space of Dirac spinors is given by $$\mathbb{C}^4$$, then we can obtain a representation of $$\text{Spin}(4)$$ on $$\mathbb{C}^4$$ as above. Indeed, if we choose an orthonormal basis for $$\mathbb{R}^4$$, it is not difficult to construct a faithful representation of $$\text{Cl}(n)$$ on $$\mathbb{C}^4$$, which then induces the isomorphism between the complex Clifford algebra. In particular, such a representation, when restricted to $$\text{Spin}(n)$$ has image in the subgroup of block diagonal matrices who's entries lie in $$\SU(2)$$. By construction, this representation satisfies the property above

However, we can also construct a representation of $$\text{Spin}(4)$$ in a different way, due to the exceptional isomorphism: \begin{align*} \text{Spin}(4)\cong \SU(2)\times \SU(2) \end{align*} We can then embed $$\SU(2)\times \SU(2)$$ in $$\GL_4(\mathbb{C})$$ by mapping $$\SU(2)\times \SU(2)$$ to block diagonal matrices, which naturally acts on $$\mathbb{C}^4$$. By identifying $$\mathbb{R}^4$$ with the real subspace of $$2\times 2$$ complex matrices spanned by the Pauli spin matrices, we can specify how $$\mathbb{R}^4$$ acts on $$\mathbb{C}^4$$. This action can be made to be the same as the one obtained by construction a faithful representation of $$\text{Cl}(n)$$ on $$\mathbb{C}^4$$. This representation of $$\text{Spin}(4)$$ on $$\mathbb{C}^4$$ then also respects Clifford multiplication with a vector.

Both of these representations are seemingly used interchangeably, though the latter is used more frequently when one wishes to discuss the specific case of $$n=4$$. Due to this, I suspect that the representations should be isomorphic, but I am unsure of how to show it.

If they are not isomorphic, then how do we justify using the latter representation when the former is derived in such generality?

• This follows from the classification of irreducible representations of SU(2) for n=4. Jul 29, 2023 at 0:07
• @KentaSuzuki sorry my representation isn’t the best, could you elaborate? Jul 29, 2023 at 1:57

For each integer $$n\ge1$$ the Lie group $$SU(2)$$ has a unique irreducible representation $$V_n$$ of dimension $$n$$, namely the symmetric power $$\mathrm{Sym}^{n-1}(\mathbb C^2)$$. Thus $$Spin(4)\cong SU(2)\times SU(2)$$ has irreducible representations of the form $$V_{(m,n)}:=V_m\boxtimes V_n$$ of dimension $$mn$$, where $$m,n\ge1$$. A four-dimensional representation of $$Spin(4)$$ is one of: $$V_{(4,1)}$$, $$V_{(2,2)}$$, $$V_{(1,4)}$$, $$V_{(3,1)}+V_{(1,1)}$$, $$V_{(1,3)}+V_{(1,1)}$$, $$V_{(1,2)}+V_{(1,2)}$$, $$V_{(1,2)}+V_{(2,1)}$$, $$V_{(2,1)}+V_{(2,1)}$$, $$V_{(2,1)}+2V_{(1,1)}$$, $$V_{(1,2)}+2V_{(1,1)}$$, or $$4V_{(1,1)}$$. Only $$V_{(1,2)}+V_{(2,1)}$$ is faithful, which must be the spinor representation.
• What does $\boxtimes$ denote here? Do you know of a source which goes into this? Does this imply that any faithful representations of $SU(2)\times SU(2)$ on $\mathbb{C}^4$ must be isomorphic? Also, I'm assuming this doesn't generalize as I thought ? Jul 29, 2023 at 4:30
• @Chris FYI, regarding the general use of $\boxtimes$, please see the Math SE post What is the operation $\boxtimes$?. Jul 29, 2023 at 4:46
• Also, how is this representation of $\text{Spin}(4)$ irreducible? Don't the projections onto the upper and lower block diagonal matrices determine sub representations of on $\mathbb{C}^2$? Jul 29, 2023 at 4:59