Representation of $SU(4)$

group-theory representation-theory spin-geometry

Solution 1:

We know that $\mathrm{SU}(4)\simeq \mathrm{Spin}(6)$. If we identify their root systems, the positive spinor representation of $\mathrm{Spin}(6)$ corresponds to the usual $4$-dimensional representation $V$ of $\mathrm{SU}(4)$, the natural $6$-dimensional representation of $\mathrm{Spin}(6)$ corresponds to the wedge product $\wedge^{2}V$ and the negative spinor representation corresponds to $\wedge^{3}V$. This can be verified easily by checking their highest weight. I think this is an explicit description of the negative spinor. The dual of the positive spinor representation is the negative spin representation.

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