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For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real representation? (The adjoint representation of $SU(N)$ is real).

A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\mathbb{C})$ is said to be pseudo-real if there exists a matrix $C$ such that, for all $g\in SU(N)$ $$\bar{R}(g)=CR(g)C^{-1},$$ where $\bar{R}(g)$ means complex conjugation. The representation $R$ is said to be real if there exists a matrix $D$ such that $DR(g)D^{-1}$ is real for all $g$.

I would appreciate any comment or reference. Thank you!

For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).

A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\mathbb{C})$ is said to be pseudo-real if there exists a matrix $C$ such that, for all $g\in SU(N)$ $$\bar{R}(g)=CR(g)C^{-1},$$ where $\bar{R}(g)$ means complex conjugation. The representation $R$ is said to be real if there exists a matrix $D$ such that $DR(g)D^{-1}$ is real for all $g$.

I would appreciate any comment or reference. Thank you!

Does $SU(N)$ have a pseudo-real representation? If so, how can we construct them explicitly? I am looking specifically for pseudo-real representation instead of real representation (The adjoint representation of $SU(N)$ is real).

If $g\in SU(N)$ and $R(g)$ is a pseudo-real representation, then there exists a matrix $C$ such that

 $$\bar{R}(g)=CR(g)C^{-1},$$

where $\bar{R}(g)$ means complex conjugation and $C$ is not the identity matrix.

I would appreciate any comment or reference. Thank you!

For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real representation? (The adjoint representation of $SU(N)$ is real).

A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\mathbb{C})$ is said to be pseudo-real if there exists a matrix $C$ such that, for all $g\in SU(N)$ $$\bar{R}(g)=CR(g)C^{-1},$$ where $\bar{R}(g)$ means complex conjugation. The representation $R$ is said to be real if there exists a matrix $D$ such that $DR(g)D^{-1}$ is real for all $g$.

I would appreciate any comment or reference. Thank you!

Does $SU(N)$ have a pseudo-real representation? If so, how can we construct them explicitly? I am looking specifically for pseudo-real representation instead of real representation (The adjoint representation of $SU(N)$ is real).

If $g\in SU(N)$ and $R(g)$ is a pseudo-real representation, then there exists a matrix $C$ such that

$$\bar{R}(g)=CR(g)C^{-1},$$

where $\bar{R}(g)$ means complex conjugation.

I would appreciate any comment or reference. Thank you!

Does $SU(N)$ have a pseudo-real representation? If so, how can we construct them explicitly? I am looking specifically for pseudo-real representation instead of real representation (The adjoint representation of $SU(N)$ is real).

If $g\in SU(N)$ and $R(g)$ is a pseudo-real representation, then there exists a matrix $C$ such that

$$\bar{R}(g)=CR(g)C^{-1},$$

where $\bar{R}(g)$ means complex conjugation and $C$ is not the identity matrix.

I would appreciate any comment or reference. Thank you!