How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By specifying the embedding, I mean that we can determine precisely the way how the Lie group embedding $G_1 \subset G_2$ is fixed as an embedding of two differentiable manifolds (since Lie groups are differentiable manifolds). Of course, as Lie group embedding, $G_1 \subset G_2$, the two groups must share the common identity element $\mathbf{1}$. So their group identity element $ \mathbf{1}_{G_1} =\mathbf{1}_{G_2} $ are the same point on two manifolds.
Previously, I asked, is it enough to give some irreducible representation (irrep) of $G_1$ called $\mathbf{R}_{1,j}$ and some irrep of $G_2$ called $\mathbf{R}_2$, then we dictate the map $$ \mathbf{R}_{1} = \bigoplus_j \mathbf{R}_{1,j} \text{ in } G_1 \mapsto \mathbf{R}_2 \text{ in } G_2, \text{ and } G_1 \subset G_2 \tag{1} $$ would the above be precisely enough to specify the embedding? Is this a necessary and sufficient condition? If not, what else data is needed? See Specify the embedding of Lie groups precisely as the embedding of two differentiable manifolds
Here I would like to use a specific example to demonstrate whether we can uniquely specify the embedding or whether we can enumerate possible differnt embedding given the eq.(1).
Let us take a special unitary group $G_1=SU(5)$ into a Spin group $G_2=Spin(10)$. The is a lift map from $SU(5) \to SO(10)$ to $SU(5) \to Spin(10)$ which the universal cover $\pi_1(Spin(10))=0$ consistent with the lift map with $\pi_1(SU(5))=0$, $$ \begin{array}{ccc} SU(5) & \longrightarrow & Spin(10)\\ &\searrow & \downarrow\\ & & SO(10). \end{array} $$
Let us specify a first possible way of embedding $SU(5) \subset Spin(10)$ via $$ \mathbf 5 \oplus \overline{\mathbf{10}} \oplus \mathbf 1 \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).$$ or $$ \bigwedge{}^{1}\mathbb{C}^5 \oplus \bigwedge{}^{3}\mathbb{C}^5 \oplus \bigwedge{}^{5}\mathbb{C}^5 \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).\tag{a} $$ Here ${\mathbf 5}$ is the fundamental representation of $SU(5)$ in $\bigwedge{}^{1}\mathbb{C}^5$. Here $\bigwedge$ is the wedge product of vectors in the vector space $\mathbb{C}^5$.
Question 1: Does eq.(a) specify a unique embedding of $SU(5) \subset Spin(10)$? Or is it possible to have two or more such distinct $SU(5)$ embedding in $Spin(10)$ with the map given eq.(a)? If so, how are these $SU(5)$ different from each other?
- Let us specify a second possible way of embedding $SU(5) \subset Spin(10)$ via $$ \mathbf 1 \oplus \mathbf{10} \oplus \overline{\mathbf{5}} \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10).$$ or $$ \bigwedge{}^{0}\mathbb{C}^5 \oplus \bigwedge{}^{2}\mathbb{C}^5 \oplus \bigwedge{}^{4}\mathbb{C}^5 \text{ in } SU(5) \mapsto \mathbf{16} \text{ in } Spin(10). \tag{b}$$
Question 2: Does eq.(b) specify a unique embedding of $SU(5) \subset Spin(10)$? Or is it possible to have two or more such distinct $SU(5)$ embedding in $Spin(10)$ with the map given eq.(b)? If so, how are these $SU(5)$ different from each other?
Question 3: How are the embedding of eq.(a) and eq.(b) related to each other? I suppose, they are related by the outer automorphism of $SU(5)$ which is a $\mathbf{Z}/2$. Then, if so, how does this $\mathbf{Z}/2$ outer automorphism of $SU(5)$ act on the $Spin(10)$?
