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LSpice
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Consider$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $SU(5)$$\SU(5)$ and the unitary group $U(16)$$\U(16)$.

Below I specify a specfic way to embed the $SU(5) \subset U(16)$$\SU(5) \subset \U(16)$:

  1. First we can embed the spin group $Spin(10)\subset U(16)$$\Spin(10)\subset \U(16)$. Here we choose the ${\bf 16}$$\mathbf{16}$-dimensional spinor representation of $Spin(10)$$\Spin(10)$ to be also the ${\bf 16}$$\mathbf{16}$-dimensional fundamental representation of $U(16)$$\U(16)$. Thus, the data for the representation: $${\bf 16} \text{ in } Spin(10) \text{ as } {\bf 16} \text{ in } U(16)$$$$\text{$\mathbf{16}$ in $\Spin(10)$ as $\mathbf{16}$ in $\U(16)$}$$ gives us an action of $Spin(10)$$\Spin(10)$ and $U(16)$$\U(16)$ on the complex vector space $\mathbb{C}^{16}$.

  2. Then, we can embed the $SU(5)$ to$\SU(5)$ into $Spin(10)$ next, see$\Spin(10)$. See the discussion of this embedding of $SU(n)$ to$\SU(n)$ into $Spin(2n)$$\Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\bf 5}$$\overline{\mathbf 5}$, the anti-symmetric representation ${\bf 10}$$\mathbf{10}$ and the 1-dimensional representation ${\bf 1}$$\mathbf 1$ of $SU(5)$$\SU(5)$: $$\overline{\bf 5} \oplus {\bf 10} \oplus {\bf 1} \text{ in } SU(5) \text{ as } {\bf 16} \text{ in } Spin(10).$$$$\text{$\overline{\mathbf 5} \oplus \mathbf{10} \oplus \mathbf 1$ in $\SU(5)$ as $\mathbf{16}$ in $\Spin(10)$}.$$ Again, the data for the representation gives us an action of $SU(5)$$\SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $SU(5)$$\SU(5)$ into $U(16)$$\U(16)$.

questionQuestion

Then my question is about the normalizer of this $SU(5)$ into$\SU(5)$ inside $U(16)$$\U(16)$, which certainly depends on the embedding that I provided above. So what is thisthe normalizer of this $SU(5)$ into$\SU(5)$ inside $U(16)$$\U(16)$? My suspicioussuspicion is that it may be $U(5)$$\U(5)$, but it could also be larger than $U(5)$$\U(5)$ with extra more $U(1)$$\U(1)$ factors if I was mistaken.. ….

Consider the special unitary group $SU(5)$ and the unitary group $U(16)$.

Below I specify a specfic way to embed the $SU(5) \subset U(16)$:

  1. First we can embed the spin group $Spin(10)\subset U(16)$. Here we choose the ${\bf 16}$-dimensional spinor representation of $Spin(10)$ to be also the ${\bf 16}$-dimensional fundamental representation of $U(16)$. Thus, the data for the representation: $${\bf 16} \text{ in } Spin(10) \text{ as } {\bf 16} \text{ in } U(16)$$ gives us an action of $Spin(10)$ and $U(16)$ on the complex vector space $\mathbb{C}^{16}$.

  2. Then, we can embed the $SU(5)$ to $Spin(10)$ next, see the discussion of this embedding of $SU(n)$ to $Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\bf 5}$, the anti-symmetric representation ${\bf 10}$ and the 1-dimensional ${\bf 1}$ of $SU(5)$: $$\overline{\bf 5} \oplus {\bf 10} \oplus {\bf 1} \text{ in } SU(5) \text{ as } {\bf 16} \text{ in } Spin(10).$$ Again, the data for the representation gives us an action of $SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $SU(5)$ into $U(16)$.

question

Then my question is about the normalizer of this $SU(5)$ into $U(16)$, which certainly depends on the embedding that I provided above. So what is this normalizer of this $SU(5)$ into $U(16)$? My suspicious may be $U(5)$, but it could also be larger than $U(5)$ with extra more $U(1)$ factors if I was mistaken...

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.

Below I specify a specfic way to embed $\SU(5) \subset \U(16)$:

  1. First we can embed the spin group $\Spin(10)\subset \U(16)$. Here we choose the $\mathbf{16}$-dimensional spinor representation of $\Spin(10)$ to be also the $\mathbf{16}$-dimensional fundamental representation of $\U(16)$. Thus, the data for the representation: $$\text{$\mathbf{16}$ in $\Spin(10)$ as $\mathbf{16}$ in $\U(16)$}$$ gives us an action of $\Spin(10)$ and $\U(16)$ on the complex vector space $\mathbb{C}^{16}$.

  2. Then, we can embed $\SU(5)$ into $\Spin(10)$. See the discussion of this embedding of $\SU(n)$ into $\Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\mathbf 5}$, the anti-symmetric representation $\mathbf{10}$ and the 1-dimensional representation $\mathbf 1$ of $\SU(5)$: $$\text{$\overline{\mathbf 5} \oplus \mathbf{10} \oplus \mathbf 1$ in $\SU(5)$ as $\mathbf{16}$ in $\Spin(10)$}.$$ Again, the data for the representation gives us an action of $\SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $\SU(5)$ into $\U(16)$.

Question

Then my question is about the normalizer of this $\SU(5)$ inside $\U(16)$, which certainly depends on the embedding that I provided above. So what is the normalizer of this $\SU(5)$ inside $\U(16)$? My suspicion is that it may be $\U(5)$, but it could also be larger than $\U(5)$ with more $\U(1)$ factors if I was mistaken ….

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wonderich
wonderich
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Consider the special unitary group $SU(5)$ and the unitary group $U(16)$.

Below I specify a \emph{specfic}specfic way to embed the $SU(5) \subset U(16)$:

  1. First we can embed the spin group $Spin(10)\subset U(16)$. Here we choose the ${\bf 16}$-dimensional spinor representation of $Spin(10)$ to be also the ${\bf 16}$-dimensional fundamental representation of $U(16)$. Thus, the data for the representation: $${\bf 16} \text{ in } Spin(10) \text{ as } {\bf 16} \text{ in } U(16)$$ gives us an action of $Spin(10)$ and $U(16)$ on the complex vector space $\mathbb{C}^{16}$.

  2. Then, we can embed the $SU(5)$ to $Spin(10)$ next, see the discussion of this embedding of $SU(n)$ to $Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\bf 5}$, the anti-symmetric representation ${\bf 10}$ and the 1-dimensional ${\bf 1}$ of $SU(5)$: $$\overline{\bf 5} \oplus {\bf 10} \oplus {\bf 1} \text{ in } SU(5) \text{ as } {\bf 16} \text{ in } Spin(10).$$ Again, the data for the representation gives us an action of $SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $SU(5)$ into $U(16)$.

question

Then my question is about the normalizer of this $SU(5)$ into $U(16)$, which certainly depends on the embedding that I provided above. So what is this normalizer of this $SU(5)$ into $U(16)$? My suspicious may be $U(5)$, but it could also be larger than $U(5)$ with extra more $U(1)$ factors if I was mistaken...

Consider the special unitary group $SU(5)$ and the unitary group $U(16)$.

Below I specify a \emph{specfic} way to embed the $SU(5) \subset U(16)$:

  1. First we can embed the spin group $Spin(10)\subset U(16)$. Here we choose the ${\bf 16}$-dimensional spinor representation of $Spin(10)$ to be also the ${\bf 16}$-dimensional fundamental representation of $U(16)$. Thus, the data for the representation: $${\bf 16} \text{ in } Spin(10) \text{ as } {\bf 16} \text{ in } U(16)$$ gives us an action of $Spin(10)$ and $U(16)$ on the complex vector space $\mathbb{C}^{16}$.

  2. Then, we can embed the $SU(5)$ to $Spin(10)$ next, see the discussion of this embedding of $SU(n)$ to $Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\bf 5}$, the anti-symmetric representation ${\bf 10}$ and the 1-dimensional ${\bf 1}$ of $SU(5)$: $$\overline{\bf 5} \oplus {\bf 10} \oplus {\bf 1} \text{ in } SU(5) \text{ as } {\bf 16} \text{ in } Spin(10).$$ Again, the data for the representation gives us an action of $SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $SU(5)$ into $U(16)$.

question

Then my question is about the normalizer of this $SU(5)$ into $U(16)$, which certainly depends on the embedding that I provided above. So what is this normalizer of this $SU(5)$ into $U(16)$? My suspicious may be $U(5)$, but it could also be larger than $U(5)$ with extra more $U(1)$ factors if I was mistaken...

Consider the special unitary group $SU(5)$ and the unitary group $U(16)$.

Below I specify a specfic way to embed the $SU(5) \subset U(16)$:

  1. First we can embed the spin group $Spin(10)\subset U(16)$. Here we choose the ${\bf 16}$-dimensional spinor representation of $Spin(10)$ to be also the ${\bf 16}$-dimensional fundamental representation of $U(16)$. Thus, the data for the representation: $${\bf 16} \text{ in } Spin(10) \text{ as } {\bf 16} \text{ in } U(16)$$ gives us an action of $Spin(10)$ and $U(16)$ on the complex vector space $\mathbb{C}^{16}$.

  2. Then, we can embed the $SU(5)$ to $Spin(10)$ next, see the discussion of this embedding of $SU(n)$ to $Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\bf 5}$, the anti-symmetric representation ${\bf 10}$ and the 1-dimensional ${\bf 1}$ of $SU(5)$: $$\overline{\bf 5} \oplus {\bf 10} \oplus {\bf 1} \text{ in } SU(5) \text{ as } {\bf 16} \text{ in } Spin(10).$$ Again, the data for the representation gives us an action of $SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $SU(5)$ into $U(16)$.

question

Then my question is about the normalizer of this $SU(5)$ into $U(16)$, which certainly depends on the embedding that I provided above. So what is this normalizer of this $SU(5)$ into $U(16)$? My suspicious may be $U(5)$, but it could also be larger than $U(5)$ with extra more $U(1)$ factors if I was mistaken...

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wonderich
wonderich
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The normalizer of SU(n) in U(m)?

Consider the special unitary group $SU(5)$ and the unitary group $U(16)$.

Below I specify a \emph{specfic} way to embed the $SU(5) \subset U(16)$:

  1. First we can embed the spin group $Spin(10)\subset U(16)$. Here we choose the ${\bf 16}$-dimensional spinor representation of $Spin(10)$ to be also the ${\bf 16}$-dimensional fundamental representation of $U(16)$. Thus, the data for the representation: $${\bf 16} \text{ in } Spin(10) \text{ as } {\bf 16} \text{ in } U(16)$$ gives us an action of $Spin(10)$ and $U(16)$ on the complex vector space $\mathbb{C}^{16}$.

  2. Then, we can embed the $SU(5)$ to $Spin(10)$ next, see the discussion of this embedding of $SU(n)$ to $Spin(2n)$. Here we choose the complex-conjugated fundamental representation $\overline{\bf 5}$, the anti-symmetric representation ${\bf 10}$ and the 1-dimensional ${\bf 1}$ of $SU(5)$: $$\overline{\bf 5} \oplus {\bf 10} \oplus {\bf 1} \text{ in } SU(5) \text{ as } {\bf 16} \text{ in } Spin(10).$$ Again, the data for the representation gives us an action of $SU(5)$ on $\mathbb{C}^{16}$, which further gives an embedding of $SU(5)$ into $U(16)$.

question

Then my question is about the normalizer of this $SU(5)$ into $U(16)$, which certainly depends on the embedding that I provided above. So what is this normalizer of this $SU(5)$ into $U(16)$? My suspicious may be $U(5)$, but it could also be larger than $U(5)$ with extra more $U(1)$ factors if I was mistaken...