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gr.group idea – The centralizer and normalizer of merchandise of (whirl(n) $instances dots$) in U(m) retort

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gr.group idea – The centralizer and normalizer of merchandise of (whirl(n) $instances dots$) in U(m)

$DeclareMathOperatorSU{SU}DeclareMathOperatorU{U}DeclareMathOperatorwhirl{whirl}$

assume the whirl group $whirl(n)$ and the unitary group $U(16)$.

Below I specify a specfic path to embed $(whirl(6)instances whirl(4))/mathbb{Z}_2 subset U(16)$:

  1. First we will simply embed the whirl group $whirl(10)subset U(16)$. Here we prefe the $mathbf{16}$-dimensional spinor illustration of $whirl(10)$ to breathe too the $mathbf{16}$-dimensional elementary illustration of $U(16)$. Thus, the info for the illustration:
    $mathbf{16}$ in $whirl(10)$ as $mathbf{16}$ in $U(16)$
    offers us an motion of $whirl(10)$ and $U(16)$ on the sophisticated vector area $mathbb{C}^{16}$.

  2. Then, we will simply embed $(whirl(6)instances whirl(4))/mathbb{Z}_2 subset whirl(10) subset U(16)$. The mod $mathbb{Z}_2$ issue signifies that the main focus $Z(whirl(6))=Z(SU(4))=mathbb{Z}_2$ is recognized with the diagonal $mathbb{Z}_2$ subgroup of the main focus $Z(whirl(4))=Z(SU(2)instances SU(2))=mathbb{Z}_2 oplus mathbb{Z}_2$.

Here we denote:

  • the irreducible spinor illustration of $whirl(6)$ as ${mathbf 4}$
    (which is just too the basic illustration of $SU(4)$).

  • the complicated-conjugated illustration (of ${mathbf 4}$) as $overline{mathbf 4}$.

  • too $whirl(4)=SU(2)instances SU(2)$, the place we lookout the basic illustration of every SU(2).

Then, the info for the next illustration offers us an motion of $(whirl(6)instances whirl(4))/mathbb{Z}_2$ on $mathbb{C}^{16}$, which additional offers an embedding of $(whirl(6)instances whirl(4))/mathbb{Z}_2$ into $U(16)$:
$$textual content{$({mathbf 4},{mathbf 2},{mathbf 1}) oplus (overline{mathbf 4},
{mathbf 1}, {mathbf 2})$ in $(whirl(6)instances whirl(4))/mathbb{Z}_2$ as a decomposition of $mathbf{16}$ in $U(16)$}.$$

Question

1). Then my query is in regards to the centralizer and normalizer of this $$(whirl(6)instances whirl(4))/mathbb{Z}_2 subset U(16),$$
which actually is determined by the embedding that I offered above. My distrust is that apart from some $U(1)$ components, there may breathe extra discrete finite teams.

2). Another refined loom is that given the an identical clique of illustration $({mathbf 4},{mathbf 2},{mathbf 1}) oplus (overline{mathbf 4}, {mathbf 1}, {mathbf 2})$,
it isn’t solely a illustration of $(whirl(6)instances whirl(4))/mathbb{Z}_2$ however too a illustration of $(whirl(6)instances whirl(4))$. Thus we will query too the centralizer and normalizer of this embedding $$(whirl(6)instances whirl(4)) subset U(16)$$
apart from the $(whirl(6)instances whirl(4))/mathbb{Z}_2 subset U(16).$

assassinate we maintain the an identical or completely different solutions of centralizer and normalizer for the illustration 1) versus 2)?

we’ll proffer you the answer to gr.group idea – The centralizer and normalizer of merchandise of (whirl(n) $instances dots$) in U(m) query by way of our community which brings all of the solutions from a number of reliable sources.

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