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)?

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