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Let $G$ and $H$ be two primitive unitary groups, is $G\times H$ necessarily a primitive unitary group? If not, is there any counterexample?

edited Apr 17 2011 at 10:37

Andreas Thom
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asked Apr 15 2011 at 5:34

Huangjun Zhu
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Can you give an example of a primitive unitary group action please? – Abel Stolz Apr 15 2011 at 8:41

A unitary group on a finite-dimensional Hilbert space is imprimitive if the space can be decomposed as a direct sum of two or more subspaces such that under the action of any element in the group, any vector in any subspace is still contained in one of the subspaces. A unitary group is primitive if it is not imprimitive. In dimension 2, the group generated by the following two elements is primitive, [i,0;0,−i], [−1+i,−1+i;1+i,−1−i]/2. This group can be seen as the symmetry group of the regular tetrahedron. – Huangjun Zhu Apr 15 2011 at 10:25

I am mainly concerned with the following scenario. $G$ and $H$ are two groups consisting of unitary matrices of size $d_1$ and $d_2$, respectively, and they can be taken as representations of themselves. Let $K$ be the group composed of all unitary matrices that are tensor product of matrices in $G$ and that in $H$. I just realized $K$ may not be a direct product due to the overall phase factors, for example, $1\times i=i\times 1$. However, if we identity unitary matrices differing by overall phase factors, then $K$ can be seen as a direct product. Is $K$ primitive if both $G$ and $H$ are? – Huangjun Zhu Apr 17 2011 at 2:12

Abel Stolz · Answer 1 · 2011-04-15 10:14:56Z UTC

I try to give an answer based on my interpretation of your question. I guess by "unitary" group you mean "subgroup of a unitary group", because $U(n)\times U(m)$ is certainly not isomorphic to $U(k)$ for any $k$. Then any finite group is fine for an example. Whenever a group $G$ acts primitively on a set $X$, it acts transitively and therefore the cardinalty of $X$ is less than the order of $G$. When $G$ acts primitively on $X$, the cardinality of $X$ is called the degree of $G$. The only finite primitive group of degree $2$ is $S_2$. Assume $S_2\times S_2$ is primitive. Then the degree of $S_2\times S_2$ must be less than $4$. The only primitive finite groups of degree $3$ are $S_3$ and $A_3$. The only ones of degree $4$ are $S_4$ and $A_4$, according to the List of small primitive groups implemented in GAP. All are not isomorphic to $S_2\times S_2$, so the answer to your question is no (assuming my interpretation is correct).

Is the direct product of two primitive unitary groups necessarily a primitive unitary group?

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