Geodesic
Homomorphisms from the unitary group to the general linear group over complex number field and applications
Archivum mathematicum, Tome 38 (2002) no. 3, pp. 209-217

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Let $M_n$ be the multiplicative semigroup of all $n\times n$ complex matrices, and let $U_n$ and $GL_n$ be the $n$–degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from $U_n$ to $GL_m$ when $n>m\ge 1$ or $n=m\ge 3$, and thereby determine multiplicative homomorphisms from $U_n$ to $M_m$ when $n>m\ge 1$ or $n=m\ge 3$. This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if $f:U_n\rightarrow M_n$ is a spectrum–preserving multiplicative homomorphism, then there exists a matrix $R$ in $GL_n$ such that $ f(A)={R}AR$ for any $A\in U_n$.
Classification : 15A30, 20E36, 20G20
Keywords: homomorphism; unitary group; general linear group 
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     title = {Homomorphisms from the unitary group to the general linear group over complex number field and applications},
     journal = {Archivum mathematicum},
     pages = {209--217},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
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     zbl = {1068.20048},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2002__38_3_a4/}
}
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Cao, Chong-Guang; Zhang, Xian. Homomorphisms from the unitary group to the general linear group over complex number field and applications. Archivum mathematicum, Tome 38 (2002) no. 3, pp. 209-217. http://geodesic.mathdoc.fr/item/ARM_2002__38_3_a4/