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 $A\in U_n$.
@article{107834,
author = {Chong-Guang Cao and Xian Zhang},
title = {Homomorphisms from the unitary group to the general linear group over complex number field and applications},
journal = {Archivum Mathematicum},
volume = {038},
year = {2002},
pages = {209-217},
zbl = {1068.20048},
mrnumber = {1921592},
language = {en},
url = {http://dml.mathdoc.fr/item/107834}
}
Cao, Chong-Guang; Zhang, Xian. Homomorphisms from the unitary group to the general linear group over complex number field and applications. Archivum Mathematicum, Tome 038 (2002) pp. 209-217. http://gdmltest.u-ga.fr/item/107834/
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