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Keywords:
homomorphism; unitary group; general linear group
Summary:
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 $A\in U_n$.
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