Xian Zhang, $^1$Department of Mathematics, Heilongjiang University, Harbin, 150080, P. R. C.
$^2$School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Stranmillis Road, Belfast, BT9 5AH, United Kingdom
E-mail: X.Zhang@Queens-Belfast.AC.UK
Abstract. 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 [{\it 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)=\inv{R}AR$ for any $A\in U_n$.
AMSclassification. 20G15.
Keywords. Homomorphism, unitary group, general linear group.