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Homomorphisms from the unitary group to the general linear group over complex
number field and applications

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*Chong-Guang Cao and Xian Zhang*

**Address.** Chong-Guang Cao, Department of Mathematics, Heilongjiang
University, Harbin, 150080, P. R. C.
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.