central character

《Introduction to Representation Theory》 · 印刷第 8 页 Section 2.2 的底域约定及第 14 页 Section 2.3，Problem 2.3.16(b) 中的定义（省略中心的例子及其他小问）

Let $k$ be a field. Unless stated otherwise, we will always assume that $k$ is algebraically closed, i.e., any nonconstant polynomial with coefficients in $k$ has a root in $k$. […] Let $A$ be an algebra over a field $k$. The center $Z(A)$ of $A$ is the set of all elements $z\in A$ which commute with all elements of $A$. […] (b) Show that if $V$ is an indecomposable finite dimensional representation of $A$, then for any $z\in Z(A)$, the operator $\rho(z)$ by which $z$ acts in $V$ has only one eigenvalue $\chi_V(z)$, equal to the scalar by which $z$ acts on some irreducible subrepresentation of $V$. Thus $\chi_V:Z(A)\to k$ is a homomorphism, which is again called the central character of $V$.