I was reading S. Mukai's- An Introduction to Invariants and Moduli, and I came across Proposition 6.16 on page 195 (see the screenshot below)

[![enter image description here][1]][1]


It says that "every invariant rational function can be expressed as ratio of semiinvariants of weight $\chi$."

>Should it be $n\chi$, for $n\in\mathbb{N}$?

The reason I'm confused is because- say I consider the weight space determined by $\chi$, i.e., all the semiinvariants of weight $\chi$. It has the structure of a $\mathbb{K}$-vector space. Say I choose two of the basis element $p,q\in\mathfrak{B}$. Now consider the invariant rational function $\frac{p^2}{q^2}$.

>I don't understand how can I write $\frac{p^2}{q^2}$ as ratio of semiinvariants of weight $\chi$?

>Also, does the Definition 6.15 about "stable with respect to $\chi$" coincide with the King's stability condition ([see here][2]) in the context of quiver representations?


  [1]: https://i.sstatic.net/H0tnm.png
  [2]: https://www.math.uni-bielefeld.de/~sek/sem/stability/king.pdf