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)

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 $\frac{p^2}{q^2}$, which is ratio of semiinvariants of weight $2\chi$.

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) in the context of quiver representations?

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)

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

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

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 $\frac{p^2}{q^2}$, which is ratio of semiinvariants of weight $2\chi$.

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) in the context of quiver representations?

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)

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) in the context of quiver representations?

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)

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 $\frac{p^2}{q^2}$, which is ratio of semiinvariants of weight $2\chi$.

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) in the context of quiver representations?