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It'sMe
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A basis of the weight space in "An introductionthe semi-invariant ring corresponding to quiver representations"the weight $\langle(2,3,2),\cdot\rangle$

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)

enter image description here

I want to understand the line underlined in red.

How does the choice of vectors $(\lambda,\mu,\nu)$ being $(1,0,0),(0,1,0),(0,0,1)$ give us a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$?

They are basically saying that the Schofield semi-invariants $c^{V}$ that you get by choosing these three vectors form a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$ in the semi-invariant ring. This is the fact that I don't understand.

A basis of weight space in "An introduction to quiver representations"

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)

enter image description here

I want to understand the line underlined in red.

How does the choice of vectors $(\lambda,\mu,\nu)$ being $(1,0,0),(0,1,0),(0,0,1)$ give us a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$?

A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)

enter image description here

I want to understand the line underlined in red.

How does the choice of vectors $(\lambda,\mu,\nu)$ being $(1,0,0),(0,1,0),(0,0,1)$ give us a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$?

They are basically saying that the Schofield semi-invariants $c^{V}$ that you get by choosing these three vectors form a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$ in the semi-invariant ring. This is the fact that I don't understand.

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YCor
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A Basisbasis of Weight Spaceweight space in "An introduction to quiver representations"

I'm trying to understand Example 10.11.1 on page 225 of the book "An Introduction to Quiver Representations""An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)

enter image description here

I want to understand the line underlined in red.

How does the choice of vectors $(\lambda,\mu,\nu)$ being $(1,0,0),(0,1,0),(0,0,1)$ give us a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$?

A Basis of Weight Space

I'm trying to understand Example 10.11.1 on page 225 of the book "An Introduction to Quiver Representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)

enter image description here

I want to understand the line underlined in red.

How does the choice of vectors $(\lambda,\mu,\nu)$ being $(1,0,0),(0,1,0),(0,0,1)$ give us a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$?

A basis of weight space in "An introduction to quiver representations"

I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)

enter image description here

I want to understand the line underlined in red.

How does the choice of vectors $(\lambda,\mu,\nu)$ being $(1,0,0),(0,1,0),(0,0,1)$ give us a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$?

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It'sMe
It'sMe
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A Basis of Weight Space

I'm trying to understand Example 10.11.1 on page 225 of the book "An Introduction to Quiver Representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)

enter image description here

I want to understand the line underlined in red.

How does the choice of vectors $(\lambda,\mu,\nu)$ being $(1,0,0),(0,1,0),(0,0,1)$ give us a basis for the weight space $\mathrm{SI}(Q,\beta)_{\langle(2,3,2),\cdot\rangle}$?