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I am reading Dummit and Foote on Representation Theory CH 18

I am struggling with the following text on page 843 - see attachment and need some help.

The text I am referring to reads as follows - see attachment page 843 for details

[itex] \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v + \beta w ) [/itex]

[tex] = g \cdot ( \alpha v ) + g \cdot ( \beta w ) [/tex]

[tex] = \alpha ( g \cdot v ) + \beta ( g \cdot w ) [/tex]

[tex] = \alpha \phi ( g ) ( v ) + \beta \phi ( g ) (w) [/tex]

Now my problem with the above concerns [tex] g \cdot ( \alpha v ) + g \cdot ( \beta w ) [/tex] [tex] = \alpha ( g \cdot v ) + \beta ( g \cdot w ) [/tex]

This looks like it is just using the fact that elements of F commute with elements of g as in [tex] g \cdot ( \alpha v ) = \alpha ( g \cdot v ) [/tex]

BUT ... this is not just an element of F commuting with an element of G as in [tex] (1_F h ) ( \alpha 1_G) = ( \alpha 1_G ) ( 1_F h ) [/tex] ...

the statement above involves the [tex] \cdot [/tex] operation which is (to quote D&F) " the given action of the ring element g on the element v of V" { why "ring" element? }

Doesn't the fact that we are dealing with an action mixed with terms like [tex] \alpha v [/tex] involving a field element multiplied by a vector complicate things ...

how do we formally and explicitly justify [tex] g \cdot ( \alpha v ) = \alpha ( g \cdot v )[/tex]?

How do we justify taking [tex] \alpha [/tex] out through the the action [tex] \cdot [/tex] ? Why are we justified in doing this?

Peter

I am struggling with the following text on page 843 - see attachment and need some help.

The text I am referring to reads as follows - see attachment page 843 for details

[itex] \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v + \beta w ) [/itex]

[tex] = g \cdot ( \alpha v ) + g \cdot ( \beta w ) [/tex]

[tex] = \alpha ( g \cdot v ) + \beta ( g \cdot w ) [/tex]

[tex] = \alpha \phi ( g ) ( v ) + \beta \phi ( g ) (w) [/tex]

Now my problem with the above concerns [tex] g \cdot ( \alpha v ) + g \cdot ( \beta w ) [/tex] [tex] = \alpha ( g \cdot v ) + \beta ( g \cdot w ) [/tex]

This looks like it is just using the fact that elements of F commute with elements of g as in [tex] g \cdot ( \alpha v ) = \alpha ( g \cdot v ) [/tex]

BUT ... this is not just an element of F commuting with an element of G as in [tex] (1_F h ) ( \alpha 1_G) = ( \alpha 1_G ) ( 1_F h ) [/tex] ...

the statement above involves the [tex] \cdot [/tex] operation which is (to quote D&F) " the given action of the ring element g on the element v of V" { why "ring" element? }

Doesn't the fact that we are dealing with an action mixed with terms like [tex] \alpha v [/tex] involving a field element multiplied by a vector complicate things ...

how do we formally and explicitly justify [tex] g \cdot ( \alpha v ) = \alpha ( g \cdot v )[/tex]?

How do we justify taking [tex] \alpha [/tex] out through the the action [tex] \cdot [/tex] ? Why are we justified in doing this?

Peter

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