What Does KW Represent in the Context of a Permutation Module?

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Homework Help Overview

The discussion revolves around the concept of permutation modules in the context of group actions, specifically focusing on the notation KW where K is a field and G is a finite group acting on a set W. The original poster expresses confusion regarding the meaning of KW and the appropriate ring for multiplication, given that W is not necessarily a group.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of KW as a permutation module and question the nature of G's action on W. There is discussion about the implications of G not being a ring and how that affects the interpretation of KG-modules.

Discussion Status

The conversation is ongoing, with some participants providing clarifications about the relationship between G and KG, while others express their misunderstandings. There is an indication that some productive insights have been shared, but no consensus has been reached.

Contextual Notes

Participants are navigating the complexities of group actions and module theory, with specific attention to the definitions and assumptions underlying the notation used in the problem statement.

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Homework Statement


K is a field with finite characteristic p, G is a finite group, and W is a set that G acts on transitively (so for all x,y in W, there exists g s.t. gx=y). It then says consider M=KW the permutation module.

What is KW supposed to mean? I know for a group G that KG is the group algebra, but we don't know that W is a group (in fact, it probably isn't). Furthermore, what ring is intended to be used for multiplication? I'm confused out of my mind. I've looked back in my lecture notes so far but haven't seen anything to resolve the issue


The Attempt at a Solution


Asking here
 
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I think in this setting the permutation module is the KG-module you obtain by letting G act on KW = set of formal linear combinations of elements of W with coefficients in K (which is basically the free K-module generated by W).
 
But G isn't a ring. Unless we just use formal addition in G to make it one?
 
G isn't; KG is.

This sort of stuff comes up when you talk about things like "G-modules". See http://planetmath.org/encyclopedia/GModule.html .
 
Last edited by a moderator:
Oh, I misunderstood what you wrote originally. That makes sense now
 

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