1. The problem statement, all variables and given/known data
My online class notes:
"Along the same vein as linear maps between vector spaces and group homomorphisms between groups we have maps between group representations that respect the algebraic structure.
Definition 3.1: Let (p,V) and (q,W) be two representations of a group G. a lineaer transformation
ø: V → W is an intertwining map if ø(p(g)v) = q(g)ø(v) for all v in V and g in G."
okay so my first question is what exactly does the arguments in the notation for the representations mean? For example, the (p,V) representation; I know the V is a vector space, but what is the p? Is it the permutation or mapping that results from the group action on the vector space? If so, why do two representations of the same group (p,V) and (q,W) have different mappings if the group acting on them only has one binary operation?
2. Relevant equations
3. The attempt at a solution
My online class notes:
"Along the same vein as linear maps between vector spaces and group homomorphisms between groups we have maps between group representations that respect the algebraic structure.
Definition 3.1: Let (p,V) and (q,W) be two representations of a group G. a lineaer transformation
ø: V → W is an intertwining map if ø(p(g)v) = q(g)ø(v) for all v in V and g in G."
okay so my first question is what exactly does the arguments in the notation for the representations mean? For example, the (p,V) representation; I know the V is a vector space, but what is the p? Is it the permutation or mapping that results from the group action on the vector space? If so, why do two representations of the same group (p,V) and (q,W) have different mappings if the group acting on them only has one binary operation?
2. Relevant equations
3. The attempt at a solution
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