- #1
PsychonautQQ
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Homework Statement
Let V have dimension 3 and consider P_1(V ) = P(1,0,0) = span of {x,y,z}.Let I denote the subspace of all polynomials in P_1 of the form {rx+ry +rz|r any scalar}.Let W denote the subspace of all polynomials in P_1 of the form {rx+sy+tz|r+s+t = 0}. I and W are S_3 invariant subspaces
Show that I and W are subspaces that are irreducible. Find a basis for the subspaces I and W. Show that P1 = I⊕W.
The Attempt at a Solution
Math newb here, going to need a lot of help on this.
Where my head is at:
P_1(V) is a representation of S_3, which means that S_3 performed it's group action of composition on a vector space V, and the resulting representation was a polynomial with degree 1 (P_1). P_1 is a reducible representation and can be decomposed into two non-irreducible sub-representations I and W.
My first job is to show that I and W are irreducible. This would mean that there are no G-invariant subspaces for representations I and W?
My second job is to find the basis for the subspaces I and W. Is a basis a vector that is able to span the vector space? So I'm looking for vectors that span the representations I and W? Are representations still considered vector spaces?
Finally I must show and I+W = P_1. P_1 is a sort of abstract notion right? It's a three dimensional vector that has been acted upon by the S_3 symmetric group, which means that the representation P_1 contains different permutations of the basis vectors of V that include (1), (1,2), (1,3), (2,3) (1,2,3), (1,3,2) in cycle notation.
Is anything I typed remotely on the right track here? Trying to wrap my head around this stuff.