Schur’s Lemma

This is a standard result that any beginner in the study of the representation theory would be aware of. I merely present restate the theorem a little(essentially I don’t) and prove it.

Theorem:

Let V,W be representations of a group G. If f:V \mapsto W is a G-linear map, then the following are true assuming that V and W are not isomorphic:

1)If both V,W are irreducible, then f is either an isomorphism or the zero map.

2)If only V is irreducible, then f is either injective or the zero map.

3)If only W is irreducible, then f is either surjective or the zero map


If V,W are isomorphic as representations, then f is a scalar multiplication map.

Proof:

Let \rho_{V},\rho_{W} be the respective representations. Here, V,W are vector spaces over an algebraically closed field.

We tackle the first part of the theorem.

Consider the kernel of f, say Ker(f), which is set of all v \in V such that f(v)=0. Since f is a G-linear map, f(g.v)=g.f(v)=0 implies that if v \in Ker(f), then g.v \in Ker(f). Hence, Ker(f) is stable under the action of G which makes it a sub-representation. Since V is irreducible, this means that either Ker(f) must be zero or f must be the zero map. This proves that f is injective.

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Coxeter Groups(intro)

Reflection groups

In order to understand the intuition underlying the theory of Coxeter groups(and Weyl groups in particular groups in particular), you can go through the theory of reflection groups which I’ll only superficially treat preceding my exposition of Coxetr systems and the associated representation theory.

Consider some Euclidean space V and a vector \alpha \in V. A reflection associated with \alpha is a linear operator which sends \alpha to -\alpha and point-wise fixes the orthogonal hyperplane/subspace H_{\alpha}.

If the reflection is s_{\alpha}, then it can represented as:

s_{\alpha}(x)=x-\frac{2(x,\alpha)}{(\alpha,\alpha)}\alpha=x-2 proj_{\alpha}(x)

Clearly s_{\alpha} is an orthogonal transformation and the set of all reflections in V can be recognized as the subgroup of O(V)(orthogonal group of V) consisting of elements of order 2.

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