Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
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where ν ε (g) is the trace of the diagonal block of δ(g) corresponding <strong>to</strong> V ε .<br />
Since g(ε) = εg is a right H-coset <strong>for</strong> any right H-coset ε, ν ε (g) = 0 if ε ⇒ = εg.<br />
Now assume that ε = εg. Then x ε g = hx ε where h = x ε gx − ε 1 H. Consider the vec<strong>to</strong>r space<br />
homomorphism ϕ : V ε ⊃ V with ϕ(f) = f(x ε ). Since f V ε is uniquely determined by f(x ε ), ϕ<br />
is an isomorphism. We have<br />
ϕ(gf) = g(f)(x ε ) = f(x ε g) = f(hx ε ) = δ V (h)f(x ε ) = hϕ(f),<br />
and gf = ϕ −1 hϕ(f). This means that ν ε (g) = ν V (h). There<strong>for</strong>e<br />
<br />
ν(g) =<br />
ν V (x ε gx − ε 1 ).<br />
εH\G,εg=ε<br />
4.10 Frobenius reciprocity<br />
A very important result about induced representations is the Frobenius Reciprocity Theorem which<br />
connects the operations Ind and Res.<br />
Theorem 4.33. (Frobenius Reciprocity)<br />
Let H → G be groups, V be a representation of G and W a representation of H. Then<br />
G<br />
Hom G (V, IndH W ) is naturally isomorphic <strong>to</strong> Hom H (Res G V, W ).<br />
Proof. Let E = Hom G (V, Ind G H W ) and E = Hom H (Res G H V, W ). Define F : E ⊃ E<br />
and F : E<br />
⊃<br />
E as follows: F (ϕ)v = (ϕv)(e) <strong>for</strong> any ϕ E and (F (α)v)(x) = α(xv) <strong>for</strong> any α E .<br />
In order <strong>to</strong> check that F and F are well defined and inverse <strong>to</strong> each other, we need <strong>to</strong> check<br />
the following five statements.<br />
Let ϕ E, α E , v V , and x, g G.<br />
(a) F (ϕ) is an H-homomorphism, i.e., F (ϕ)hv = hF (ϕ)v.<br />
Indeed, F (ϕ)hv = (ϕhv)(e) = (hϕv)(e) = (ϕv)(he) = (ϕv)(eh) = h · (ϕv)(e) = hF (ϕ)v.<br />
(b) F (α)v IndH G W , i.e., (F (α)v)(hx)<br />
= h(F (α)v)(x).<br />
Indeed, (F (α)v)(hx) = α(hxv) = hα(xv) = h(F (α)v)(x).<br />
(c) F (α) is a G-homomorphism, i.e. F (α)gv = g(F (α)v).<br />
Indeed, (F (α)gv)(x) = α(xgv) = (F (α)v)(xg) = (g(F (α)v))(x).<br />
(d) F ∞ F = Id E ⊗ .<br />
This holds since F (F (α))v = (F (α)v)(e) = α(v).<br />
(e) F ∞ F = Id E , i.e., (F (F (ϕ))v)(x) = (ϕv)(x).<br />
Indeed, (F (F (ϕ))v)(x) = F (ϕxv) = (ϕxv)(e) = (xϕv)(e) = (ϕv)(x), and we are done.<br />
Exercise. The purpose of this exercise is <strong>to</strong> understand the notions of restricted and induced<br />
representations as part of a more advanced framework. This framework is the notion of tensor<br />
products over k-algebras (which generalizes the tensor product over k which we defined in Definition<br />
H<br />
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