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By Hanspeter Kraft, Claudio Procesi

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Then ρ detr ·ρ for some r ∈ Z. (Hint: HomSL(V ) (W , W ) is a GL(V )-stable one-dimensional subspace of Hom(W , W ). 5 Frobenius Reciprocity. Let G be a finite group and H ⊂ G a subgroup. If W is a H-module we denote by IndG H W the induced G-module (or induced representation) which is usually defined by IndG H W := KG ⊗KH W. where KG, KH denote the group algebras. On the other hand, every G-module V can be regarded as an H-module by restriction; we will denote it by V |H . Proposition (Frobenius reciprocity).

Ik )(j1 , . . , jr ) · · · (l1 , . . , ls ). Define a function Trσ : End(V )m → K by Trσ (A1 , . . , Am ) := Tr(Ai1 · · · Aik ) Tr(Aj1 · · · Ajr ) · · · Tr(Al1 · · · Als ). Clearly, Trσ is a multilinear invariant. It is easy to see that it does not depend on the presentation of σ as a product of disjoint cycles (cf. Exercise 2 below). 5). Theorem (Multilinear FFT for matrices). Assume char K = 0. The multilinear invariants on End(V )m are linearly generated by the functions Trσ , σ ∈ S m.

The sum over all monomials of degree j) are called the complete symmetric polynomials: xi1 xi2 · · · xij . hj (x1 , . . , xn ) := χSj V = i1 ≤i2 ≤··· We obviously have the following generating function: n 1 = 1 − xi t i=1 ∞ hj · tj . j=0 These polynomials hj are special cases of Schur polynomials which we are going to introduce and discuss in the next section. Exercise 2. The representation ρ : GLn → GL(W ) is polynomial if and only if its character χρ is a polynomial. 2 Schur polynomials. Let P denote the set of decreasing finite sequences of natural numbers: P := {λ = (λ1 , λ2 , .

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