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Download CLASSICAL INVARIANTTHEORY A Primer by Hanspeter Kraft, Claudio Procesi PDF

By Hanspeter Kraft, Claudio Procesi

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For m = m , V ⊗m and V ⊗m do not contain isomorphic submodules. Exercises ∼ 10. There is a canonical isomorphism ϕ : End(V ) → End(V )∗ . It is induced by the bilinear form (A, B) → Tr(AB) and is GL(V )-equivariant. Moreover, ϕ∗ = ϕ. ∼ 11. There is a natural GL(V )-equivariant isomorphism V ⊗ V ∗ → End(V ). Which element of V ⊗ V ∗ corresponds to id ∈ End(V ) and which elements of End(V ) correspond to the “pure” tensors v ⊗ λ? 30 Polarization and Restitution §4 Polarization and Restitution §4 In this paragraph we study the multilinear invariants of vectors and covectors and of matrices.

Xn ) = i

Assume char K = 0 and let V be a finite dimensional representation of a group G. Then every homogeneous invariant f ∈ K[V ]G of degree d is the full restitution of a multilinear invariant F ∈ K[V d ]G . 4 above. 1 d! Pf , which is a multihomogeneous invariant Exercises 5. 6 Generalization to several representations sym F (v1 , . . , vd ) := 35 F (vσ(1) , . . , vσ(d) ). σ∈S d 6. Let f ∈ K[V ] be homogeneous of degree d and write d si td−i fi (v, w), f (sv + tw) = s, t ∈ K, v, w ∈ V. i=0 Then the polynomials fi are bihomogeneous of degree (i, d−i) and the linear operators f → fi are GL(V )-equivariant.

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