CLASSICAL INVARIANTTHEORY A Primer by Hanspeter Kraft, Claudio Procesi

By Hanspeter Kraft, Claudio Procesi

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1 +n−1 λ2 +n−2 n sgn σ xλσ(1) xσ(2) · · · xλσ(n) . = σ∈S n Clearly, vλ is an alternating homogeneous polynomial ∈ Z[x1 , . . , xn ] of degree |λ| + n2 . Moreover, (xi − xj ) v(0,... ,0) (x1 , . . , xn ) = i

Are simple with highest weight kε1 . ) j V )Un = K(e1 ∧ . . ∧ ej ) and that (Sk V )Un = (2) If W is a simple GLn -module of highest weight λ = p1 ε1 +· · ·+pn εn then the dual module W ∗ is simple of highest weight λ∗ := −pn ε1 − pn−1 ε2 − · · · − p1 εn . 8. ) Exercises 22. (char K = 0) Every rational representation of GL(V ) × GL(V ) is completely reducible. Every irreducible representation is of the form W ⊗ W with irreducible representations W and W of GL(V ) and GL(V ), respectively. (Cf.

A simple module W as in the corollary above is called a module of highest weight λ. Exercise 21. Assume that char K = 2 and consider the representation on W := S2 V where V = K 2 is the standard representation. (a) W U2 = Ke21 , and GL2 e21 = Ke21 ⊕ Ke22 is isomorphic to V . (b) (W ∗ )U2 = Kx1 x2 ⊕Kx22 , Kx1 x2 is the determinant representation and GL2 x22 = W ∗ . Corollary 2. Assume char K = 0. (a) If W is a simple GLn (K)-module then EndGLn (W ) = K. (b) Every rational representation of GLn (K) is defined over Q.

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