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3-Characterizations of finite groups by Makhnev A. A.

By Makhnev A. A.

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Let W be a representation of a Lie group G. Define Vn (W ) to be the space of holomorphic polynomials on W that are homogeneous of degree n and let (g P) (η) = P(g −1 η). Show that there is an equivalence of representations S n (W ∗ ) ∼ = Vn (W ) induced by viewing T1 · · · Tn , Ti ∈ W ∗ , as a function on W . 18 (a) Show that the map π : t → 1 t 0 1 produces a representation of R on C2 . (b) Show that this representation is not unitary. (c) Find all invariant submodules. (b) Show that the representation is reducible and yet not completely reducible.

Ym ) in Rm and u ∈ R, let (x, y, u) = (x1 , . . , xm , y1 , . . , ym , u) ∈ Rn . In particular, (x, y, u) = 12 (x −i y, i(x −i y), 0)+ 1 (x +i y, −i(x +i y), 0) + (0, 0, u). 35) 1 (x − i y) − 2ι(x + i y) + (−1)deg m iu . 36. For n even, the half-spin representations S ± of Spinn (R) are irreducible. For n odd, the spin representation S of Spinn (R) is irreducible. 3 Examples of Irreducibility 45 Proof. Using the standard basis {e j }nj=1 , calculate (e j ± ie j+m )(ek ± iek+m ) = e j ek ± i(e j ek+m + e j+m ek ) − e j+m ek+m for 1 ≤ j, k ≤ m.

Namely, if a group G acts on a space M, then G can be made to act on the space of functions on M (or various generalizations of functions). Begin with the standard two-dimensional representation of SU (2) on C2 where gη is simply left multiplication of matrices for g ∈ SU (2) and η ∈ C2 . Let Vn (C2 ) be the vector space of holomorphic polynomials on C2 that are homogeneous of degree n. A basis for Vn (C2 ) is given by {z 1k z 2n−k | 0 ≤ k ≤ n}, so dim Vn (C2 ) = n + 1. Define an action of SU (2) on Vn (C2 ) by setting (g · P)(η) = P(g −1 η) for g ∈ SU (2), P ∈ Vn (C2 ), and η ∈ C2 .

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