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The most function of those lectures is first to in brief survey the elemental con­ nection among the illustration idea of the symmetric workforce Sn and the idea of symmetric capabilities and moment to teach how combinatorial tools that come up clearly within the thought of symmetric services result in effective algorithms to precise quite a few prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

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We note that the automorphism θ is in the center of Sp(2d, C) and θ is the center of Sp(2d, C). Therefore, Sp(2d, C)/ θ faithfully acts on SF + . We shall prove that Aut (SF + ) ∼ = Sp(2d, C)/ θ . We see that the characters SM (τ ) for M = SF ± , SF(θ )± are mutually distinct. This implies that for any g ∈ Aut (SF + ) and irreducible SF + -module M, the SF + -module (Mg , Y g ( · , z)) with Mg = M and Y g ( · , z) = Y(g(·) , z) is isomorphic to itself because SMg (τ ) = SM (τ ). In particular, for any g ∈ Aut (SF + ), there exists a unique SF + module isomorphism fg : SF − → (SF − )g up to nonzero scalar multiple.

26]). 6 There exists a nonzero constant αg ∈ C such that ( · , · )g = αg ( · , · ). − − − Proof The bilinear forms ( · , · ) and ( · , · )g satisfy that (SF − m , SF n ) = (SF m , SF n )g = 0 if m = n. This implies that the linear maps γ and γg from SF − to the contragredi∞ − ∗ − ent SF + -module (SF − ) = n=1 (SF n ) ⊂ D(SF ) defined by γ (u) = (u, · ) and + γg (u) = (u, · )g respectively are SF -module isomorphisms. Hence γ −1 ◦ γg = αg idSF − for some αg ∈ C − {0}. This proves the proposition.

This implies that u ∈ SF (0) ⊕ SF (1) . Hence (SF) ⊂ SF (0) ⊕ SF (1) . It is clear that SF (0) ⊂ (SF). We claim that (SF) ∩ SF (1) = SF[2d](1) . 1) Let v ∈ SF (1) . For a canonical basis {(ei , f i )} of h, we can find vectors vj ∈ SF (0) (1 ≤ j j ≤ 2d) such that v = di=1 ei(−1) vi + dj=1 f(−1) vd+j . Then for any ψ ∈ h and 1 ≤ i ≤ d, ((ei(−1) ψ)(2) + (ei(−2) ψ)(3) )v = −ψ(0) vd+i , i i ((f(−1) ψ)(2) + (f(−2) ψ)(3) )v = ψ(0) vi . Therefore, if v ∈ (SF), then for each i, vi is annihilated by the action of ψ(0) for any ψ ∈ h.

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