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The most goal of those lectures is first to in brief survey the basic con­ nection among the illustration concept of the symmetric staff Sn and the idea of symmetric features and moment to teach how combinatorial tools that come up certainly within the concept of symmetric capabilities bring about effective algorithms to precise a variety of prod­ ucts of representations of Sn by way of sums of irreducible representations.

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Let L be a Lie algebra over K, and let S be a subspace of L. 3 {; € s 1 [;. L 1 c s} . Using this, we define inductively a descending sequence {Di,(S)}p_ >O of subspaces of S, by (1. 6) for convenience, we set Di,(S} = L, for p < 0 . We record the following properties of the derived subspace: Proposition 1. 2.. Let L be a Lie algebra, and let S be a subspace of L. (i) S. The derived subspace DL(S) is a subalgebra of L contained in Moreover, if S is a subalgebra of L, then DL(S) is an ideal of S.

Subalgebra of L. (Z; of L by subalgebras, with for p < 0 , M, LP = Di,(M) Then, the filtration {LP} "O €Hu for p > 0 . endows L with a structure of filtered Lie 27 algebra, that is, if p s; q, then (1. 8) and, for all p, q e. Z:, (1. 9) Furthermore, if L is a linearly compact Lie algebra, and M is an open (resp. closed) subalgebra of L, then each of the subalgebras LP is open (resp. closed) in L, for all p e. Z:. Proof: The first inclusion (1. 8) is obvious. by induction on p + q; the case p + q < 0 is trivial, From the definition of the filtration {Lr}r e.

Let V be a finite-dimensional vector space over K. Recall that, for any vector space W over K, we have defined a natural structure of * S(V)-module on the tensor product W IC\ ~K S(V ) ; if vE V ov for multiplication by v in the module W@K S(V*). G = © Gp' with = S 1 (V), we write Suppose that Gp c W(8)K sP(v'~) ' pEZ is a graded S(V)-submodule of W(8)K S(V,~). For all pEZ, there is a natural mapping such that, for all a E G if p > 0, p and vE V, the mapping 6 is injective. Assume that dim(V) = n; we write Aq(V1' ) for the q-th exterior power of v'~, for q ~ 0.