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By Krasil'shchik J., Verbovetsky A.

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The most function of those lectures is first to in short survey the basic con­ nection among the illustration concept of the symmetric team Sn and the idea of symmetric capabilities and moment to teach how combinatorial tools that come up certainly within the concept of symmetric capabilities bring about effective algorithms to specific numerous prod­ ucts of representations of Sn by way of sums of irreducible representations.

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And of their representative objects J k (P ), Λk , etc. For example, to any operator ∇ : Q1 → Q2 there correspond operators F ⊗ ∇ : F ⊗A Q1 → F ⊗A Q2 and F∆ ⊗ ∇ : F∆ ⊗A Q1 → F∆ ⊗A Q2 . These natural transformations allow us to lift the theory of linear differential operators from A to F and to restrict the lifted theory to F∆ . They are in parallel to the theory of C-differential operators (see the next section). The natural embeddings sym Diff sym (k) (P, R) ֒→ Diff (k−1) (P, Diff(P, R)) generate the map ℓ : F ⊗A R → F ⊗A Diff(P, R), ϕ → ℓϕ , which is called the universal linearization.

Proof. Let us choose local coordinates in M in such a way that the vectors ∂/∂x1 , . . , ∂/∂xr form a basis in H. Then, in the corresponding special system in J k (π), coordinates along Ann(H) will consist of those functions ujσ , |σ| = k, for which multi-index σ does not contain indices 1, . . , r. Let N ⊂ J k (π) be a maximal integral manifold of the Cartan distribution and θk ∈ N. Then the tangent plane to N at θk is a maximal involutive plane. Let its type be equal to r(θk ). 15. The number tp(N) = max r(θk ) is called the type of the θk ∈N maximal integral manifold N of the Cartan distribution.

4. The Euler operator. Let P and Q be A-modules. Introduce the notation Diff (k) (P, Q) = Diff(P, . . , Diff(P , Q) . . ) k times ∞ k=0 Diff (k) (P, Q). A differential operator ∇ ∈ and set Diff (∗) (P, Q) = Diff (k) (P, Q) satisfying the condition ∇(p1 , . . , pi , pi+1 , . . , pk ) = σ∇(p1 , . . , pi+1 , pi , . . , pk ) is called symmetric, if σ = 1, and skew-symmetric, if σ = −1 for all i. The modules of symmetric and skew-symmetric operators will be denoted alt by Diff sym (k) (P, Q) and Diff (k) (P, Q), respectively.

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