By Yuri Tschinkel (Ed.)

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Topics in Computational Algebra

The most function of those lectures is first to in short survey the elemental con­ nection among the illustration idea of the symmetric workforce Sn and the idea of symmetric features and moment to teach how combinatorial equipment that come up clearly within the concept of symmetric features bring about effective algorithms to precise a number of prod­ ucts of representations of Sn by way of sums of irreducible representations.

Extra resources for Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005

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E, . . , e](2n-times) = [e, . . , e], (2n − 1-times). . ∂(e, . . , e)(2n − 1-times) = 0. This complex is a cellular decomposition of an infinite-dimensional ball, hence contractible. For an arbitrary G, the complex EG contains complexes E g , g ∈ G isomorphic to E e and we can contract all of them independently E g → g ∈ EG where g is a 0-dimensional subcomplex G ⊂ EG. Now we can define a contraction of EG to the simplex ∆ = (g 1 , . . , g n ), where g i runs through G once without repetitions.

It is isomorphic to the image of H ∗ (X , F˜ ) in H ∗ (G al (k(X ), F )). 34 Mathematisches Institut, Seminars, 2005 In fact, this new object is closely related to stable group cohomology, so that the space V L /G and the cohomology groups HS∗ (G, F ) play the role of universal objects in this theory. 2. Let X , F˜ be as above. Then there exists an open subvariety X ⊂ X and regular map X → V L /G for some finite group G, such that a sheaf F˜ is induced from a finite module F over G and the image of H s∗ (X , F˜ ) ⊂ H ∗ (G al (k(X ), F ) is contained in the image of H s∗ (G, F ) ⊂ H ∗ (G al (k(X ), F )).

G r ] = G r as a quotient G r +1 /G by the diagonal. The r -simplices in EG are indexed by G r +1 (before taking the quotient by G) with simplices of dimension n given by ordered sets of vertices (g 1 , g 2 , . . , g n+1 ). We define a natural boundary operator on these simplices via identification with a standard simplex in Rn+1 given by equations x i 0 and x i = 1 with vertices g i given by equations x j = 0, j = i , x i = 1 with induced orientation. The space EG is contractible. e] ∈ ...