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Algebraic Geometry IV: Linear Algebraic Groups Invariant by T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich

By T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

The difficulties being solved through invariant thought are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is nearly a similar factor, projective geometry. items of linear algebra or, what Invariant idea has a ISO-year heritage, which has noticeable alternating classes of development and stagnation, and alterations within the formula of difficulties, tools of resolution, and fields of software. within the final 20 years invariant conception has skilled a interval of development, encouraged by way of a prior improvement of the speculation of algebraic teams and commutative algebra. it truly is now seen as a department of the idea of algebraic transformation teams (and less than a broader interpretation will be pointed out with this theory). we'll freely use the idea of algebraic teams, an exposition of that are came across, for instance, within the first article of the current quantity. we'll additionally suppose the reader knows the fundamental strategies and least difficult theorems of commutative algebra and algebraic geometry; whilst deeper effects are wanted, we'll cite them within the textual content or supply appropriate references.

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Let X = G/B, the flag manifold of G and denote for W by Xw the image of C(w) in X. This is a Bruhat cell. The closure Sw = Xw is a Schubert variety. These are interesting projective varieties, in general singular. 2 then imply: WE Proposition. (i) X is the disjoint union of the X w , WE W; (ii) Xw is a locally closed subvariety of X, isomorphic to k1(W); (iii) Sw = Uw'';;w Xw" In particular, X = UW€ W Xw is a "paving of X by affine spaces". One can view the Bruhat cells in X as the B-orbits (or the V-orbits) in X, the groups acting via left translations.

It operates faithfully on the torus T. 4. Examples (a) G = GL n. Let T be the subgroup of diagonal matrices. Then clearly ZG(T) = T, which shows that T is a maximal torus, coinciding with its Cartan subgroup. The normalizer N is the subgroup whose matrices have in each row and each column only one non-zero entry. The Weyl group is isomorphic to the symmetric group Sn' the operation on T corresponding to permutation of diagonal elements. The intersection T (\ SLn is a maximal torus in SLn and its normalizer is N n SL n.

Z/(2) C. I----+--i •••••• ~ Z/(2) D. I----+--i •••••• ! E6 I I E7 I I I Es I I I ~ I 1 I Z/(2) EEl Z/(2), n even Z/(4), n odd I I 1 Z/(3) Z/(2) I I I 1 F4 I---t:¢I---i I G2 ~ 1 Then C = k 2 EB V EB V v, the multiplication being given by (a, b, x, xV)(a l , bl , XI' x{) = (aa l - XV), aX I + blx + XV /\ x{, alx v + bx{ + X /\ xd, for a, b, a I' bl E k, x, XI E V, X v , X{ E V v • Let G be the automorphism group of C, this is a linear algebraic group. e. does not stabilize a non-trivial subspace) and using the following lemma.

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