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

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

The difficulties being solved through invariant concept are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of assorted is nearly a similar factor, projective geometry. gadgets of linear algebra or, what Invariant conception has a ISO-year background, which has noticeable alternating sessions of progress and stagnation, and alterations within the formula of difficulties, tools of answer, and fields of software. within the final twenty years invariant conception has skilled a interval of progress, prompted through a prior improvement of the idea of algebraic teams and commutative algebra. it truly is now seen as a department of the speculation of algebraic transformation teams (and less than a broader interpretation may be pointed out with this theory). we are going to freely use the speculation of algebraic teams, an exposition of which might be came upon, for instance, within the first article of the current quantity. we are going to additionally think the reader knows the elemental innovations and least difficult theorems of commutative algebra and algebraic geometry; while deeper effects are wanted, we'll cite them within the textual content or offer appropriate references.

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Extra resources for Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory

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The corresponding basis is D = rei - ei + l ll ~ i ~ n - I}. (b) G = SL n, T again being the torus of diagonal matrices. Now the root datum of(G, T) is (Xl' R l , Xt, Rl), where (X, Xv being as in (a)) Xl = Xjll(e l + ... + en), Xl = {(Xl, ... ,Xn ) E Xv li~ Xi = o}. If rr: X -> Xl is the canonical map then Rl = rrR and R{ = R v c Xl (R and R v as in (a)). For G = PGL n the root datum is the dual of the one for SL n. (c) G = SP2m (m ~ 2). 3(d) the root datum (X, R, X v, R V), where again X = X v = lln andR={±2e i , ±ei ± ejll ~ i,j ~ n,i =lj},RV = {±ei, ±ei ± ejll ~ i, j ~ n, i =I j}.

One can take for g an element in T such that 27 I. Linear Algebraic Groups ZG(g) = ZG(T), Such elements exist, for an arbitrary subtorus T of G (to prove this it suffices to consider the case G = GLn). The finite group W is the Weyl group of (G, T). 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 notion of isomorphism of root data being defined in the obvious manner, it is clear from the conjugacy of maximal tori that the root datum is determined by G up to isomorphism. Let (G, T) and (G', T) be two pairs as in the preceding paragraph and let

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