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

The difficulties being solved by way of invariant concept are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is sort of an identical factor, projective geometry. items of linear algebra or, what Invariant concept has a ISO-year heritage, which has obvious alternating sessions of progress and stagnation, and adjustments within the formula of difficulties, tools of resolution, and fields of software. within the final 20 years invariant concept has skilled a interval of progress, motivated by way of a prior improvement of the idea of algebraic teams and commutative algebra. it really is now considered as a department of the idea of algebraic transformation teams (and less than a broader interpretation might be pointed out with this theory). we are going to freely use the idea of algebraic teams, an exposition of which are stumbled on, for instance, within the first article of the current quantity. we'll additionally think the reader is aware the elemental thoughts and easiest theorems of commutative algebra and algebraic geometry; while deeper effects are wanted, we'll cite them within the textual content or supply appropriate references.

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**Additional resources for Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory**

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The algebraic closure of Fin k is denoted by F and the separable closure of F in k by Fs (so Fs is the set of elements of F which are separably algebraic over F). Then Fs is a Galois extension of F, whose Galois group r = Gal(Fs/F) is a profinite group. We put p = char k. § 1. Recollections from Algebraic Geometry We review briefly some known results about algebraic varieties over arbitrary fields. It is not always easy to give accessible references. Several results can be found in [B2]. 1. 1. An F-structure on the k-vector space V is a subspace of V which spans V.

L = LaED mana restricted if 0 ~ m < p. lm i= O. l. Theorem (Steinberg's tensor product theorem). ld(l) ® ... lm)(m). The notations are as above. There is an efficient proof of the theorem using Frobenius kernels, see [J, II. 3]. Example. G = SL 2 . The dominant weights are the natural numbers. If is the p-adic expansion of m then 47 I. 2(c)). We thus have an explicit description of all irreducible representations. The representations insert L«p" - l)p) are the Steinberg representations. Proposition.

2). A subspace W of V is totally isotropic for this bilinear form if (x, y) = 0 for all x, YEW One knows that then dim W ~ m. e. flags (V1' ... , Vm ) with all V; totally isotropic, and dim V; = i. 4). (c) G = SOn (char(k) =f. 2). Now define a symmetric bilinear form on V = k n by (x, Y> = m L (XiYm+i + Xm+iY;) i=l if n = 2m is even, respectively (x, Y> = m L (XiYm+i + xm+iY;) + X~m+1 i=l if n = 2m + 1 is odd. The G is the subgroup of SLn fixing this form. One has again a description of Borel subgroups and parabolic subgroups involving isotropic flags.