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

ISBN-10: 3642081193

ISBN-13: 9783642081194

ISBN-10: 366203073X

ISBN-13: 9783662030738

The difficulties being solved by way of invariant idea are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is nearly an identical factor, projective geometry. items of linear algebra or, what Invariant idea has a ISO-year background, which has obvious alternating classes of development and stagnation, and alterations within the formula of difficulties, tools of resolution, and fields of software. within the final twenty years invariant thought has skilled a interval of progress, prompted via a prior improvement of the idea of algebraic teams and commutative algebra. it's now considered as a department of the idea of algebraic transformation teams (and below a broader interpretation might be pointed out with this theory). we'll freely use the speculation of algebraic teams, an exposition of that are chanced on, for instance, within the first article of the current quantity. we'll additionally suppose the reader knows the elemental innovations 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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Similar data for R v are obtained by replacing the roots 2ei by ei (1 ~ i ~ n). (d) G = S02n+l (char(k) =I 2, n > 2). The root datum is the dual of the one of the previous example. (e) G = S02n (char(k) =I 2, n>2). We now obtain the root datum (X, R, Xv, RV) with X = Xv = lln, R = R V = {±ei ± ejli =lj}. A system of positive roots is {ei±e)1~i

Is a sum of positive roots. > I. l) a fixed line exists for any rational representation ifJ. The essential point of (i) is the uniqueness for ifJ irreducible. 1 E X with the property of (ii) is said to be dominant. This dominant weight is the highest weight of the irreducible representation ifJ. It follows from (iii) that it is unique (B and T being fixed). Theorem. 1. It is unique up to isomorphism. For details about the preceding results see [Hu, Ch. XI]. Let Q c X be the root lattice and define the weight lattice P c V = X ® lR by P = {v E Q ® lRl (v, RV) c Z}.

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. 3. Maximal Tori and Cartan Subgroups. 2. Theorem. Two maximal tori of G are conjugate. This follows from the conjugacy of Borel subgroups and the conjugacy of maximal tori in solvable groups.

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

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