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 via invariant conception are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is sort of an identical factor, projective geometry. gadgets of linear algebra or, what Invariant thought has a ISO-year background, which has noticeable alternating classes of development and stagnation, and adjustments within the formula of difficulties, equipment of answer, and fields of software. within the final twenty years invariant concept has skilled a interval of development, inspired through a prior improvement of the speculation of algebraic teams and commutative algebra. it truly is now considered as a department of the idea of algebraic transformation teams (and lower than a broader interpretation may 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 imagine the reader understands the elemental 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 offer appropriate references.

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

**Sample text**

In fact, (w, S) is a Coxeter system. For these facts see [B02, Ch. VI]. 4. Examples (a) G = GL .. Let T be the torus of diagonal matrices. Define characters O(ij of Tby O(ij(diag(t l " ' " tn)) = titj-l (1 ~ i, j ~ n, i =I j). Then R is the set of O(ij' The corresponding groups G~ are non-solvable, so R = R. The groups G~jKer 0( are all isomorphic to PGL 2. Now O(il is the homomorphism k* -> G with O(il(t) a diagonal matrix with i lh entry t,ph entry t- l and all other diagonal entries 1. The root datum IJ'(G, T) = (X, R, Xv, RV) is given by X = XV = lln with standard pairing, R = R v = {e i - ejl i =I j}, where (eJ is the canonical basis.

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. The dimension of a maximal torus of G is the rank of G. Let T be a maximal torus, C = ZG(T) its centralizer and N = NG(T) its normalizer. C is called a Cartan subgroup. Lemma. (i) C is a normal subgroup of N and the quotient W = NIC is finite; (ii) The identity component CO is nilpotent; (iii) Let X = {(xCO, y) E GICo X Glx- 1 yx E CO}.

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}. 6. The fundamental weights 1t~ E P (a E D) are defined by for a, f3 E D. 1o,RV) =0. We define P E Vby 2p Then p = L aeR+ a. 2. 2 resp. 4. 4(a).

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

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