Springer, T.A.: Linear Algebraic Groups, Birkhauser, Boston 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by the University of Buenos Aires and CNRS, France. A large part of our computations in Section 4 were performed with Maple. 1. Main results 1.1. Semisimple groups. We begin by introducing general terminology and notation (mostly standard) for semisimple Lie groups and algebras (cf. e.g. [22]) Let g be a semisimple complex Lie algebra of rank r with the Cartan decomposition g = n Gamma Phi h Phi n. Let e i ; h i ; f i , for i = 1; r, be the standard generators of g, and let A = a ij ) be the Cartan matrix. Thus a ij = ff j (h i ) where ff 1 ; ff r 2 h are ....

T. A. Springer, Linear algebraic groups, Progress in Mathematics 9, Birkhauser, 1981.

....scheme X over a field k, which is non singular everywhere and X x is singular. Exercise 3.31. Prove Proposition 3.13. 4. ALGEBRAIC GROUPS We shall define and discuss examples of algebraic groups in this section. Basic ref erences for the theory of algebraic groups are Borel [2] Springer [25], and Humphreys [9] Definition 4.1. 1) A group scheme G over k is a a scheme of finite type over k with a section e: Speck G called the identity, a morphism : W W called the inverse, and a morphism : W xk W G called the product which satisfies the usual group laws. 2) If a group ....

....we point out that if X T(G) the corresponding element O(X) L(G) can be constructed as follows. Let A: A A A be the homomorphism which corresponds to the multiplication (x, y) xy, f A, and A(f) i fi gi. Then define o(x) y) y, xg, It is shown in 3.4 Theorem [2] or in 4.4. 5 Proposition [25] that 0 is indeed the inverse of the natural map L(G)k T(G) From now on, we identify L(G)k with T(G) It is easy to see that if 01, 02 [01, 02] 01 o 02 02 o 01 is again an element of L(G) The operation [ is obviously k linear and [D,D] 0 for all D. It is easy to verify that ....

Springer, T.A. Linear algebraic groups. Birkh/user, Boston, 2nd edition, 1998.

....Theorem. For any ring R and any reduced decomposition i = i 1 ,i 2 , i N) of the permutation w # S n , the map P i : x 1 ,x 2 , x N)## P i 1 (x 1 ) P i 2 (x 2 ) P i N (x N) is a bijection from R N to the Schubert cell Cw # GL n (R) Theorem 1. 1 is known when R is a field (cf. [Sp], Lemma 10.2.6; see also [FZ] Proposition 2.11) The proof we give for an arbitrary ring in Section 2.5 is based on Relations (1.4a b) and on a matrix identity (given in Proposition 2.1) which is of independent interest. Let us illustrate this theorem with the permutation w = # 12345 ....

....element and a unique maximal element, which we describe explicitly. The paper [BFZ] by Berenstein, Fomin, and Zelevinsky was a source of inspiration to us, despite the fact that we deal with di#erent questions and di#erent parametrizations. We thank Robert Bedard for pointing out Lemma 10.2. 6 of [Sp] and for showing us the experimental data he collected on the posets C(w 0 ) of Section 6. 2. A Matrix Identity Let w be a permutation of the set 1, n and Mw be its permutation matrix. We complete Mw with stars at all places (w(k) j) such that j kand w(j) w(k) As we saw in the ....

T. A. Springer, Linear algebraic groups, Progr. In Math., vol. 9, Birkhauser, Boston, 1981.

.... l Q (C) be the Weyl gr oup for LQ (x 0 ) For each w # WQ set # (w) # # # w 1 # # # Then # (w) #(w) is the length of w and the set W 1 Q = w # WQ # (w) # #(H, NQ (C) consists of the unique element of minimal length fr# m each of the cosets WQx # WQ W ([Sp] 10.2, V] 3.2.1) Let # : G # GL(E) be an algebr# : ir#: ducible r#ev q: tation of G with highest weight #. Let V be the ir# ducible LQ (C) module with highest weight . Then, Kostant s theor em ( K] 5.14, V] 3.2.16) states that, as ar epr#v## tation of LQ , the Lie algebr# cohomology ....

T. A. Springer, Linear algebraic groups., Birkhauser Boston, Boston MA, 1981.

....the topological type or topological class of . It is functorial with respect to extensions G Gamma Gamma Gamma G 0 of the structure group. Let us have a closer look at 1 (G) For details on the structure theory of reductive groups we refer, here and in the sequel, to the textbooks [3] 15] [27]. Let H = G; G) denote the semisimple derived subgroup of G. The corresponding factor commutator group G=H is then a torus A, say of dimension s. Since 2 (A) is trivial, we obtain an exact sequence 0 Gamma Gamma Gamma 1 (H) Gamma Gamma Gamma 1 (G) Gamma Gamma Gamma 1 (A) ....

T. A. Springer, Linear algebraic groups, 2nd ed. Progress in Math. 9, Birkhauser, Basel, Boston 1998.

.... (y 0 Delta y) x [19 :22, flip] 28 (x Delta y) Delta y 0 = x [21 5] 35,34 (x Delta y) Delta z = x Delta (y Delta z) 16 7 :27,27,17] 44,43 x Delta (y Delta y 0 ) x [28 :35] 45 B Delta B 0 = A Delta A 0 2 [2 :44,35 :1,1] 51 x Delta x 0 = y Delta y 0 [43 3] 52 2 [51,45] Theorem #DUAL GT 8. The set 8 : 1) x Delta (x 0 Delta y) y (2) y Delta x 0 ) Delta x = y (3) x Delta x 0 = y 0 Delta y (4) x Delta (y Delta z) Delta y) Delta u = x Delta (y Delta ( z Delta y) Delta u) 9 = is an independent ....

T. A. Springer. Linear Algebraic Groups. Birkhauser, Boston, 1980.

....and symmetry of polynomials under linear group actions. The computational part of this project leads me into the exciting area of the Grobner basis theory, multivariable resultants, as well as some other areas of commutative algebra and algebraic geometry [Sturmfels98] Cox Little OShea96] [Springer98]. It would be worthwhile to reformulate the method of moving frames in the language of algebraic geometry for actions of algebraic groups on varieties, and investigate how it can be applied to the problems of symmetries and equivalence. I am also interested in applications of geometric invariants ....

Springer, T. A., Linear algebraic groups, 2nd ed,. Progress in Mathematics, 9. Birkhuser Boston, Inc., Boston, MA, 1998.

....In this section we briefly recall the notation used in the sequel and the notion of a basic representation. All background information on root systems, Lie algebras, algebraic groups, Chevalley groups and representations may be found in [Bo1] Bo2] Bu1] Bu2] Ca] H e] H1] H3] Ja] [Sp1], Sp2] Sb] 1 o . Root systems and Weyl groups. Let Phi be a reduced irreducible root system of rank l, Q( Phi) be the root lattice, P ( Phi) be the weight lattice. Fix an order on Phi, and let Phi , Phi Gamma and Pi = fff 1 ; ff l g be the sets of positive, negative and ....

Springer T.A, Linear algebraic groups. 2nd ed., Birkhauser, Boston et al., 1981.

....subgroup of the connected linear algebraic group G. Clearly, the derivative d of the embedding : H G at the identity of H is the embedding L(H) L(G) of the Lie algebra of H in the Lie algebra of G. As D linearly spans L, we have L(H) L = L(G) and so d is surjective. By Theorem 3.2. 21 of [13], this implies that is dominant, that is, H is dense in G. But H is closed as well, so H = G. This establishes that G is generated by at most t(L) long root subgroups. The converse is handled by the previous lemma. 2 ....

T. A. Springer, Linear Algebraic Groups, Birkhauser, 1984.

....that F is also a pure algebraically closed field. In out case this implies that the group G is definable in F . Since G is definable in F , by theorem 5.6, G is definably isomorphic to an algebraic group over F . It is easy to see that G is one dimensional and connected. It is well known (see [Sp]) that any such group is either an elliptic curve or isomorphic to the additive or multiplicative group of the field. Since G is torsion free it must be isomorphic to the additve group of the F . In particular in A, there is a definable isomorphism between K and F . We identify F and K ....

T.A. Springer, Linear Algebraic Groups, Birkhauser (1980).

....g is minimal as above, that r dim(A) To see this, one proceeds by induction on r, the case r = 1 being clear. By the minimality assumption, A 2 = a 1 ; a r 1 has dimension at most N 1, for otherwise we could throw away a r . As the product of algebraic groups is algebraic (see, e.g. [Sp], p.31) the Zariski closure of a product is the product of Zariski closures. Hence fa 1 ; a r 1 g is minimal for A 2 , so by induction r 1 N 1. Hence r N . Since in an algebraic group the centralizer of a single element is clearly algebraic, and the intersection of algebraic groups ....

T.A. Springer, Linear Algebraic Groups, Progress in Math., Vol. 9, Birkhauser, 1981.

.... LQ (x 0 ) For each w 2 WQ set Phi (w) fff 2 Phi jw Gamma1 ff 2 Phi Gamma g Then j Phi (w)j = w) is the length of w and the set W 1 Q = fw 2 WQ j Phi (w) ae Phi(H; NQ (C ) g consists of the unique element of minimal length from each of the cosets WQx 2 WQ nW ([Sp] x10.2, V] x3.2.1) Let : G GL(E) be an algebraic irreducible representation of G with highest weight . Let V be the irreducible LQ (C ) module with highest weight . Then, Kostant s theorem ( K] x5.14, V] x3.2.16) states that, as a representation of LQ , the Lie algebra cohomology H (NQ ; ....

T. A. Springer, Linear algebraic groups., Birkhauser Boston, Boston MA, 1981.

....for a number of helpful suggestions. 1. Preliminaries and Recollections In this section we set the notations and recall a few results from [23] 16] and [17] We use as our basic reference for reductive groups the papers of Borel and Tits [5, 6] and also the books of Humphreys [24] and Springer [34]. We shall follow their notations and terminology. 1.1. Given an algebraic group G, the identity component is denoted by G 0 . We use L(G) or #, the corresponding lower case German letter for the Lie algebra of G.IfH is a subset of G, N G (H) resp. Z G (H) is the normalizer (resp. ....

T. A. Springer, Linear algebraic groups, Progr. Math., vol. 9, Birkh auser, Boston/Basel/Stuttgart, 1981.

....of the orbits, which is geared more toward the conjugacy classes of # stable maximal (quasi) k split tori. We will also prove a number of additional results. Our basic reference for reductive groups will be the papers of Borel and Tits [1, 2] and also the books of Humphreys [16] and Springer [26]. We shall follow their notations and terminology. 1.1. Notations. Given an algebraic group G, the identity component is denoted by G 0 . We use L(G) resp. #, the corresponding lower case German letter) for the Lie algebra of G.IfH is a subset of G, then we write NG (H) resp. ZG (H) for the ....

T. A. Springer, Linear algebraic groups, Progr. Math., vol. 9, Birkh auser, Boston/Basel/Stuttgart, 1981.

....conjugacy classes. 2. Preliminaries and Recollections In this section we set the notations and recall a few results from [HW93] Hel88] and [Hel91] We use as our basic reference for reductive groups the papers of Borel and Tits [BT65, BT72] and also the books of Humphreys [Hum75] and Springer [Spr81] We shall follow their notations and terminology. 2.1. Given an algebraic group G, the identity component is denoted by G 0 .We use L(G) resp. #, the corresponding lower case German letter) for the Lie algebra ON THE CONJUGACY OF CARTAN SUBSPACES 5 of G.IfS is a subset of G and H a closed ....

T. A. Springer, Linear algebraic groups, Progr. Math., vol. 9, Birkh auser, Boston/Basel/Stuttgart, 1981.

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Springer, T.A.: Linear Algebraic Groups, Birkhauser, Boston 1998.

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T. A. Springer, Linear Algebraic Groups, Progr. Math. 9, Birkha user, Boston, MA, 1981.

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Springer, T. A. Linear algebraic groups, Progr. Math., vol. 9, Birkh auser, Boston/Basel/Stuttgart, 1981. E-mail address:loek@math##h##0#0# E-mail address: lwu@unity.ncsu.edu

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T. A. Springer, Linear algebraic groups, 2nd ed., Birkhauser, Boston, 1998.

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T. Springer, Linear Algebraic Groups, Birkhauser, 1981.

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T. A. Springer, Linear Algebraic Groups, Birkhuser, Boston, 1981.

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T. A. Springer. Linear Algebraic Groups. Birkhauser, Boston, 1980.

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T. A. Springer. Linear Algebraic Groups. Birkhauser, Boston, 1980.

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T.A. Springer, Linear Algebraic Groups, Birkhauser, Boston, 1981.

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T.A. Springer, Linear Algebraic Groups (2nd edition), Birkhauser, 1998.

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