Showing papers on "Reductive group published in 1970"
TL;DR: In this article, the question of birational equivalence of linear algebraic groups is investigated and necessary conditions for rationality of a group over the field of definition are derived, which in application to algebraic tori are fairly delicate.
Abstract: This paper investigates the question of birational equivalence of linear algebraic groups. Certain necessary conditions are derived for rationality of a group over the field of definition, which in application to algebraic tori are fairly delicate. In particular, it is shown that the condition is a necessary criterion of rationality for tori and for semisimple groups defined over an algebraic number field.
64 citations
TL;DR: For the regular action of a semisimple irreducible algebraic group G on an affine space, the existence of a closed orbit of maximum dimension is equivalent to an invariant open set at any point of which the stationary subgroup is reductive.
Abstract: In this work it is proved that, for the regular action of a semisimple irreducible algebraic group G on an affine space, the existence of a closed orbit of maximum dimension is equivalent to the existence of an invariant open set at any point of which the stationary subgroup is reductive. This result is established for the action of G on manifolds of a special type (the so-called factorial manifolds). There are given several other conditions equivalent to the existence of a closed orbit of maximum dimension for the action of G on an arbitrary affine manifold.
50 citations
TL;DR: In this article, the question of whether the ideal of m+1 by m + 1 minors of an r by 5 matrix is perfect if the grade is as large as possible, (r −m)(s −m) for the special cases m = 0, 1, and r −1 (r^s) is closed.
Abstract: In recent years several authors [ l ] , [2], [4], [13], [17], [18] have studied the special homological properties of ideals generated by the subdeterminants of a matrix or \"determinantal\" ideals. The question of whether the ideal of m + 1 by m + 1 minors of an r by 5 matrix is perfect if the grade is as large as possible, (r—m)(s — m), has remained open, although the special cases m = 0, 1, and r—1 (r^s) are known. The general result is Corollary 4 of Theorem 1. For purposes of the induction argument used to prove the theorem it is necessary to consider a larger class of ideals somewhat complicated to describe.
26 citations
22 citations
9 citations
01 Mar 1970
TL;DR: In this article, the Clifford algebra of a quaternion skew-hermitian form H over a sesquilinear form on a finite-dimensional right vector space over Z is described.
Abstract: Introduction. Let K be an infinite perfect field of characteristic different from 2, let e3 be a quaternion division algebra over K, and let tt-denote the canonical involution of the first kind on Z. Let V be a finite-dimensional right vector space over Z. A quaternion skew-hermitian form H over Z is a sesquilinear form on VX V, i.e., H is a map from VX V to Z such that (i) H(x, Yl+Y2) =H(x, yi) +H(x, Y2) and H(x, yax) =H(x, y)a for all x, y, Y1, y2 in V and ai in ; (ii) H(x, y) = -H(y, x) for all x, y in V. Let {xl, . . *, x.) be a basis for V over Q. We say that H is nondegenerate if the reduced norm in Mn(Z) of the matrix (H(xi, xj)) is not zero. Associated to such a nondegenerate form H are 3 invariants, the dimension of V over 0, dime(V), the discriminant of H, b(H), and the Clifford algebra of H, G. Let G be a simply connected semisimple algebraic group (in some GL(m, X)) which is defined over K and let p:G-?GL(V/Z) be an absolutely irreducible representation of G defined over K into the group of all nonsingular Z-linear endomorphisms of V. We shall assume that there is a nondegenerate quaternion skew-hermitian form H on V which is invariant with respect to p (G). The purpose of this paper is to describe the Clifford algebra of the invariant form H in terms of p, G, and the Steinberg group associated to G. In a previous paper, we have described dimsz(V) and b(H) in such a way and have indicated how representations such as p arise [2, Theorem I.2 ]. The invariant -y(G) plays an important role in our study and so we recall some of its properties in ?1. In 2, we define the invariant Y using the representation p. Jacobson first constructed the Clifford algebra of a quaternion skew-hermitian form [4]. In this paper, however, we shall follow a method due to Satake [6]. We give some examples in ?3 with special emphasis on the case where G is absolutely simple.
4 citations
2 citations
TL;DR: In this article, the structure of Lie groups SL(n, C) · R(n2) is studied from a mathematical point of view, i.e., their structure and unitary irreducible representations.
Abstract: The family of Lie groups SL(n, C) · R(n2) is studied from the mathematical point of view, i.e., their structure and unitary irreducible representations. The first steps in the study of representations are based on the Wigner‐Mackey‐Bruhat method. The homogeneous spaces and little groups are determined by introducing a (generally) pseudo‐Hermitian metric on vector spaces of finite dimension. However, the presence of null vectors does not allow this approach to be an exhaustive one; there are cases where the stabilizer is a semidirect product of a reductive group by a nilpotent one. In such cases, results are available if the reductive subgroup is compact, but it seems that these results can be generalized. The connection is also examined, both in structure and in representations, between this family and creation‐annihilation operators. A complete list of inequivalent classes of representations is given in terms of orbits and little groups for n = 2, 3, 4, 6.