Quillen–Suslin theory for classical groups: Revisited over graded rings

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters.

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.

Classical groups over finite fields, including symplectic groups, unitary groups and orthogonal groups, are finite simple groups of crucial importance. The structure and order of classical groups are the basic problems in group theory. Geometry of classical groups has been found extremely useful in many fields, such as combinatorial mathematics, coding and cryptographic theory. In this paper, a class of quasi-classical groups are introduced, which partially generalize the concepts of symplectic groups and orthogonal groups. We determine the structure and order of quasi-classical groups. As an application, we discuss the relationship between quasi-classical groups and self-dual normal bases over finite fields.

We classify the irreducible complex characters of the symplectic groups Sp 2n (q) and the orthogonal groups , Spin 2n+1(q) of degrees up to the bound D, where D = (q n − 1)q 4n−10/2 for symplectic groups, D = q 4n−8 for orthogonal groups in odd dimension, and D = q 4n−10 for orthogonal groups in even dimension.

We are concerned with finite-dimensional classical groups over arbitrary commutative fields. In an orthogonal group a Siegel transvection, that is, an element centralizing l⊥ for some totally singular 2-dimensional subspace l, plays the same role as a transvection in the linear, symplectic or unitary groups. The Main Theorem of this paper describes the possible embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections.

By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.

Bulk Witten indices and the number of normalizable ground states in supersymmetric quantum mechanics of orthogonal, symplectic and exceptional groups

Maximal Overgroups of Singer Elements in Classical Groups

We consider twisted conjugacy classes and the R ∞-property for classical linear groups. In particular, it is stated that the general linear group GL n (K) and the special linear group SL n (K), where $$ n \geqslant 3 $$ , possess the R ∞-property if either K is an infinite integral domain with trivial automorphism group or K is an integral domain containing a subring of integers, whose automorphism group Aut(K) is finite. By an integral domain we mean a commutative ring with identity which has no zero divisors.

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

Let Γ be chosen from the orthogonal group O n according to Haar measure, and let A be an n x n real matrix with non-random entries satisfying TrAA* = n.We show that TrAΓ converges in distribution to a standard normal random variable as n -> oo uniformly in A. This extends a theorem of E. Borel.The result is applied to show that if entries βi, ,βk n are selected from Γ where k n -• oo as n -• oo, then J T~ Ί2\ k =i^ A, 0 < ί < 1 converges to Brownian motion.Partial results in this direction are obtained for the unitary and symplectic groups.

The purpose of this work is to describe some new connections between the characteristic-free representation theories of the symplectic group and the corresponding general linear group (Theorem 2.2 and Theorem 2.6).

Bireflectionality of the weak orthogonal and the weak symplectic groups

Let be a finite-dimensional complex vector space.Assume that is a direct sum of subspaces each of which is equipped with a nondegenerate symmetric or skew-symmetric bilinear form.In this paper, we introduce a stratification of the Grassmannian Gr ( ) related to the action of the appropriate product of orthogonal and symplectic groups, and we study the topology of this stratification.The main results involve sheaves with coefficients in a field of characteristic other than 2. We prove that there are "enough" parity sheaves, and that the hypercohomology of each parity sheaf also satisfies a parity-vanishing property.This situation arises in the following context: let be a nilpotent element in the Lie algebra of either = Sp (C) or = SO (C), and let = ker C .Our stratification of Gr ( ) is preserved by the centralizer , and we expect our results to have applications in Springer theory for classical groups.