Bruhat Decompositions Research Articles

Tenfold way for holography: AdS/CFT and beyond

The main purpose of this paper is to lay the foundations of generalizing the AdS/CFT (holography) idea beyond the conformal setting. The main tool is to find suitable realizations of the bulk and boundary via group theory. We use all ten families of classical real semisimple Lie groups [Formula: see text] and Lie algebras [Formula: see text]. For this are used several group and algebra decompositions: the global Iwasawa decomposition and the local Bruhat and Sekiguchi-like decompositions. The same analysis is applied to the exceptional real semisimple Lie algebras.

Subgroups of Chevalley groups of types $B_l$ and $C_l$ containing the group over a subring, and corresponding carpets

We continue study of subgroups of a Chevalley group $G_P(\Phi,R)$ over a ring $R$ with a root system $\Phi$ and a weight lattice $P$, containing the elementary subgroup $E_P(\Phi,K)$ over a subring $K$ of $R$. Recently A. Bak and A. Stepanov considered the symplectic case (i. e. the case of simply connected group of type $\Phi=C_l$) in characteristic 2. In this article we extend their result for groups with arbitrary weight lattice of types $B_l$ and $C_l$. Similarly to the work of Nuzhin that handles the case of an algebraic extension $R$ of a nonperfect field $K$ of bad characteristic, we use in the description a special kind of carpet subgroups. In the second half of the article we study Bruhat and Gauss decompositions for these carpet subgroups.

DECOMPOSITION THEOREMS FOR AUTOMORPHISM GROUPS OF TREES

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

Let G be a finite reductive group defined over $\mathbb {F}_{q}$ , with q a power of a prime p. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of G into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants with respect to a standard basis of the endomorphism algebra of a Gelfand-Graev representation of G when G = PGL3(q) for an arbitrary prime p, and when G = SO5(q) for p odd.

Affine Grassmannians of group schemes and exotic principal bundles over ${\Bbb A}^1$

Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.

Positroid varieties: juggling and geometry

AbstractWhile the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

Linear groups over general rings. I. Generalities

This paper is the first part of a systematic survey on the structure of classical groups over general rings. We intend to cover various proofs of the main structure theorems, commutator formulas, finiteness and stability conditions, stability and prestability theorems, the nilpotency of K 1, the centrality of K 2, automorphisms and homomorphisms, etc. This first part covers background material such as one-sided inverse, elementary transformations, definitions of obvious subgroups, Bruhat and Gaus decompositions, relative subgroups, finitary phenomena, and transvections. Bibliography: 674 titles.

Mod p classification of Shimura F -crystals

Let k be an algebraically closed field of positive characteristic p. We first classify the D -truncations mod p of Shimura F -crystals over k and then we study stratifications defined by inner isomorphism classes of these D -truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat F -decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over k (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Extending the Bruhat order and the length function from the Weyl group to the Weyl monoid

For a symmetrizable Kac–Moody algebra the category of admissible representations of the category O is an analogue of the category of finite-dimensional representations of a semisimple Lie algebra. The monoid which is associated to this category and the category of restricted duals by a Tannaka–Krein reconstruction contains the Kac–Moody group as open dense unit group. It has similar properties as a reductive algebraic monoid. In particular, there are Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid [C. Mokler, An analogue of a reductive algebraic monoid, whose unit group is a Kac–Moody group, arXiv: math.AG/0204246, Mem. Amer. Math. Soc., in press]. We determine the closure relations of the Bruhat and Birkhoff cells, which give extensions of the Bruhat order from the Weyl group to the Weyl monoid. We show that the Bruhat and Birkhoff cells are irreducible and principal open in their closures. We give product decompositions of the Bruhat and Birkhoff cells. We define extended length functions, which are compatible with the extended Bruhat orders. We show a generalization of some of the Tits axioms for twin BN-pairs.

Applications of Bruhat decompositions to complex hyperbolic geometry

The double coset space AΛ (n, ℂ) / U (n − 1, 1) is studied, where A consists of the diagonal matrices in GL (n, ℂ). This space naturally arises in the harmonic analysis on the hermitian symmetric space GL (n, ℂ) / U (n − 1, 1). It is shown here that these double cosets also represent a class of basic invariants related to complex hyperbolic geometry. An algebraic parametrization for the double cosets is given and it is shown how this may be used to conveniently compute the geometric invariants.

Connected components of real double Bruhat cells

Double Bruhat cells in a semisimple group are intersections of cells in two Bruhat decompositions corresponding to two opposite Borel subgroups. They form a geometric framework for the study of total positivity in semisimple groups; they are also closely related to symplectic leaves in the corresponding Poisson-Lie groups. The term “cells” might be misleading because their topology can be quite nontrivial. As a first step towards understanding this topology, we enumerate the connected components of real double Bruhat cells. This result extends (from the simply-laced case to the general one) and proves the conjecture made in a joint work with B. Shapiro, M. Shapiro, and A. Vainstein; it also extends earlier work by B. Shapiro, M. Shapiro, A. Vainstein, and K. Rietsch.

Double Bruhat cells and total positivity

We study the totally nonnegative variety G ≥ 0 G_{\ge 0} in a semisimple algebraic group G G . These varieties were introduced by G. Lusztig, and include as a special case the variety of unimodular matrices of a given order whose all minors are nonnegative. The geometric framework for our study is provided by intersecting G ≥ 0 G_{\ge 0} with double Bruhat cells (intersections of cells of the two Bruhat decompositions of G G with respect to opposite Borel subgroups).

Matrix Bruhat decompositions with a remark on the QR (GR) algorithm

In a simple and systematic way we present matrix Bruhat decompositions of two kinds: basic and modified. We show that it is the modified Bruhat decomposition that governs the eigenvalue disorder in the QR (GR) algorithm. This paper can be considered as a commentary on a previous observation about the QR algorithm made by Wilkinson.

Structural analysis and elementary representations of SL(4, R) and GL(4, R) and their covering groups

The structure of the groups SL(4,R) and GL(4,R) and their universal cover− ing groups SL(4,R) and GL(4,R), respectively, and Lie algebras sl(4,R) and gl(4,R), respectively, are studied. The parabolic subgroups and subalgebras are identified and the cuspidal parabolic subgroups singled out. The Iwasawa and Bruhat decompositions are given explicitly. All elementary representations (ER) of SL(4,R) are explicitly given in two equivalent realizations. Using the preceding detailed structural analysis the SL(4,R) constructions are used for the explicit realization of all ER of SL(4,R), GL(4,R), and GL(4,R). The results shall be applied (among other things) elsewhere for the construction of all irreducible representations of the above groups.