Borel Group Research Articles

Borel edge colorings for finite-dimensional groups

AbstractWe study the potential of Borel asymptotic dimension, a tool introduced recently in [2], to help produce Borel edge colorings of Schreier graphs generated by Borel group actions. We find that it allows us to recover the classical bound of Vizing in certain cases, and also use it to exactly determine the Borel edge chromatic number for free actions of abelian groups.

The Σ-invariants of 𝑆-arithmetic subgroups of Borel groups

Given a Chevalley group G \mathcal {G} of classical type and a Borel subgroup B ⊆ G \mathcal {B} \subseteq \mathcal {G} , we compute the Σ \Sigma -invariants of the S S -arithmetic groups B ( Z [ 1 / N ] ) \mathcal {B}(\mathbb {Z}[1/N]) , where N N is a product of large enough primes. To this end, we let B ( Z [ 1 / N ] ) \mathcal {B}(\mathbb {Z}[1/N]) act on a Euclidean building X X that is given by the product of Bruhat–Tits buildings X p X_p associated to G \mathcal {G} , where p p is a prime dividing N N . In the course of the proof we introduce necessary and sufficient conditions for convex functions on C A T ( 0 ) CAT(0) -spaces to be continuous. We apply these conditions to associate to each simplex at infinity τ ⊂ ∂ ∞ X \tau \subset \partial _\infty X its so-called parabolic building X τ X^{\tau } and to study it from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential n n -connectivity rather than actual n n -connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building Δ \Delta contains an apartment, provided Δ \Delta is thick enough and A u t ( Δ ) Aut(\Delta ) acts chamber transitively on Δ \Delta .

Abelian group actions and hypersmooth equivalence relations

We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.

On the Finiteness length of some soluble linear groups

AbstractGiven a commutative unital ringR, we show that the finiteness length of a groupGis bounded above by the finiteness length of the Borel subgroup of rank one$\textbf {B}_2^{\circ }(R)=\left ( \begin {smallmatrix} * & * \\ 0 & * \end {smallmatrix}\right )\leq \operatorname {\textrm {SL}}_2(R)$wheneverGadmits certainR-representations with metabelian image. Combined with results due to Bestvina–Eskin–Wortman and Gandini, this gives a new proof of (a generalization of) Bux’s equality on the finiteness length ofS-arithmetic Borel groups. We also give an alternative proof of an unpublished theorem due to Strebel, characterizing finite presentability of Abels’ groups$\textbf {A}_n(R) \leq \operatorname {\textrm {GL}}_n(R)$in terms ofnand$\textbf {B}_2^{\circ }(R)$. This generalizes earlier results due to Remeslennikov, Holz, Lyul’ko, Cornulier–Tessera, and points out to a conjecture about the finiteness length of such groups.

GIT quotient of a Bott–Samelson–Demazure–Hansen variety by a maximal torus

Let G be an almost simple, simply connected algebraic group over the field $$\mathbb {C}$$ of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G, and let W be the Weyl group defined by T. The Borel group B determines a subset of simple reflections in W. For w in W, we let $$Z(w,\underline{i})$$ be the Bott–Samelson–Demazure–Hansen variety corresponding to a reduced expression $$\underline{i}$$ of w as a product of these simple reflections. In this article, we study the geometric invariant theoretic quotient of $$Z(w,\underline{i})$$ for the T-linearized ample line bundles.

Minimalist approach to the classification of symmetry protected topological phases

A number of proposals with differing predictions (e.g. Borel group cohomology, oriented cobordism, group supercohomology, spin cobordism, etc) have been made for the classification of symmetry protected topological (SPT) phases. Here we treat various proposals on an equal footing and present rigorous, general results that are independent of which proposal is correct. We do so by formulating a minimalist generalized cohomology hypothesis, which is satisfied by existing proposals and captures essential aspects of SPT classification. From this hypothesis alone, formulas relating classifications in different dimensions and/or protected by different symmetry groups are derived. Our formalism is expected to work for fermionic as well as bosonic phases, Floquet as well as stationary phases, and spatial as well as on-site symmetries. As an application, we predict that the complete classification of 3D bosonic SPT phases with space group symmetry G is , where the H1 term classifies phases beyond the Borel group cohomology proposal.

On the intersection pairing between cycles in SU(p,q)-flag domains and maximally real Schubert varieties

An SU(p,q)-flag domain is an open orbit of the real Lie group SU(p,q) acting on the complex flag manifold associated to its complexification SL(p+q,C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and representation theoretical properties of the flag domain. This article is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. Equivalently, the Borel group contains the AN-factor of some Iwasawa decomposition. We consider the Schubert varieties of this type which are of complementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements. Much of our work is of an algorithmic nature, but, for example in the case of maximal parabolics, i.e. Grassmannians, formulas are derived.

Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups

Let G denote a linear algebraic group over \bold{Q} and K and L two number fields. We establish conditions on the group G , related to the structure of its Borel groups, under which the existence of a group isomorphism G(\bold{A}_{K,f}) \cong G(\bold{A}_{L,f}) over the finite adeles implies that K and L have isomorphic adele rings. Furthermore, if G satisfies these conditions, K or L is a Galois extension of \bold{Q} , and G(\bold{A}_{K,f}) \cong G(\bold{A}_{L,f}) , then K and L are isomorphic as fields. We use this result to show that if for two number fields K and L that are Galois over \bold{Q} , the finite Hecke algebras for \mathrm{GL}(n) (for fixed n \geq 2 ) are isomorphic by an isometry for the L^1 -norm, then the fields K and L are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field, if it is Galois over \bold{Q} .

FIELD OF INVARIANTS OF BORELEAN GROUP OF ADJOINT REPRESENTATION OF GL(n,K)

The paper is devoted to invariant theory problems, in particular to the problem of finding generators of invariant fields in an explicit form. The set of generators is given for invariant field of unitriangular group concerning the ad-joint representation of GL(n, K) group. Moreover, the set of generators of Borel group for the field of invariants is constructed and their algebraic independence is proved. Lie group;adjoint representation;field of invariant;generators of the field of invariants;Borel group;

Matrix divisibility sequences

The plan for the paper is as follows: we shall first introduce the general notion of a matrix divisibility sequence indexed by a semigroup. Then we will see how a faithful representation of the semigroup by endomorphisms of an affine space gives rise to a matrix divisibility sequence, by considering the Jacobian matrices of the endomorphisms. We will show how most of the commonly known divisibility sequences (mentioned brie y above) arise as determinants of matrix divisibility sequences through interesting semigroups of endomorphisms of affine spaces, often associated to a representation of addition in an algebraic group. For example, Lucas sequences are associated to the 2 x 2 Borel group. We also construct the elliptic matrix divisibility sequence that underlies the usual elliptic divisibility sequences, and prove that it has primitive right matrix divisor classes.

Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials

We study the equivariant K-group of the affine flag manifold with respect to the Borel group action. We prove that the structure sheaf of the (infinite-dimensional) Schubert variety in the K-group is represented by a unique polynomial, which we call the affine Grothendieck polynomial

The hypercentral structure of the group of unitriangular automorphisms of a polynomial algebra

We describe the hypercentral structure of the group of unitriangular automorphisms of the polynomial algebra K[x1, …, xn] for n ≥ 3 and an arbitrary field K. This group is sometimes called the Borel group. The description implies the nonlinearity for this group over any field.

Biorthogonal Laurent Polynomials, Toplitz Determinants, Minimal Toda Orbits and Isomonodromic Tau Functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae. For all these classes of polynomials a 2 × 2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo-measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) Toplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.

Compactness of the congruence group of measurable functions in several variables

We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let \(f:\prod\limits_{i = 1}^n {X_i \to Z} \) be a measurable function defined on the product of finitely many standard probability spaces (Xi, \(\mathfrak{B}_i \), μi), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g1, ..., gn) of measure preserving automorphisms of the respective spaces (Xi, \(\mathfrak{B}_i \), μi) such that f(g1 x 1, ..., gnxn) = f(x1, ..., xn) almost everywhere and prove that this group is compact, provided that its “trivial” symmetries are factored out. As a consequence, we are able to characterize all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which was solved by the first author but is interesting in itself. Bibliography: 5 titles.

Coordinatewise decomposition, Borel cohomology, and invariant measures

Given Polish spaces X and Y and a Borel set S ⊆ X × Y with countable sections, we describe the circumstances under which a Borel function f : S → R is of the form f(x, y) = u(x) + v(y), where u : X → R and v : Y → R are Borel. This turns out to be a special case of the problem of determining whether a real-valued Borel cocycle on a countable Borel equivalence relation is a coboundary. We use several Glimm-Effros style dichotomies to give a solution to this problem in terms of certain σ-finite measures on the underlying space. The main new technical ingredient is a characterization of the existence of type III measures of a given cocycle. Suppose that S ⊆ X×Y and G is a group. A coordinatewise decomposition of a function f : S → G is a pair (u, v), where u : X → G, v : Y → G, and ∀(x, y) ∈ S (f(x, y) = u(x)v(y)). If X and Y are Polish spaces, G is a standard Borel group, and u and v are Borel, then we say that (u, v) is a Borel coordinatewise decomposition of f . Our main goal here is to show that when S is a Borel set with countable sections, f : S → G is Borel, and G = 〈R,+〉, the existence of a Borel coordinatewise decomposition can be characterized in terms of certain σ-finite measures on the disjoint union of X and Y (by a measure on a Polish space, we shall mean always a measure on its Borel subsets). Before getting to this, however, we consider first the existence of coordinatewise decompositions, without imposing any definability restrictions. ∗The author was supported in part by NSF VIGRE Grant DMS-0502315.

Quasiregular representations of the infinite-dimensional Borel group

The notion of quasiregular (Representation of Lie groups, Nauka, Moscow, 1983) or geometric (Grundlehren der Mathematischen Wissenschaften, Band 220, Springer, Berlin, New York, 1976; Encyclopaedia of Mathematical Science, Vol. 22, Springer, Berlin, 1994, pp. 1–156) representation is well known for locally compact groups. In the present work an analog of the quasiregular representation for the solvable infinite-dimensional Borel group G=Bor 0 N is constructed and a criterion of irreducibility of the constructed representations is presented. This construction uses G-quasi-invariant Gaussian measures on some G-spaces X and extends the method used in Kosyak (Funktsional. Anal. i Priložhen 37 (2003) 78–81) for the construction of the quasiregular representations as applied to the nilpotent infinite-dimensional group B 0 N .

Invariants of a reduced decomposition

Let G be a semi-simple group, provided with an involutorial automorphism whose fixed-point group is K . A Borel group B of G has finitely many orbits in the variety G / K . This paper discusses some invariants attached to a B -orbit: a numerical invariant introduced by R.W. Richardson, and two subsets of the Weyl group. These sets are described in terms of root system data.

Perverse Sheaves on Grassmannians

AbstractWe compute the category of perverse sheaves on Hermitian symmetric spaces in types A and D, constructible with respect to the Schubert stratification. The calculation is microlocal, and uses the action of the Borel group to study the geometry of the conormal variety Λ.

ACTION OF THE BOREL GROUP ON MONOMIAL IDEALS

ABSTRACT Let be the Borel group of upper triangular matrices. In this paper we want to study the action of on the set of monomial ideals of ( a field of characteristic zero) from a computational point of view. More specifically, we show that the stabilizer of a monomial ideal in is a purely combinatorial object and we give an algorithm for computing it. Then we characterize the subgroups of that are stabilizers of monomial ideals, we give an algorithm which finds if a given ideal is in the orbit of a monomial ideal under the action of and in the affirmative case, finds the matrices such that . We show that the entries of can be directly obtained from the coefficients of the generators of , so in particular no solutions of polynomial equations are required.

Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2ε(G,K) the maximal number of functionally independent functions from A\C. We prove that ε(G,K) is equal to the codimension δ(G,K) of maximal dimension orbits of the Borel subgroup B⊂G. Moreover, if δ(G,K)=1, then all G-invariant Hamiltonian systems on T*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.