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Vol 232, No 2 (2018)
- Year: 2018
- Articles: 8
- URL: https://journals.rcsi.science/1072-3374/issue/view/14929
Article
Nonunitary Representations of the Groups of U(p, q)-currents for q ≥ p > 1
Abstract
The purpose of this paper is to give a construction of representations of the group of currents for semisimple groups of rank greater than one. Such groups have no unitary representations in the Fock space, since the semisimple groups of this form have no nontrivial cohomology in faithful irreducible representations. Thus we first construct cohomology of the semisimple groups in nonunitary representations. The principal method is to reduce all constructions to Iwasawa subgroups (solvable subgroups of the semisimple groups), with subsequent extension to the original group. The resulting representation is realized in the so-called quasi-Poisson Hilbert space associated with natural measures on infinite-dimensional spaces.
99-120
Infinite Geodesics in the Discrete Heisenberg Group
Abstract
We give an exhaustive description of the family of infinite geodesics in the discrete Heisenberg group (with respect to the standard generating set). The classification of infinite geodesics is needed to describe the so-called absolute (exit boundary) of a group. The absolute of the discrete Heisenberg group will be described in a forthcoming paper.
121-128
Discrete Morse Theory for the Barycentric Subdivision
Abstract
Let F be a discrete Morse function on a simplicial complex L. We construct a discrete Morse function Δ(F) on the barycentric subdivision Δ(L). The constructed function Δ(F) “behaves the same way” as F, i.e., has the same number of critical simplices and the same gradient path structure.
129-137
Combinatorial Encodings of Infinite Symmetric Groups and Descriptions of Semigroups of Double Cosets
Abstract
Spaces of double cosets of infinite symmetric groups with respect to some special subgroups admit natural structures of semigroups. Elements of such semigroups can be interpreted in combinatorial terms. We present a description of such constructions in a relatively wide degree of generality.
138-156
157-163
Regularity of Maximum Distance Minimizers
Abstract
We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality maxyϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline.
In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.
164-169
On the Dual Complexity and Spectra of Some Combinatorial Functions
Abstract
In a recent paper, A. M. Vershik and the author started the study of representation-theoretic aspects of well-known combinatorial functions on the symmetric groups ????n. The note presents a series of further results in this direction.
170-176
Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. I
Abstract
Consider a system of polynomial equations with parametric coefficients over an arbitrary ground field. We show that the variety of parameters can be represented as a union of strata. For values of parameters from each stratum, the solutions of the system are given by algebraic formulas depending only on this stratum. Each stratum is a quasiprojective algebraic variety with degree bounded from above by a subexponential function in the size of the input data. Also, the number of strata is subexponential in the size of the input data. Thus, here we avoid double exponential upper bounds on the degrees and solve a long-standing problem.
177-203