A note on the circuit polynomials and characteristic polynomials of wheels and ladders

Let G be a graph. The circuit polynomial of G is the polynomial Σ Π_αw_α, where w_α is a weight given to the circuit α in G, Π_αw_α is the weight of a spanning subgraph of G, consisting of circuits, nodes and edges, and the summation is taken over all such spanning subgraphs in G. The circuit polynomial of a graph is a generalization of its characteristic polynomial. The characteristic polynomials of wheels and ladders are deduced from their corresponding circuit polynomials.     

Moufang polygons and irreducible spherical BN-pairs of rank 2, I

Let G be a group with an irreducible spherical BN-pair of rank 2 satisfying the additional condition: (∗) There exists a normal nilpotent subgroup U of B with B=TU, where T=B∩N and |W|≠16 for the Weyl group W=N/B∩N. We show that G corresponds to a Moufang polygon and hence is essentially known.     

Heckman-Opdam's Jacobi polynomials for the BCn root system and generalized spherical functions

We prove that the radial part of the Laplacian on the space of generalized spherical functions on the symmetric space GL(m+n)/GL(m)×GL(n) is the Sutherland differential operator for the root system BC_n and the radial parts of the differential operators corresponding to the higher Casimirs yield the integrals of the quantum Calogero-Moser system. It allows us to give a representation theoretical construction for the three parameter family of Heckman-Opdam's Jacobi polynomials for the BC_n root system.     

Low rank perturbation of regular matrix polynomials

Let A(λ) be a complex regular matrix polynomial of degree ? with g elementary divisors corresponding to the finite eigenvalue λ₀. We show that for most complex matrix polynomials B(λ) with degree at most ? satisfying rank the perturbed polynomial (A+B)(λ) has exactly elementary divisors corresponding to λ₀, and we determine their degrees. If does not exceed the number of λ₀-elementary divisors of A(λ) with degree greater than 1, then the λ₀-elementary divisors of (A+B)(λ) are the elementary divisors of A(λ) corresponding to λ₀ with smallest degree, together with rank(B(λ)-B(λ₀)) linear λ₀-elementary divisors. Otherwise, the degree of all the λ₀-elementary divisors of (A+B)(λ) is one. This behavior happens for any matrix polynomial B(λ) except those in a proper algebraic submanifold in the set of matrix polynomials of degree at most ?. If A(λ) has an infinite eigenvalue, the corresponding result follows from considering the zero eigenvalue of the perturbed dual polynomial.     

Arbitrary rank jumps for A-hypergeometric systems through Laurent polynomials

We investigate the solution space of hypergeometric systemsof differential equations in the sense of Gel’fand, Graev,Kapranov and Zelevinski. For any integer d 2, we constructa matrix A_(d) ^{d x 2d} and a parameter vector ß_(d)such that the holonomic rank of the A-hypergeometric systemH_A(d)(ß_(d)) exceeds the simplicial volume vol(A_(d))by at least d – 1. The largest previously known gap betweenrank and volume was 2. Our construction gives evidence to the general observation thatrank jumps seem to go hand in hand with the existence of multipleLaurent (or Puiseux) polynomial solutions.     

Extremal polynomials for obtaining bounds for spherical codes and designs

We investigate two extremal problems for polynomials giving upper bounds for spherical codes and for polynomials giving lower bounds for spherical designs, respectively. We consider two basic properties of the solutions of these problems. Namely, we estimate from below the number of double zeros and find zero Gegenbauer coefficients of extremal polynomials. Our results allow us to search effectively for such solutions using a computer. The best polynomials we have obtained give substantial improvements in some cases on the previously known bounds for spherical codes and designs. Some examples are given in Section 6. This research was partially supported by the Bulgarian NSF under Contract I-35/1994.     

Quantum zonal spherical functions and Macdonald polynomials

A unified theory of quantum symmetric pairs is applied to q-special functions. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions. Here a distinguished family of such functions, invariant under the Weyl group associated to the restricted roots, is shown to be a family of Macdonald polynomials, as conjectured by Koornwinder and Macdonald. Our results place earlier work for Lie algebras of classical type in a general context and extend to the exceptional cases.     

Decompositions of root systems of Kac-Moody algebras of rank 2

A classifications is presented of the parabolic decompositions of the system of roots of Kac- Moody algebras of rank 2. New series of irreducible representations are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1712–1715, December, 1992.     

Poincaré polynomials and level rank dualities in the N=2 coset construction

We review the coset construction of conformal field theories; the emphasis is on the Hilbert spaces for these models, especially if fixed points occur. This is applied to theN=2 superconformal cosets constructed by Kazama and Suzuki. To calculate heterotic string spectra we reformulate the Gepner construction in terms of simple currents and introduce the so-called extended Poincaré polynomial. We finally comment on the various equivalences arising between models of this class, which can be expressed as level rank dualities.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 467–478, March, 1994.     

Hodge polynomials of the moduli spaces of rank 3 pairs

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple on X consists of two holomorphic vector bundles E ₁ and E ₂ over X and a holomorphic map . There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E ₁) = 3, rk(E ₂) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.      

On the subdifferential regularity of max root functions for polynomials

In 2001, Burke and Overton showed that the abscissa mapping on polynomials is subdifferentially regular on the monic polynomials of degree n. We extend this result to the class of max polynomial root functions which includes both the polynomial abscissa and the polynomial radius mappings. The approach to the computation of the subgradient simplifies that given by Burke and Overton and provides new insight into the variational properties of these functions.     

Chapter 2:rank and tensor rank

Let Vbe a vector space of matrices over a field and ka fixed positive integer. In this chapter we first survey results concerning linear maps on certain types of Vthat preserve one of the following:(a) the set of rank kmatrices, (b) the set of matrices of rank less than k. We next survey results concerning linear maps on certain symmetry classes of tensors that preserve nonzero decomposable elements.