Zonal polynomials for wreath products

The pair of groups, symmetric group S _2n and hyperoctohedral group H _n , form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions of (S _2n ,H _n ) into the ring of symmetric functions. The images of the zonal spherical functions under this map are called the zonal polynomials. A wreath product generalization of the Gelfand pair (S _2n ,H _n ) is discussed in this paper. Then a multi-partition versions of the theory is constructed. The multi-partition version of zonal polynomials are products of zonal polynomials and Schur functions and are obtained from a characteristic map from the graded Hecke algebra into a multipartition version of the ring of symmetric functions. Dedicated to Professor Eiichi Bannai on his 60th birthday.     

Boole–De Morgan Algebras and Quasi-De Morgan Functions

In this paper we establish a Stone-type and a Birkhoff-type representation theorems for Boole–De Morgan algebras and prove that the free Boole–De Morgan algebra on n free generators is isomorphic to the Boole–De Morgan algebra of quasi-De Morgan functions of n variables. Also we introduce the concept of Zhegalkin polynomials for quasi-De Morgan functions and consider the representation problem of those functions by polynomials.     

New lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-Hamiltonian structure

We study restricted multiple three wave interaction system by the inverse scattering method. We develop the algebraic approach in terms of classical r-matrix and give an interpretation of the Poisson brackets as linear r-matrix algebra. The solutions are expressed in terms of polynomials of theta functions. In particular case for n = 1 in terms of Weierstrass functions.      

Preserving Zeros of a Polynomial

We study nonlinear surjective mappings on ? _n () and its subsets, which preserve the zeros of some fixed polynomials in noncommuting variables.     

Polynomial Lie Algebras

We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x ₁,...,x _n]/(f ₁,...,f _n) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A,B,C, D, and E ₆.     

Dvoretzky Type Theorems for Multivariate Polynomials and Sections of Convex Bodies

In this paper we prove the Gromov–Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on \mathbbRⁿ{\mathbb{R}^{n}}, and improve bounds on the number n(d, k) in the analogous conjecture for odd degrees d (this case is known as the Birch theorem) and complex polynomials.     

Lie and Jordan Products in Interchange Algebras

We study Lie brackets and Jordan products derived from associative operations ○, ? satisfying the interchange identity (w?x) ○ (y?z) ≡ (w ○ y)?(x ○ z). We use computational linear algebra, based on the representation theory of the symmetric group, to determine all polynomial identities of degree ≤7 relating (i) the two Lie brackets, (ii) one Lie bracket and one Jordan product, and (iii) the two Jordan products. For the Lie–Lie case, there are two new identities in degree 6 and another two in degree 7. For the Lie–Jordan case, there are no new identities in degree ≤6 and a complex set of new identities in degree 7. For the Jordan–Jordan case, there is one new identity in degree 4, two in degree 5, and complex sets of new identities in degrees 6 and 7.     

Pseudozero Set of Real Multivariate Polynomials

The pseudozero set of a system P of polynomials in n variables is the subset of C ⁿ consisting of the union of the zeros of all polynomial systems Q that are near to P in a suitable sense. This concept arises naturally in Scientific Computing where data often have a limited accuracy. When the polynomials of the system are polynomials with complex coefficients, the pseudozero set has already been studied. In this paper, we focus on the case where the polynomials of the system have real coefficients and such that all the polynomials in all the perturbed polynomial systems have real coefficients as well. We provide an explicit definition to compute this pseudozero set. At last, we analyze different methods to visualize this set.      

Computation of Minkowski Values of Polynomials over Complex Sets

As a generalization of Minkowski sums, products, powers, and roots of complex sets, we consider the Minkowski value of a given polynomial P over a complex set X. Given any polynomial P(z) with prescribed coefficients in the complex variable z, the Minkowski value P(X) is defined to be the set of all complex values generated by evaluating P, through a specific algorithm, in such a manner that each instance of z in this algorithm varies independently over X. The specification of a particular algorithm is necessary, since Minkowski sums and products do not obey the distributive law, and hence different algorithms yield different Minkowski value sets P(X). When P is of degree n and X is a circular disk in the complex plane we study, as canonical cases, the Minkowski monomial value P _m (X), for which the monomial terms are evaluated separately (incurring n(n+1) independent values of z) and summed; the Minkowski factor value P _f (X), where P is represented as the product (z–r ₁)(z–r _n ) of n linear factors – each incurring an independent choice zX – and r ₁,...,r _n are the roots of P(z); and the Minkowski Horner value P _h (X), where the evaluation is performed by nested multiplication and incurs n independent values zX. A new algorithm for the evaluation of P _h (X), when 0X, is presented.     

Constructing affine Lie algebras

The ?-grading determined by a long simple root of a rank n+1 a?ne Lie algebra over ? arises from a representation of a rank n semi-simple complex Lie algebra. Analysis of the relationship between the grading and the representation yields constructions that generalize the minuscule and adjoint algorithms as well as Kac’s construction of nontwisted a?ne Lie algebras.     

The classification of n-dimensional anticommutative algebras with (n − 3)-dimensional annihilator

We give the classification of all n-dimensional anticommutative complex algebras with (n???3)-dimensional annihilator. Namely, we describe all central extensions of all 3-dimensional anticommutative complex algebras.     

Centralisers of spaces of symmetric tensor products and applications

We show that the centraliser of the space of n-fold symmetric injective tensors, n≥2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials. This project was supported in part by Enterprise Ireland, International Collaboration Grant – 2004 (IC/2004/009). The second author was also partially supported by PIP 5272,UBACYTX108 and PICT 03-15033     

An extension of an inequality of I. Schur

Denote by π_n the set of all algebraic polynomials of degree at most n with complex coefficients. An inequality of I. Schur asserts that the first derivative of the transformed Tchebycheff polynomial has the greatest uniform norm in [?1, 1] among all f ∈ ??_n, where (1) Here we show that this extremal property of persists in the wider class of polynomials f ∈ π_n which vanish at ±1, and for which there exist n ? 1 points separating the zeros of and such that for j = 1, …, n ? 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D _n is shown to be semisimple and free of rank (2 ⁿ  + 1)n!! ? (2 ^n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D _n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ^±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D _n is a subalgebra of the BMW algebra of the same type.     

Universal Poisson Envelope for Binary-Lie Algebras

In this article the universal Poisson enveloping algebra for a binary-Lie algebra is constructed. Taking a basis 𝔹 of a binary-Lie algebra B, we consider the symmetric algebra S(B) of polynomials in the elements of 𝔹. We consider two products in S(B), the usual product of polynomials fg and the braces {f, g}, defined by the product in B and the Leibniz rule. This algebra is a general Poisson algebra. We find an ideal I of S(B) such that the factor algebra S(B)/I is the universal Poisson envelope of B. We provide some examples of this construction for known binary-Lie algebras.     

Rational Functions and Association Scheme Parameters

The parameters of metric, cometric, symmetric association schemes with q ± 1 (the same as the parameters of the underlying orthogonal polynomials) can be given in general by evaluating a single rational function of degree (4, 4) in the complex variable q ^j. But in all known examples, save the simple n-gons, these reduce to polynomials of degree at most 2 in q ^j with q an integer. One reason this occurs is that the rational function can have singularities at points which would determine some of the parameters. This paper deals with the case in which not all of the singularities are removable, thus giving some reason why the n-gons might naturally be the only exceptions to schemes with parameters being polynomials of degree at most 2 in q ^j , except possibly for schemes of very small diameter.     

Crystal planes and reciprocal space in Clifford geometric algebra

This paper discusses the geometry of kD crystal cells given by (k+ 1) points in a projective space ?^{n+ 1}. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual representation are related (and geometrically interpreted) in the projective geometric algebra Cl(?^{n+ 1}) (see (Die Ausdehnungslehre von 1844 und die Geom. Anal. Teubner: Leipzig, 1894)) and in the conformal algebra Cl(?^{n+ 1, 1}). The crystallographic notions of d‐spacing, phase angle, structure factors, conditions for Bragg reflections, and the interfacial angles of crystal planes are obtained in the same context. Copyright © 2011 John Wiley & Sons, Ltd.     

On balanced bases

It is proved that either a given balanced basis of the algebra (n + 1)M₁ M_n or the corresponding complementary basis is of rank n + 1. This result enables us to claim that the algebra (n + 1)M₁ M_n is balanced if and only if the matrix algebra M_n admits a WP-decomposition, i.e., a family of n + 1 subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form tr XY) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra (n + 1)M₁ M_n is balanced is equivalent to the well-known Winnie-the-Pooh problem on the existence of an orthogonal decomposition of a simple Lie algebra of type A_n–1 into the sum of Cartan subalgebras.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 213–218.Original Russian Text Copyright © 2005 by D. N. Ivanov.This revised version was published online in April 2005 with a corrected issue number.     

Laguerre matrix polynomial series expansion: Theory and computer applications

This paper deals with the expansion of matrix functions in series of Laguerre matrix polynomials of a complex matrix parameter. A new asymptotic expression for positive and large x and n is obtained for this family of matrix polynomials that improves previous existing expressions. This expression is applied to improving the conditions imposed on the matrix function that are to be expanded in a series of non-hermitian Laguerre matrix polynomials, providing a new series expansion theorem that is more general than previous works. An application to solving linear differential systems without matrix exponentials is included.     

Finite free resolutions of length two and koszul generalized complex

Abstract

In 1956, Ehrenfeucht proved that a polynomial f ₁(x ₁) + · + f _n (x _n ) with complex coefficients in the variables x ₁, …, x _n is irreducible over the field of complex numbers provided the degrees of the polynomials f ₁(x ₁), …, f _n (x _n ) have greatest common divisor one. In 1964, Tverberg extended this result by showing that when n ≥ 3, then f ₁(x ₁) + · + f _n (x _n ) belonging to K[x ₁, …, x _n ] is irreducible over any field K of characteristic zero provided the degree of each f _i is positive. Clearly a polynomial F = f ₁(x ₁) + · + f _n (x _n ) is reducible over a field K of characteristic p ≠ 0 if F can be written as F = (g ₁(x ₁)) ^p  + (g ₂(x ₂)) ^p  + · + (g _n (x _n )) ^p  + c[g ₁(x ₁) + g ₂(x ₂) + · + g _n (x _n )] where c is in K and each g _i (x _i ) is in K[x _i ]. In 1966, Tverberg proved that the converse of the above simple fact holds in the particular case when n = 3 and K is an algebraically closed field of characteristic p > 0. In this article, we prove an extension of Tverberg's result by showing that this converse holds for any n ≥ 3.