We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schiitzenberger, and provide simplified proofs of their basic propert...

The main objective of this article is to construct a totally new, effective algorithm, for the calculations of all series of elementary symmetric polynomials at once for the arbitrary order of the general polynomial. We proved that the effectiveness of the advanced algorithm proposed in this article is by the linear term better than the algorithms existing in the literature so far. Moreover, we...

An explicit expansion for the Macdonald polynomials of the form H (32 a 1 b) [X; q, t] and H (41 k) [X; q, t] in terms of Hall-Littlewood symmetric functions is presented. The expansion gives a proof of a symmetric function operator that adds a row of size 3 to Macdonald polynomials of the form H (2 a 1 b) [X; q, t] and another operator that adds a row of size 4 to Macdonald polynomials of only...

Let f(x) ∈ Z[x] be a totally real polynomial with roots α1 ≤ . . . ≤ αd. The span of f(x) is defined to be αd − α1. Monic irreducible f(x) of span less than 4 are special. In this paper we give a complete classification of those small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. As one application, we find some low-degree polynomials that do not aris...

Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by differential-difference (“Dunkl”) operators, multiplication by coordinate functions and the group algebra. By specializing Griffeth’s (arχiv:0707.0251) results for the ...

Various Nyquist polynomials and their spectral factorizations to obtain suitable finite impulse response (FIR) vestigial sideband (VSB) filters are summarized here. The filters of interest have transfer functions that are polynomials with complex coefficients which are symmetric, due to Lawton, and polynomials whose coefficients are conjugate symmetric or conjugate antisymmetric, which are also...

Let D be any integral domain of any characteristic. A polynomial p(x) ∈ D[x] is D-nice if p(x) and its derivative p′(x) split in D[x]. We give a complete description of all D-nice symmetric polynomials with four roots over integral domains D of any characteristic not equal to 2 by giving an explicit formula for constructing these polynomials and by counting equivalence classes of such D-nice po...

In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painlevé hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials ar...

In this paper we define a family of polynomials closely related to the modified R-polynomials of the symmetric group and begin work toward a classification of the polynomials by using a combinatorial interpretation involving subwords of the maximal element in the Bruhat order. The problem of determining the precise conditions which make one of these polynomials zero motivates our work. We state...

The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–...