A set of n homogeneous polynomials in n variables is a regular sequence if the associated polynomial system has only the obvious solution (0, 0, . . . , 0). Denote by pk(n) the power sum symmetric polynomial in n variables x k 1 +x 2 + · · ·+xk n . The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variabl...

Noncommutative symmetric functions have many properties analogous to those of classical (commutative) symmetric functions. For instance, ribbon Schur functions (analogs of the classical Schur basis) expand positively in noncommutative monomial basis. More of the classical properties extend to noncommutative setting as I will demonstrate introducing a new family of noncommutative symmetric funct...

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for po...

Recently Lapointe et. al. [3] have expressed Jack Polynomials as determinants in monomial symmetric functions mλ. We express these polynomials as determinants in elementary symmetric functions eλ, showing a fundamental symmetry between these two expansions. Moreover, both expansions are obtained indifferently by applying the Calogero-Sutherland operator in physics or quasi Laplace Beltrami oper...

This behavior has been seen in some notable cases. Kirillov [3] shows that elementary symmetric polynomials in noncommuting variables commute (and, in some cases, all Schur functions) when elementary symmetric polynomials of degree at most three commute when restricted to at most three of the variables. Generalizing this, Blasiak and Fomin [1] give a wider theory for rules of three of generatin...

The method of symmetric polynomials (MSP) was developed for computation analytical functions of matrices, in particular, integer powers of matrix. MSP does not require for its realization finding eigenvalues of the matrix. A new type of recurrence relations for symmetric polynomials of order n is found. Algorithm for the numerical calculation of high powers of the matrix is proposed.This comput...

The pair of groups, symmetric group S2n and hyperoctohedral group Hn , form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions of (S2n, Hn) 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 (S2n, ...

We characterize the class of ultraspherical polynomials in between all symmetric orthogonal polynomials on [−1, 1] via the special form of the representation of the derivatives pn+1(x) by pk(x), k = 0, ..., n.

In this paper, the authors consider the Carlitz’s generalized twisted q-Bernoulli polynomials attached to χ and investigate some novel symmetric identities for these polynomials arising from the p-adic q-integral on Zp under S3.

We study the parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs. In particular, we show that these polynomials are always either zero or a monic power of q, and that they are combinatorial invariants.