Related papers: Thom polynomials and Schur functions I

For the Schur polynomials bounded and unbounded generalizations of the Cauchy identities are found.

An element [\Phi] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis {\phi} for C(x). The Pl\"ucker coordinates…

This is a report on recent work of Chalupnik and Touze. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we…

Each degree $n+k$ polynomial of the form $(x+1)^k(x^n+c_1x^{n-1}+\cdots +c_n)$, $k\in \mathbb{N}$, is representable as Schur-Szeg\H{o} composition of $n$ polynomials of the form $(x+1)^{n+k-1}(x+a_j)$. We study properties of the affine…

We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains…

We suggest to associate with each knot the set of coefficients of its HOMFLY polynomial expansion into the Schur functions. For each braid representation of the knot these coefficients are defined unambiguously as certain combinations of…

In this paper, using the theory of category, we generalize known properties of symmetric polynomials and functions and characterize the multi-indicial symmetric functions. Examples have been given on Schur functions.

We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the…

This is a note on my mini-course in the International Workshop on Real and Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here we introduce a new branch of the Thom polynomial theory for singularities of…

The theory of Schur functors provides a powerful and elegant approach to the representation theory of GL_n - at least to the so-called polynomial representations - especially to questions about how the theory varies with n. We develop…

We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than $k$. We introduce symmetric functions called the…

Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…

We apply a theorem of Geronimus to derive some new formulas connecting Schur functions with orthogonal polynomials on the unit circle. The applications include the description of the associated measures and a short proof of Boyd's result…

We introduce and study a generalization $s_{(\mu|\lambda)}$ of the Schur functions called the almost symmetric Schur functions. These functions simultaneously generalize the finite variable key polynomials and the infinite variable Schur…

Recently, the functional equation \[ \sum_{i=0}^mf_i(b_ix+c_iy)= \sum_{i=1}^na_i(y)v_i(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in\mathbf{GL}_d(\mathbb{C})$, was studied by Almira and Shulman, both in the classical context of continuous…

Let $(r_{A,n}(x))_{n \in \mathbb{N}}$ be a sequence of polynomials with coefficients from a field $K$ satisfying the recurrence relation $r_{A,n}(x)= \sum_{|\alpha|\leq m} t_{\alpha,n}(x)\textbf{r}_{A,n}^\alpha(x)$ of order $d+1 \in…

It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper we study…

The Schur functions in superspace $s_\Lambda$ and $\bar s_\Lambda$ are the limits $q=t=0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We prove Pieri rules for the bases $s_\Lambda$ and $\bar s_{\Lambda}$ (which…

Plethysm of two Schur functions can be expressed as a linear combination of Schur functions, and monomial symmetric functions. In this paper, we express the coefficients combinatorially in the case of monomial symmetric functions. And by…

We introduce and study a generalization of Schur's $P$-/$Q$-functions associated to a polynomial sequence, which can be viewed as ``Macdonald's ninth variation'' for $P$-/$Q$-functions. This variation includes as special cases Schur's…