Related papers: A note on the map expansion of Jack polynomials
In the theory of symmetric Jack polynomials the coefficients in the expansion of the $p$th elementary symmetric function $e_p(z)$ times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this…
Given a family $\{ f_{\lambda} \}_{\lambda \in \Lambda}$ of polynomial maps of degree $d$ where $\Lambda$ is the set of parameters, a polynomial map $f_{\lambda_0}$ is called {\it $J$-stable in $\Lambda$} if there exists a neighborhood of…
We extend the family non-symmetric Macdonald polynomials and define general-basement Macdonald polynomials. We show that these also satisfy a triangularity property with respect to the monomials bases and behave well under the…
We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal…
Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi(x, y, z; t, 1+\beta)$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They…
Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular…
We give a Hecke algebra derivation of Macdonald's expansion formula for Hall-Littlewood polynomials in terms of semistandard Young tableaux. This is accomplished by first obtaining a Hecke algebra lift of the expansion coefficients and then…
The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald…
We derive combinatorial formulae for the modified Macdonald polynomial $H_{\lambda}(x;q,t)$ using coloured paths on a square lattice with quasi-cylindrical boundary conditions. The derivation is based on an integrable model associated to…
This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the \emph{Jackson algebra}, that will be used in sequels to this paper to study non-commutative arithmetic…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the…
In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ and integral Macdonald polynomials $J_{\lambda}(X;q,t)$, in terms of several new statistics and the major index for a partition…
The possibility for the Jacobi equation to admit in some cases general solutions that are polynomials has been recently highlighted by Calogero and Yi, who termed them para-Jacobi polynomials. Such polynomials are used here to build seed…
For K a field of characteristic 0 and d any integer number greater than or equal to 2, we prove the invertibility of polynomial endomorphisms of the affine space of dimension d over K of the form F=Id+H, where each coordinate of H is the…
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and…
In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some…
We study several fundamental operators in harmonic analysis related to Jacobi expansions, including Riesz transforms, imaginary powers of the Jacobi operator, the Jacobi-Poisson semigroup maximal operator and Littlewood-Paley-Stein square…
Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.
A novel polynomial expansion method of symmetric Boolean functions is described. The method is efficient for symmetric Boolean function with small set of valued numbers and has the linear complexity for elementary symmetric Boolean…