Related papers: Supersymmetric Calogero-Moser-Sutherland models an…
We give the explicit analytic development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments…
We construct linear operators factorizing the three bases of symmetric polynomials: monomial symmetric functions m(x), elementary symmetric polynomials E(x), and Schur functions s(x), into products of univariate polynomials.
We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under…
A new family of A_N-type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and…
The kinematical and dynamical symmetries of equations describing the time evolution of quantum systems like the supersymmetric harmonic oscillator in one space dimension and the interaction of a non-relativistic spin one-half particle in a…
We provide a mathematically rigorous framework for supersymmetric fermion lattice systems. We construct supersymmetric C*-dynamics in terms of a nilpotent superderivation and a one-parameter group of automorphisms on the CAR-algebra. (We do…
We discuss several applications and extensions of our previous operator solution of the $N$-body Calogero problem, \ie N particles in 1 dimension subject to a two-body interaction of the form $\half \sum_{i,j}[ (x_i - x_j)^2 + g/ {(x_i -…
We give a construction for three parameter family of Jack polynolials for the root system $BC_n$ through the generalized spherical functions on the symmetric space $GL(m+n)/GL(m)\times GL(n)$.
We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric…
In the Supersymmetric SU(5) Model of Unification with the Missing Partner Mechanism, we present a renormalizable model using the Georgi-Jarlsog mechanism to describe the fermion masses and mixing. At the meantime the proton decay rates are…
Starting from the Hamiltonian formulation of supersymmetric Calogero models associated with the classical $A_n$, $B_n$, $C_n$ and $D_n$ series we construct the ${\cal N}{=}\,2$ and ${\cal N}{=}\,4$ supersymmetric extensions of the their…
In this paper we present a generalization of the classical Hermite polynomials to the framework of Clifford-Dunkl operators. Several basic properties, such as orthogonality relations, recurrence formulae and associated differential…
The paper considers a Clifford extension of the Grassmann algebra, in which operators are built from Grassmann variables and by the derivatives with respect to them. It is shown that a subalgebra which is isomorphic to the usual matrix…
We consider the gl(N)-invariant Calogero-Sutherland Models with N=1,2,3,... in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the gl(N)-invariant…
For each irreducible module of the symmetric group $\mathcal{S}_{N}$ there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a…
Majorization inequalities for symmetric polynomials have interested mathematicians for centuries, from the AM-GM inequality for two variables going back at least to Euclid, through classical results of Newton, Muirhead and Gantmacher, to…
We explore $\mathcal{N}=1$ supersymmetric extensions of algebras going beyond the Poincar\'e and AdS ones in three spacetime dimensions. Besides reproducing two known examples, we present new superalgebras, which all correspond to…
We develop the general theory of Jack-Laurent symmetric functions, which are certain generalisations of the Jack symmetric functions, depending on an additional parameter p_0.
Through an algebraic method using the Dunkl--Cherednik operators, the multivariable Hermite and Laguerre polynomials associated with the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles are…
We construct the supersymmetric $\beta$ and $(q,t)$-deformed Hurwitz-Kontsevich partition functions through $W$-representations and present the corresponding character expansions with respect to the Jack and Macdonald superpolynomials,…