相关论文: q-Analogs of symmetric function operators

Macdonald defined two-parameter Kostka functions K_{\lambda\mu}(q,t) where \lambda, \mu are partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture…

We investigate the concept of $q$-replicated arguments in symmetric functions with its connection to spectral functions of hyperbolic geometry. This construction suffices for vector generation functions in the form of $q$-series, and string…

We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which…

Inhomogeneous analogues of symmetric and nonsymmetric Macdonald polynomials were introduced by F. Knop and the author. In the symmetric case A. Okounkov has recently proved a beautiful expansion formula which can be viewed as a…

We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes.…

We investigate the homogeneous symmetric Macdonald polynomials $P_\lambda(\X;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_\lambda(\X;q,q^k)$ and $P_\lambda(\frac{1-q}{1-q^k}\X;q,q^k)$. As a…

In this paper, a link between $q$-difference equations, Jacobi operators and orthogonal polynomials is given. Replacing the variable $x$ by $ q^{-n}$ in a Sturm-Liouville $q$-difference equation we discovered the Jacobi operator. With…

We study the quantum symmetric spaces for quantum general linear groups modulo symplectic groups. We first determine the structure of the quotient quantum group and completely determine the quantum invariants. We then derive the…

The Macdonald polynomials with prescribed symmetry are obtained from the nonsymmetric Macdonald polynomials via the operations of $t$-symmetrisation, $t$-antisymmetrisation and normalisation. Motivated by corresponding results in Jack…

We study three different $q$-analogues of the harmonic numbers. As applications, we present some generating functions involving number theoretical functions and give the $q$-generalization of Gosper's exponential generating function of…

The additive analogues of Pseudo-Smarandache, Smarandache-simple functions and their duals have been recently studied by J. Sandor. In this note, we obtain q-analogue of Sandor's theorems.

One more model of a q-harmonic oscillator based on the q-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier…

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula is general enough to give new expressions for all known types of Schubert polynomials. In the present paper we…

The quantum cohomology of Grassmannians exhibits two symmetries related to the quantum product, namely a \Bbb {Z}/n action and an involution related to complex conjugation. We construct a new ring by dividing out these symmetries in an…

In the 1940s Littlewood formulated three fundamental correspondences for the immanants and Schur symmetric functions on the general linear group, which establish deep connections between representation theory of the symmetric group and the…

In this paper, we explore a consequence of symplectic duality (also known as 3d mirror symmetry) in the setting of enumerative geometry. The theory of quasimaps allows one to associate hypergeometric functions called vertex functions to…

We obtain the specialization of monomial symmetric functions on the alphabet (a-b)/(1-q). This gives a remarkable algebraic identity, and four new developments for the Macdonald polynomial associated with a row. The proofs are given in the…

We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of…

A generalization of Newton's identity on symmetric functions is given. Using the generalized Newton identity we give a unified method to show the existence of Hall-Littlewood, Jack and Macdonald polynomials. We also give a simple proof of…

By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of…