Related papers: Clebsch-Gordan coefficients for Macdonald polynomi…
In a seminal paper Richard Stanley derived Pieri rules for the Jack symmetric function basis. These rules were extended by Macdonald to his now famous symmetric function basis. The original form of these rules had a forbidding complexity…
We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew…
In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining…
In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial…
We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there…
We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra.…
For any reduced crystallographic root system, we introduce a unitary representation of the (extended) affine Hecke algebra given by discrete difference-reflection operators acting in a Hilbert space of complex functions on the weight…
We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, which depend on $d$ additional parameters and specialize to all…
Pascal routines are provided that generate representations of the group $SU(3)$ and tabulate the Clebsch-Gordan coefficients in the products of representations.
We establish a connection between (degenerate) nonsymmetric Macdonald polynomials and standard bases and dual standard bases of maximal parabolic modules of affine Hecke algebras. Along the way we prove a (weak) polynomiality result for…
We prove a sharp bound for the average value of the triple product of modular functions for the Hecke subgroup \Gamma_0(N). Our result is an extension of the main result in {Bernstein&Reznikov-2004} to a fixed cuspidal representation of the…
We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner…
We study the Gaussent-Littelmann formula for Hall-Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type A_n, B_n and C_n. This description is in terms of Young tableaux and…
We investigate the summability of the coefficients of $m$-homogeneous polynomials and $m$-linear mappings defined on $\ell_{p}$-spaces. In our research we obtain results on the summability of the coefficients of $m$-linear mappings defined…
The purpose of this paper is to prove a Pieri-type multiplication formula for quantum Grothendieck polynomials, which was conjectured by Lenart-Maeno. This formula would enable us to compute explicitly the quantum product of two arbitrary…
The $SL(2,\mathbb Z)$-symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a…
A method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation…
The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In…
In this paper, we derive some explicit expansion formulas associated to Brenke polynomials using operational rules based on their corresponding generating functions. The obtained coefficients are expressed either in terms of finite double…
A generalization of the well-known Fibonacci sequence is the $k$-Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,0, \ldots, 1$, and each term afterwards is the sum of the preceding $k$ terms.…