For each pair (k,r) of positive integers with r>1, we consider an ideal I^(k,r)_n of the ring of symmetric polynomials in n variables. The ideal I_n^(k,r) has a basis consisting of Macdonald polynomials P(x_1,...,x_n;q,t) at t^{k+1}q^{r-1}=1, and is a deformed version of the one studied earlier in the context of Jack polynomials.
In this paper we give a characterization of I^(k,r)_n in terms of explicit zero conditions on the k-codimensional shifted diagonals of the form x_{2}=tq^{s_1}x_1,...,x_{k+1}=tq^{s_k}x_k.
The ideal I^(k,r)_n may be viewed as a deformation of the space of correlation functions of an abelian current of the affine Lie algebra \hat{sl_r}. We give a brief discussion about this connection.
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 polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.
We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We obtain two explicit formulas for these polynomials: a $q$-integral representation and a combinatorial formula. Our main tool is a $q$-integral representation for ordinary Macdonald polynomials. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.
It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald-Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced, the combinatorial formulas for them are given.
We study a pair of commuting difference operators arising from the elliptic solution of the dynamical Yang-Baxter equation of type C_2. The operators act on the space of meromorphic functions on the weight space of sp(4,C). We show that these operators can be identified with the system by van Diejen and by Komori-Hikami with special parameters. It turns out that our case can be related to the difference Lame operator (two-body Ruijsenaars operator) and thereby we diagonalize the system on the finite dimensional space spanned by the level one characters of the C_2^{(1)}-affine Lie algebra.
The Macdonald process is a stochastic process on the collection of partitions that is a $(q,t)$-deformed generalization of the Schur process. In this paper, we approach the Macdonald process identifying the space of symmetric functions with a Fock representation of a Heisenberg algebra. By using the free field realization of operators diagonalized by the Macdonald symmetric functions, we propose a method of computing several correlation functions with respect to the Macdonald process. It is well-known that expectation value of several observables for the Macdonald process admit determinantal e…
We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special value of parameter $α= -2.$ As a corollary we present an explicit formula for the Hilbert-Poincarè series of this ideal and the generator of the minimal degree as a special Jack polynomial. A generalization to the case of the symmetric polynomials vanishing on the double shifted diagonals and the Macdonald polynomials specialized at $t^2 q = 1$ is also presente…
We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of U_q(gl_n). In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture.
For a pair of positive integers (k,r) with r>1 such that k+1 and r-1 are relatively prime, we describe the space of symmetric polynomials in variables x_1,...,x_n which vanish at all diagonals of codimension k of the form x_i=tq^{s_i}x_{i-1}, i=2,...,k+1, where t and q are primitive roots of unity of orders k+1 and r-1.
We introduce a $q,t$-enumeration of Dyck paths which are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla$ operator applied to a Hall-Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math. $\mathbf {126}$ (2005), pp. 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall-Littlewood polynomials indexed by compositions are the building blocks for the a…
Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly-graded S_n module can be expressed using the Frobenius characteristic map as \nabla e_n, where e_n is the n-th elementary symmetric function, and \nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for \nabla e_n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. Th…
We study the $q$-analogue of the Haldane-Shastry model, a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of Uglov, via 'freezing' from the affine Hecke algebra. We obtain explicit expressions for the spin-analogue of Macdonald operators. Through freezing these yield the higher Hamiltonians of the spin chain, including a Hamiltonian of the opposite chirality. The sum of the two chiral Hamiltonians has a real spectrum also for $q$ …
The classical algebra $Λ$ of symmetric functions has a remarkable deformation $Λ^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s^*_μ$, where $μ$ ranges over the set of all partitions. The main significance of the shifted Schur functions is that they determine a natural basis in $Z(\frak{gl}(n))$, the center of the universal enveloping algebra $U(\frak{gl}(n))$, $n=1,2,\ldots$. The functions $s^*_μ$ are closely related to the factorial Schur functions introduced by Biedenharn and Louck and f…
Elementary properties of the Koornwinder-Macdonald multivariable Askey-Wilson polynomials are discussed. Studied are the orthogonality, the difference equations, the recurrence relations, and the orthonormalization constants for these polynomials. Essential in our approach are certain commuting difference operators simultaneously diagonalized by the polynomials.
The lattice cell in the ${i+1}^{st}$ row and ${j+1}^{st}$ column of the positive quadrant of the plane is denoted $(i,j)$. If $μ$ is a partition of $n+1$, we denote by $μ/ij$ the diagram obtained by removing the cell $(i,j)$ from the (French) Ferrers diagram of $μ$. We set $Δ_{μ/ij}=\det \| x_i^{p_j}y_i^{q_j} \|_{i,j=1}^n$, where $(p_1,q_1),... ,(p_n,q_n)$ are the cells of $μ/ij$, and let ${\bf M}_{μ/ij}$ be the linear span of the partial derivatives of $Δ_{μ/ij}$. The bihomogeneity of $Δ_{μ/ij}$ and its alternating nature under the diagonal action of $S_n$ gives ${\bf M}_{μ/ij}$ the structure…
