265 research outputs found

### Recurrence formulas for Macdonald polynomials of type A

We consider products of two Macdonald polynomials of type A, indexed by
dominant weights which are respectively a multiple of the first fundamental
weight and a weight having zero component on the k-th fundamental weight. We
give the explicit decomposition of any Macdonald polynomial of type A in terms
of this basis.Comment: 18 pages, LaTeX, revised version, to appear in Journal of Algebraic
Combinatoric

### Baxter operator formalism for Macdonald polynomials

We develop basic constructions of the Baxter operator formalism for the
Macdonald polynomials associated with root systems of type A. Precisely we
construct a dual pair of mutually commuting Baxter operators such that the
Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter
operators is closely related to the dual pair of recursive operators for
Macdonald polynomials leading to various families of their integral
representations. We also construct the Baxter operator formalism for the
q-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained by
degenerations of the Macdonald polynomials associated with the type A_l root
system. This note provides a generalization of our previous results on the
Baxter operator formalism for the Whittaker functions. It was demonstrated
previously that Baxter operator formalism for the Whittaker functions has deep
connections with representation theory. In particular the Baxter operators
should be considered as elements of appropriate spherical Hecke algebras and
their eigenvalues are identified with local Archimedean L-factors associated
with admissible representations of reductive groups over R. We expect that the
Baxter operator formalism for the Macdonald polynomials has an interpretation
in representation theory of higher-dimensional arithmetic fields.Comment: 22 pages, typos are fixe

### Macdonald polynomials in superspace: conjectural definition and positivity conjectures

We introduce a conjectural construction for an extension to superspace of the
Macdonald polynomials. The construction, which depends on certain orthogonality
and triangularity relations, is tested for high degrees. We conjecture a simple
form for the norm of the Macdonald polynomials in superspace, and a rather
non-trivial expression for their evaluation. We study the limiting cases q=0
and q=\infty, which lead to two families of Hall-Littlewood polynomials in
superspace. We also find that the Macdonald polynomials in superspace evaluated
at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In
particular, their expansion coefficients in the corresponding Hall-Littlewood
bases appear to be polynomials in t with nonnegative integer coefficients. More
strikingly, we formulate a generalization of the Macdonald positivity
conjecture to superspace: the expansion coefficients of the Macdonald
superpolynomials expanded into a modified version of the Schur superpolynomial
basis (the q=t=0 family) are polynomials in q and t with nonnegative integer
coefficients.Comment: 18 page

### Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle

We introduce an integrable lattice discretization of the quantum system of n
bosonic particles on a ring interacting pairwise via repulsive delta
potentials. The corresponding (finite-dimensional) spectral problem of the
integrable lattice model is solved by means of the Bethe Ansatz method. The
resulting eigenfunctions turn out to be given by specializations of the
Hall-Littlewood polynomials. In the continuum limit the solution of the
repulsive delta Bose gas due to Lieb and Liniger is recovered, including the
orthogonality of the Bethe wave functions first proved by Dorlas (extending
previous work of C.N. Yang and C.P. Yang).Comment: 25 pages, LaTe

### Quantum $W_N$ Algebras and Macdonald Polynomials

We derive a quantum deformation of the $W_N$ algebra and its quantum Miura
transformation, whose singular vectors realize the Macdonald polynomials.Comment: LaTeX file, 17-pages, no-figures, a reference adde

### Roots of Ehrhart Polynomials of Smooth Fano Polytopes

V. Golyshev conjectured that for any smooth polytope P of dimension at most
five, the roots z\in\C of the Ehrhart polynomial for P have real part equal
to -1/2. An elementary proof is given, and in each dimension the roots are
described explicitly. We also present examples which demonstrate that this
result cannot be extended to dimension six.Comment: 10 page

### Moments of a single entry of circular orthogonal ensembles and Weingarten calculus

Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$
from a circular orthogonal ensemble. In this paper, we study moments of a
single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit
values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading
and subleading terms in the asymptotic expansion with respect to a large matrix
size $N$. Our technique is to apply the Weingarten calculus for a
Haar-distributed unitary matrix.Comment: 17 page

### Proper holomorphic mappings between symmetrized ellipsoids

We characterize the existence of proper holomorphic mappings in the special
class of bounded $(1,2,...,n)$-balanced domains in $\mathbb{C}^n$, called the
symmetrized ellipsoids. Using this result we conclude that there are no
non-trivial proper holomorphic self-mappings in the class of symmetrized
ellipsoids. We also describe the automorphism groupof these domains.Comment: 10 pages, some modification

### Combinatorial interpretation and positivity of Kerov's character polynomials

Kerov's polynomials give irreducible character values in term of the free
cumulants of the associated Young diagram. We prove in this article a
positivity result on their coefficients, which extends a conjecture of S.
Kerov. Our method, through decomposition of maps, gives a description of the
coefficients of the k-th Kerov's polynomials using permutations in S(k). We
also obtain explicit formulas or combinatorial interpretations for some
coefficients. In particular, we are able to compute the subdominant term for
character values on any fixed permutation (it was known for cycles).Comment: 33 pages, 13 figures, version 3: minor modifcation

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