36 research outputs found
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
Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by
certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of
the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is
invariant under the action of certain differential operators which include half
the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit
clusters of size at most k: they vanish when k+1 of their variables are
identified, and they do not vanish when only k of them are identified. We
generalize most of these properties to superspace using orthogonal
eigenfunctions of the supersymmetric extension of the trigonometric
Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular,
we show that the Jack superpolynomials P_{\Lambda}^{(\alpha)} at
\alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span
an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N
commuting variables and N anticommuting variables. We prove that the ideal
{\mathcal I}^{(k,r)}_N is stable with respect to the action of the
negative-half of the super-Virasoro algebra. In addition, we show that the Jack
superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting
variables are equal, and conjecture that they do not vanish when only k of them
are identified. This allows us to conclude that the standard Jack polynomials
with prescribed symmetry should satisfy similar clustering properties. Finally,
we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for
the subspace of symmetric superpolynomials in N variables that vanish when k+1
commuting variables are set equal to each other.Comment: 36 pages; the main changes in v2 are : 1) in the introduction, we
present exceptions to an often made statement concerning the clustering
property of the ordinary Jack polynomials for (k,r,N)-admissible partitions
(see Footnote 2); 2) Conjecture 14 is substantiated with the extensive
computational evidence presented in the new appendix C; 3) the various tests
supporting Conjecture 16 are reporte
Orthogonality of Jack polynomials in superspace
Jack polynomials in superspace, orthogonal with respect to a
``combinatorial'' scalar product, are constructed. They are shown to coincide
with the Jack polynomials in superspace, orthogonal with respect to an
``analytical'' scalar product, introduced in hep-th/0209074 as eigenfunctions
of a supersymmetric quantum mechanical many-body problem. The results of this
article rely on generalizing (to include an extra parameter) the theory of
classical symmetric functions in superspace developed recently in
math.CO/0509408Comment: 22 pages, this supersedes the second part of math.CO/0412306; (v2) 24
pages, title and abstract slightly modified, minor changes, typos correcte
Jack polynomials in superspace
This work initiates the study of {\it orthogonal} symmetric polynomials in
superspace. Here we present two approaches leading to a family of orthogonal
polynomials in superspace that generalize the Jack polynomials. The first
approach relies on previous work by the authors in which eigenfunctions of the
supersymmetric extension of the trigonometric Calogero-Moser-Sutherland
Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by
diagonalizing the first nontrivial element of a bosonic tower of commuting
conserved charges not containing this Hamiltonian. Quite remarkably, the
expansion coefficients of these orthogonal eigenfunctions in the supermonomial
basis are stable with respect to the number of variables. The second and more
direct approach amounts to symmetrize products of non-symmetric Jack
polynomials with monomials in the fermionic variables. This time, the
orthogonality is inherited from the orthogonality of the non-symmetric Jack
polynomials, and the value of the norm is given explicitly.Comment: 28 pages. Corrected version of lemme 3 and other minor corrections
and 2 new references; version to appear in Commun. Math. Phy