Related papers: A positivity conjecture for Jack polynomials

Interpolation polynomials were introduced by Knop--Sahi in type $A$, and Okounkov in type $BC$. They are inhomogeneous polynomials whose top terms are Jack and Macdonald polynomials. Thus the expansion coefficients for the product of two…

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…

In this short note, a general result concerning the positivity, under some conditions, of the coefficients of a power series is proved. This allows us to answer positively a question raised by Guo (2010) about the sign of the coefficients…

Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack…

We prove a positivity result for interpolation polynomials that was conjectured by Knop and Sahi. These polynomials were first introduced by Sahi in the context of the Capelli eigenvalue problem for Jordan algebras, and were later shown to…

The Jack symmetric polynomials $P_\lambda^{(\alpha)}$ form a class of symmetric polynomials which are indexed by a partition $\lambda$ and depend rationally on a parameter $\alpha$. They reduced to the Schur polynomials when $\alpha=1$, and…

In the the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with $\alpha = - (r-1)/(k+1)$, $(r-1)$ and $(k+1)$…

We introduce a new operator $\Gamma$ on symmetric functions, which enables us to obtain a creation formula for Macdonald polynomials. This formula provides a connection between the theory of Macdonald operators initiated by Bergeron,…

A new type of conjectures on characters of finite groups, related to the McKay conjecture, have recently been proposed. In this paper, we study these conjectures for symmetric groups.

Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021),…

Superpolynomials consist of commuting and anti-commuting variables. By considering the anti-commuting variables as a module of the symmetric group the theory of vector-valued nonsymmetric Jack polynomials can be specialized to…

We give a construction for three parameter family of Jack polynolials for the root system $BC_n$ through the generalized spherical functions on the symmetric space $GL(m+n)/GL(m)\times GL(n)$.

We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…

We give the explicit analytic development of any Jack or Macdonald polynomial in terms of elementary (resp. modified complete) symmetric functions. These two developments are obtained by inverting the Pieri formula.

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…

The theory of total positivity for reductive groups is here extended to the case of symmetric spaces.

We describe a new formula for weight multiplicities and characters of semisimple Lie algebras. Our formula expresses these weight multiplicities as sums of positive rational numbers. In fact, the formula works more generally for the Jacobi…

We study Jack polynomials in $N$ variables, with parameter $\alpha$, and having a prescribed symmetry with respect to two disjoint subsets of variables. For instance, these polynomials can exhibit a symmetry of type AS, which means that…

It is known that the Ehrhart polynomials of cross-polytopes, as well as of pyramids over them, have positive coefficients. We give a combinatorial proof of this fact by showing that a scaled version of the Ehrhart polynomials are generating…

We prove a conjecture of Dale Peterson on positivity in the multiplication in the T-equivariant cohomology of the flag variety. The theorem follows from a more general positivity result about the equivariant cohomology of varieties with…