Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInterpolation polynomials, bar monomials, and their positivity
126
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 be related to Jack polynomials by Knop-Sahi and Okounkov-Olshanski. The positivity result proved here is an inhomogeneous generalization of Macdonalds positivity conjecture for Jack polynomials. We also formulate and prove the non-symmetric version of the Knop-Sahi conjecture, and in fact we deduce everything from an even stronger positivity result. This last result concerns certain inhomogeneous analogues of ordinary monomials that we call bar monomials. Their positivity involves in an essential way a new partial order on compositions that we call the bar order, and a new operation that we call a glissade.
rate research
Read More
LLT polynomials are $q$-analogues of product of Schur functions that are known to be Schur-positive by Grojnowski and Haiman. However, there is no known combinatorial formula for the coefficients in the Schur expansion. Finding such a formula also provides Schur positivity of Macdonald polynomials. On the other hand, Haiman and Hugland conjectured that LLT polynomials for skew partitions lying on $k$ adjacent diagonals are $k$-Schur positive, which is much stronger than Schur positivity. In this paper, we prove the conjecture for $k=2$ by analyzing unicellular LLT polynomials. We first present a linearity theorem for unicellular LLT polynomials for $k=2$. By analyzing linear relations between LLT polynomials with known results on LLT polynomials for rectangles, we provide the $2$-Schur positivity of the unicellular LLT polynomials as well as LLT polynomials appearing in Haiman-Hugland conjecture for $k=2$.
Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that [ P(x_1,ldots,x_n,x_1^{-1},ldots,x_n^{-1}) =Q(x_1+x_1^{-1},ldots,x_n+x_n^{-1}). ] We denote this polynomial $Q$ by $Phi_n(P)$ and show that $Phi_n$ is an epimorphism of algebras. We compute $Phi_n(P)$ for several families of symmetric polynomials $P$: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016). The polynomials of the form $Phi_n(operatorname{s}_{lambda/mu}^{(2n)})$, where $operatorname{s}_{lambda/mu}^{(2n)}$ is a skew Schur polynomial in $2n$ variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as $x_1,ldots,x_n,x^{-1}_1,ldots,x^{-1}_n$. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.
A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenbergs work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.
The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of the joint distribution of excedances, fixed points and cycles of permutations and derangements, signed or not, colored or not. Let $pin [0,1]$ and $qin [0,1]$ be two given real numbers. We prove that the cyc q-Eulerian polynomials of permutations are bi-gamma-positive, and the fix and cyc (p,q)-Eulerian polynomials of permutations are alternatingly increasing, and so they are unimodal with modes in the middle, where fix and cyc are the fixed point and cycle statistics. When p=1 and q=1/2, we find a combinatorial interpretation of the bi-gamma-coefficients of the (p,q)-Eulerian polynomials. We then study excedance and flag excedance statistics of signed permutations and colored permutations. In particular, we establish the relationships between the (p,q)-Eulerian polynomials and some multivariate Eulerian polynomials. Our results unify and generalize a variety of recent results.
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{lfloor n/2rfloor}$ divides $sum_{piinmathfrak{S}_n}q^{rm{altmaj}(pi)}$, where $mathfrak{S}_n$ is the set of all permutations of ${1,2,ldots,n}$ and $rm{altmaj}(pi)$ is the alternating major index of $pi$. This leads us to discover a $q$-analog of $n!=2^{ell}m$, $m$ odd, using the statistic of alternating major index. Moreover, we study the $gamma$-vectors of the alternating descent polynomials by using these two recursions and the ${textbf{cd}}$-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
