Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial$B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$element subsets of $\{1, 2, \dots , n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.
more » « less Award ID(s):
 1764012
 NSFPAR ID:
 10125811
 Publisher / Repository:
 Oxford University Press
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

We prove that the KkSchur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. LamSchillingShimozono identified the KkSchur functions as Schubert representatives for Khomology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the KkSchur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for KkSchur functions produces a second shiftinvariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of IkedaIwaoMaeno, we conjecture that this second basis gives the images of the LenartMaeno quantum Grothendieck polynomials under a Ktheoretic analog of the Peterson isomorphism.more » « less

null (Ed.)Abstract We investigate the longstanding problem of finding a combinatorial rule for the Schubert structure constants in the $K$theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux–M.P. Schützenberger (1982) serve as polynomial representatives for $K$theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $n$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $\beta$Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $K$theory, and we state our results in this more general context.more » « less

null (Ed.)Abstract We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasiLascoux basis, which is simultaneously both a $K$ theoretic deformation of the quasikey basis and also a lift of the $K$ analogue of the quasiSchur basis from quasisymmetric polynomials to general polynomials. We give positive expansions of this quasiLascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasiLascoux basis. As a special case, these expansions give the first proof that the $K$ analogues of quasiSchur polynomials expand positively in multifundamental quasisymmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $K$ theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $K$ analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.more » « less

Let ν be a positive measure supported on [1,1], with infinitely many points in its support. Let {p_{n}(ν,x)}_{n≥0} be its sequence of orthonormal polynomials. Suppose we add masspoints at ±1, giving a new measure μ=ν+Mδ₁+Nδ₋₁. How much larger can p_{n}(μ,0) be than p_{n}(ν,0)? We study this question for symmetric measures, and give more precise results for ultraspherical weights. Under quite general conditions, such as ν lying in the Nevai class, it turns out that the growth is no more than 1+o(1) as n→∞.more » « less

Abstract The superspace ring $\Omega _n$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $\Omega _n$ , the authors previously defined a family of doubly graded quotients ${\mathbb {W}}_{n,k}$ of $\Omega _n$ , which carry an action of the symmetric group ${\mathfrak {S}}_n$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules ${\mathbb {W}}_{n,k}$ in greater detail. We describe a monomial basis of ${\mathbb {W}}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions . These are ordered set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots ,n\}$ in which the nonminimal elements of any block $B_i$ may be barred or unbarred.more » « less