Abstract In this paper, we generalize a result of Cuttler, Greene, Skandera, and Sra that characterizes the majorization order on Young diagrams in terms nonnegative specializations Schur polynomials. More precisely, introduce generalized notion associated to an arbitrary crystallographic root system $\Phi $ show it admits natural characterization values spherical functions any Riemannian symme...

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SLn and a similar formula is conjectured for S...

We extend the big and p-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the p-typical case, it uses positivity with respect to an apparently new basis of the p-typical symmetric functions. We also give...

We construct a family of functors assigning an R-module to a flag of R-modules, where R is a commutative ring. As particular instances, we get flagged Schur functors and Schubert functors, the latter family being indexed by permutations. We identify Schubert functors for vexillary permutations with some flagged Schur functors, thus establishing a functorial analogue of a theorem from [6] and [1...

|S|=n a(S) m. Specifically, we present an explicit formula for F (m,n) as a product of two matrices, ultimately yielding a polynomial in q = pd. The first matrix is independent of n while the second makes no mention of finite fields. However, the complexity of calculating each grows with m. The main tools here are the Schur-Weyl duality theorem, and some elementary properties of symmetric funct...

We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Pólya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled...

An experimental design is said to be Schur optimal, if it is optimal with respect to the class of all Schur isotonic criteria, which includesKiefer’s criteria of p-optimality, distance optimality criteria andmany others. In the paper we formulate an easily verifiable necessary and sufficient condition for Schur optimality in the set of all approximate designs of a linear regression experiment w...

In this note we attempt to develop an analog of Pólya-Schur theory describing the class of univariate hyperbolicity preservers in the setting of linear finite-difference operators. We study the class of linear finite-difference operators preserving the set of real-rooted polynomials whose mesh (i.e. the minimal distance between the roots) is at least one. In particular, finitedifference version...

Compton’s method of proving monadic second-order limit laws is based on analyzing the generating function of a class of finite structures. For applications of his deeper results we previously relied on asymptotics obtained using Cauchy’s integral formula. In this paper we develop elementary techniques, based on a Tauberian theorem of Schur, that significantly extend the classes of structures fo...