The Wronski map and shifted tableau theory

The Mukhin-Tarasov-Varchenko Theorem, conjectured by B. and M. Shapiro, has a number of interesting consequences. Among them is a well-behaved correspondence between certain points on a Grassmannian - those sent by the Wronski map to polynomials with only real roots - and (dual equivalence classes of) Young tableaux. In this paper, we restrict this correspondence to the orthogonal Grassmannian OG(n,2n+1) inside Gr(n,2n+1). We prove that a point lies on OG(n,2n+1) if and only if the corresponding tableau has a certain type of symmetry. From this we recover much of the theory of shifted tableaux for Schubert calculus on OG(n,2n+1), including a new, geometric proof of the Littlewood-Richardson rule for OG(n,2n+1).Comment: 11 pages, color figures, identical to v1 but metadata correcte

From symmetric fundamental expansions to Schur positivity

We consider families of quasisymmetric functions with the property that if a symmetric function $f$ is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of $f$. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families. This poset includes the Schur functions, the quasisymmetric Schur functions, the fundamental quasisymmetric generating functions of shifted dual equivalence classes, as well as three new families of functions --- one of which is conjectured to be a basis of the vector space of quasisymmetric functions. Each of the six families is realized as the fundamental quasisymmetric generating functions over the classes of some refinement of dual Knuth equivalence. Thus, we also produce a poset of refinements of dual Knuth equivalence. In doing so, we define quasi-dual equivalence to provide classes that generate quasisymmetric Schur functions

Some remarks on sign-balanced and maj-balanced posets

Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if exactly half the linear extensions of P (regarded as permutations of 1,2,...,n) are even permutations, i.e., have an even number of inversions. This concept first arose in the work of Frank Ruskey, who was interested in the efficient generation of all linear extensions of P. We survey a number of techniques for showing that posets are sign-balanced, and more generally, computing their "imbalance." There are close connections with domino tilings and, for certain posets, a "domino generalization" of Schur functions due to Carre and Leclerc. We also say that P is maj-balanced if exactly half the linear extensions of P have even major index. We discuss some similarities and some differences between sign-balanced and maj-balanced posets.Comment: 30 pages. Some inaccuracies in Section 3 have been corrected, and Conjecture 3.6 has been adde

The forgotten monoid

We study properties of the forgotten monoid which appeared in work of Lascoux and Schutzenberger and recently resurfaced in the construction of dual equivalence graphs by Assaf. In particular, we provide an explicit characterization of the forgotten classes in terms of inversion numbers and show that there are n^2-3n+4 forgotten classes in the symmetric group S_n. Each forgotten class contains a canonical element that can be characterized by pattern avoidance. We also show that the sum of Gessel's quasi-symmetric functions over a forgotten class is a 0-1 sum of ribbon-Schur functions.Comment: 13 pages; in version 3 the proof of Proposition 3 is correcte