Jennifer Morse

Tableau atoms and a new Macdonald positivity conjecture

A snap action fluid control valve, the operation of which is controlled by a relatively slow acting thermally responsive actuator member. The valve of this invention is particularly adapted for use in controlling flow of fluid to a fluid operable clutch or the like for operation thereof. The thermally responsive actuator portion of the valve senses temperature of a fluid, the temperature of which is responsive to operation of a fan which is operated through the clutch.

Tableaux on k + 1-crores, reduced words for affine permutations, and k -Schur expansions

The k-Young lattice Yk is a partial order on partitons with no part larger than k. This weak subposet of the Young lattice originated (Duke Math. J. 116 (2003) 103-146) from the study of the k-Schur functions sλ(k), Symmetric functions that from a natural basis of the space spanned by homogeneous funtions indexed by k-bounded partitions. The chains in the k-Young lattice are induced by a Pieritype rule experimentally satisfied by the k-Schur functions. Here, using a natural bijection between k-bounded partitons and k + 1-cores, we establish an algorithm for identifying chains in the k- Young lattice with certain tableaux on k + 1 cores. This algorithm reveals that the k-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric gruop S˜k+1 by a maximal parabolic subgruop. From this, the conjectured k-Pieri rule implies that the k-Kostka matrix connecting the homogeneous basis {hλ}λ∈Yk to {sλ(k)}λ∈Yk may now be obtained by counting appropriate classes of tableaux on k + 1-cores. This suggests that the conjecturally positive k-Schur expansion coefficients for Macdonald polynomials (reducing to q, t-Kostka polynomials for large k) could be described by a q, t-statistic on these tableaux, or equivalently on reduced words for affine permutations.

QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to su(l) are shown to be k-Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k-Schur functions defined on weights of k-tableaux that, given Shimozonos conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.

Schur function analogs for a filtration of the symmetric function space

We work here with the linear span Λt(k) of Hall-Littlewood polynomials indexed by partitions whose first part is no larger than k. The sequence of spaces Λt(k) yields a filtration of the space Λ of symmetric functions in an infinite alphabet X. In joint work with Lascoux [4] we gave a combinatorial construction of a family of symmetric polynomials {Aλ(k)[X; t]}λ1 ≤ k, with N[t]-integral Schur function expansions, which we conjectured to yield a basis for Λt(k). Our primary motivation for this construction is to provide a positive integral factorization of the Macdonald q, t-Kostka matrix. More precisely, we conjecture that the connection coefficients expressing the Hall-Littlewood or Macdonald polynomials belonging to Λt(k) in terms of the basis {Aλ(k)[X; t]}λ1 ≤ k are polynomials in N [q, t]. We give here a purely algebraic construction of a new family {sλ(k)[X; t]}λ1 ≤ kof polynomials in Λt(k) which we conjecture is identical to {Aλ(k)[X; t]}λ1 ≤ k. We prove that {sλ(k) [X; t]}λ1 ≤ k is in fact a basis of Λt(k) and derive several further properties including that sλ(k)[X; t] reduces to the Schur function sλ[X] for sufficiently large k. We also state a number of conjectures which reveal that the polynomials {sλ(k)[X; t]}λ1 ≤ k are in fact the natural analogues of Schur functions for the space Λt(k).

Stanley Symmetric Functions and Peterson Algebras

This purpose of this chapter is to introduce Stanley symmetric functions and affine Stanley symmetric functions from the combinatorial and algebraic point of view. The presentation roughly follows three lectures I gave at a conference titled “Affine Schubert Calculus” held in July of 2010 at the Fields Institute in Toronto.

Schur function identities, their t-analogs, and k-Schur irreducibility

Abstract We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t -analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act trivially on the k -Schur functions, thus leading to a concept of irreducibility for these functions.

Crystal Approach to Affine Schubert Calculus

Author(s): Morse, Jennifer; Schilling, Anne | Abstract: We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-

Structure constants for K-theory of Grassmannians, revisited

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Affine Schubert Calculus

affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a

Primer on k -Schur Functions

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