We make a systematic study of a new combinatorial construction called a dual equivalence graph. Motivated by the dual equivalence relation on standard Young tableaux introduced by Haiman, we axiomatize such constructions and prove that the generating functions of these graphs are Schur positive. We construct a graph on k-ribbon tableaux which we conjecture to be a dual equivalence graph, and we...

Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a kvertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph G has a face-k-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k-flow. However, if the surface is nonorientable, then a...

The k-Schur functions were first introduced by Lapointe, Lascoux and Morse [18] in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono [17] as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Br...

Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a kvertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph G has a face-k-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k-flow. However, if the surface is nonorientable, then a...

1.1. Let G be a simply connected semisimple group with Borel subgroup B, N = [B,B] and let H = B/N . Let g, b, n and h be the respective Lie algebras of these groups. Let G be the Langlands dual group with dual Borel B, etc. Let X be a smooth curve or the formal disc D or the formal punctured disc D×. 1.2. Recall that a g-oper over X is a G-bundle FLG = F with a reduction FLB to B and a connect...

We study the class C of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, k-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur fu...

Total subspaces in dual Banach spaces which are not norming over any infinite dimensional subspace Abstract. The main result: the dual of separable Banach space X contains a total subspace which is not norming over any infinite dimensional subspace of X if and only if X has a nonquasireflexive quotient space with the strictly singular quotient mapping. Let X be a Banach space and X * be its dua...

Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions ...

In [9] it is proved that there are subsets M of the complex plane such that for any two entire functions f and g if f [M ] = g[M ] then f = g. In [3] it was shown that the continuum hypothesis (CH) implies the existence of a similar set M ⊂ R for the class Cn(R) of continuous nowhere constant functions from R to R, while it follows from the results in [5] and [7] that the existence of such a se...

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified HallLittlewood polynomials indexed by any diagram δ ⊂ Z × Z, written as H̃δ(X; q, t) and H̃δ(X; 0, t), respectively. We then give an explicit Schur expansion of H̃δ(X; 0,...