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

A nowhere-zero k-flow on a graph G is an assignment of a direction and a non-zero integer in absolute value smaller than k to each edge of G in such a way that, for each vertex, the sum of incoming values equals the sum of outgoing values. Nowhere-zero flow problems evolved from flowcolouring duality to a theory which has a central role in graph theory. There are many important flow problems wh...

In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A∗ = A − ...

Various equicontinuity properties for families of Markov operators have been – and still are used in the study existence uniqueness invariant probability these operators, asymptotic stability. We prove a general result on equivalence concepts. It allows comparing results literature switching from one view to another, which is technically convenient proofs. More precisely, characterisation based...

Given a C ⁎ -dynamical system ( A , G α ) with discrete group, Schur -multipliers and Herz–Schur are used to implement approximation properties, namely exactness the strong operator property (SOAP), of ⋊ r . The resulting characterisations SOAP generalise corresponding statements for reduced group -algebra.

Quantum Weyl reciprocity relates the representation theory of Hecke algebras of type A with that of q-Schur algebras. This paper establishes that Weyl reciprocity holds integrally (i. e., over the ringZ[q;q ] of Laurent polynomials) and that it behaves well under base-change. A key ingredient in our approach involves the theory of tilting modules for q-Schur algebras. New results obtained in th...