A function f from a countable product X = ∏ i Xi of Polish spaces into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X. We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of Xi’s. This result is used to distinguish between n-cube density notions for different n ≤ ω, where ω-cube den...

We define a new family F̃w(X) of generating functions for w ∈ S̃n which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric fu...

We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. subsequently use this geometric characterization to answer several questions in analysis. Notably, it follows the Lipschitz-free space $\mathcal{F}(M)$ over $M$ is dual if and only 1-unrectifiable. Furthermore, we...

In a recent paper, Dipper and Doty, [2], introduced certain finite dimensional algebras, associated with the natural module of the general linear group and its dual, which they call rational Schur algebras. We give a proof, via tilting modules, that these algebra are in fact generalized Schur algebras. Using the same technique we show that certain finite dimensional algebras with classical grou...

Dual equivalence puts a crystal-like structure on linear representations of the symmetric group that affords many nice combinatorial properties. In this talk, we extend this theory to type B, putting an analogous structure on projective representations of the symmetric group. On the level of generating functions, the type A theory gives a universal method for proving Schur positivity, and the t...

We establish a positivity property for the difference of products of certain Schur functions, sλ(x), where λ varies over a fundamental Weyl chamber in R n and x belongs to the positive orthant in R. Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions ...

We introduce an analogue of the q-Schur algebra associated to Coxeter systems of type b A n?1. We give two constructions of this algebra. The rst construction realizes the algebra as a certain endomorphism algebra arising from an aane Hecke algebra of type b A r?1 , where n r. This generalizes the original q-Schur algebra as deened by Dipper and James, and the new algebra contains the ordinary ...

We introduce an analogue of the q-Schur algebra associated to Coxeter systems of type A n−1. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an affine Hecke algebra of type A r−1 , where n ≥ r. This generalizes the original q-Schur algebra as defined by Dipper and James, and the new algebra contains the ordina...

We investigate Tukey functions from the ideal of all closed nowhere dense subsets of 2N. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of N. We also prove non-existence of various special types of Tukey reductions from the nowhere dense ideal to analytic P-ideals. In connection with these res...