The anti-spherical category

We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localisation procedure for the anti-spherical category, from which we construct a "light leaves" basis of morphisms. Our techniques may be used to calculate many new elements of the $p$-canonical basis in the anti-spherical module.Comment: Best viewed in colo

Physiological Mechanisms Underlying Motion-Induced Blindness

Visual disappearance illusions - such as motion-induced blindness (MIB) - are commonly used to study the neural underpinnings of visual perception. In such illusions a salient visual target becomes perceptually invisible. Previous studies are inconsistent regarding the role of primary visual cortex (V1) in these illusions. Here we provide physiological and psychophysical evidence supporting a role for V1 in generating MIB

Standard objects in 2-braid groups

For any Coxeter system we establish the existence (conjectured by Rouquier) of analogues of standard and costandard objects in 2-braid groups. This generalizes a known extension vanishing formula in the BGG category O.Comment: 23 pages. v2: final versio

Blob algebra approach to modular representation theory

Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type $\tilde{A}_1$. In this paper we take that observation far beyond its original scope. We conjecture that for $\tilde{A}_n$ there is an equivalence of categories between the characteristic $p$ diagrammatic Hecke category and a "blob category" that we introduce (using certain quotients of KLR algebras called \emph{generalized blob algebras}). Using alcove geometry we prove the "graded degree" part of this equivalence for all $n$ and all prime numbers $p$. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic $p$ give the $p$-Kazhdan Lusztig polynomials in type $\tilde{A}_n$. We prove this for $\tilde{A}_1$, the only case where the $p$-Kazhdan Lusztig polynomials are known.Comment: 43 pages, many figures, best viewed in color. Accepted for publication in the Proceedings of the LM