Blob algebra approach to modular representation theory

Two decades ago, Martin and 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 typeA similar to 1. In this paper, we take that observation far beyond its original scope. We conjecture that forA similar to nthere is an equivalence of categories between the characteristicpdiagrammatic Hecke category and a 'blob category' that we introduce (using certain quotients of KLR algebras calledgeneralized blob algebras). Using alcove geometry we prove the 'graded degree' part of this equivalence for allnand all prime numbersp. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristicpgive thep-Kazhdan-Lusztig polynomials in typeA similar to n. We prove this forA similar to 1, the only case where thep-Kazhdan-Lusztig polynomials are known. This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures.