Abstract

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 type (Formula presented.). In this paper, we take that observation far beyond its original scope. We conjecture that for (Formula presented.) there is an equivalence of categories between the characteristic (Formula presented.) diagrammatic Hecke category and a ‘blob category’ that we introduce (using certain quotients of KLR algebras called generalized blob algebras). Using alcove geometry we prove the ‘graded degree’ part of this equivalence for all (Formula presented.) and all prime numbers (Formula presented.). If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic (Formula presented.) give the (Formula presented.) -Kazhdan–Lusztig polynomials in type (Formula presented.). We prove this for (Formula presented.), the only case where the (Formula presented.) -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.