Blob algebra approach to modular representation theory

TL;DR

This paper explores the deep connection between blob algebras and representation theory, proposing a conjecture linking diagrammatic categories and generalized blob algebras, and proves a key part of this for all n and primes.

Contribution

It introduces a new 'blob category' and proves the graded degree equivalence for all n and primes, advancing the understanding of modular representation theory.

Findings

Proves graded degree part of the conjectured equivalence for all n and primes.

Suggests that decomposition numbers of generalized blob algebras relate to p-Kazhdan Lusztig polynomials.

Establishes a foundation for future verification of the conjecture in broader cases.

Abstract

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.