Graded decomposition numbers for the blob algebra

Proc. London Math. Soc.

2020

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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 A∼1. In this paper, we take that observation far beyond its original scope. We conjecture that for 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 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 A∼n. We prove this for A∼1, the only case where the p‐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.

Section: The Blob Algebramentioning

confidence: 99%

“…Given a path p we say that a subset of consecutive steps of p is a wall to wall step if these steps form a straight line between two consecutive hyperplanes. In the following lemma (proved in [38,Lemma 4.7]), for a path p(t) we think of t as a variable representing time.…”

Section: Description Of Standard Tableaux With the Same Residue As λmentioning

confidence: 99%

Blob algebra approach to modular representation theory

Libedinsky

Proc. London Math. Soc.

2020

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“…The existence of a graded cellular basis for the blob algebra allows one to define graded decomposition numbers. Recently, the first author has succeeded in calculating these graded decomposition numbers (see [20]).…”

Section: Proofmentioning

confidence: 99%

Graded cellular bases for Temperley–Lieb algebras of type A and B

Ryom-Hansen

2013

J Algebr Comb

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“…An answer to Q1 was given in [Pl13] and [PR14]. In [PR14], it was proven that b n is endowed with a non-trivial Z-grading by showing that b n admits a KLR-type presentation, that is, with homogeneous generators e(i), y r , and ψ s (see Theorem 4.3 below).…”

Section: Introductionmentioning

confidence: 99%

“…They are called graded decomposition numbers and are denoted by d λ,µ (v) ∈ Z[v, v −1 ]. In [Pl13], the graded decomposition numbers of b n were calculated explicitly. On the other hand, by taking advantage of the fact that there exists a closed formula for the Kazhdan-Lusztig polynomials associated with W , it can be concluded that for all x, y ∈ W and λ, µ ∈ Λ n such that λ ∈ A x and µ ∈ A y .…”

Section: Introductionmentioning

confidence: 99%

Blob algebra and two-color Soergel calculus

Espinoza

Journal of Pure and Applied Algebra

2019

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In 2003, Martin and Woodcock noticed a connection between the representation theory of the blob algebra and the Kazhdan-Lusztig polynomials associated with the infinite dihedral group. However, no conceptual explanation for this coincidence has yet been provided. In this study, a possible explanation of this phenomenon is suggested by enunciating a conjecture that relates the endomorphism algebra of Bott-Samelson bimodules to certain subalgebras of the blob algebra obtained by idempotent truncation. Evidence supporting this conjecture is provided.