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 (and prove for $\tilde{A}_1$, $p\neq 2$) an algorithm to produce the $p$-Kazhdan-Lusztig polynomials in type $\tilde{A}_n$, using "generalized blob algebras", that are quotients of certain KLR algebras. It is of fundamental importance for modular representation theory to understand these polynomials, as they control the modular representation theory of the symmetric group and of the special linear group, by recent work of G. Williamson and S. Riche. We believe that our conjecture is the shadow of an equivalence between the Hecke category for $\tilde{A}_n$ and a certain "Blob category", that we introduce.

中文翻译：

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Blob代数方法用于模块表示理论

二十年前，P。Martin和D. Woodcock在统计力学和表示理论之间建立了令人惊讶的（预言的）联系。他们观察到，斑点代数的分解数（出现在传递矩阵代数的上下文中）是类型为\\ tilde {A} _1 $的Kazhdan-Lusztig多项式。在本文中，我们认为该观察结果超出了其原始范围。我们猜想（并证明$ \ tilde {A} _1 $，$ p \ neq 2 $）使用“广义blob”来生成类型为$ \ tilde {A} _n $的$ p $ -Kazhdan-Lusztig多项式的算法代数”，即某些KLR代数的商。理解这些多项式对模块表示理论至关重要，因为它们通过G. Williamson和S.的最新工作控制了对称组和特殊线性组的模块表示理论。里奇 我们认为，我们的猜想是$ \ tilde {A} _n $的Hecke类与我们引入的某个“ Blob类”之间对等的阴影。