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Trihedral Soergel bimodules
Publication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Tubbenhauer, Daniel
The quantum Satake correspondence relates dihedral Soergel bimodules to the
semisimple quotient of the quantum $\mathfrak{sl}_2$ representation category.
It also establishes a precise relation between the simple transitive
$2$-representations of both monoidal categories, which are indexed by bicolored
$\mathsf{ADE}$ Dynkin diagrams.
Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$
Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$
representation category, we introduce trihedral Hecke algebras and Soergel
bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These
have their own Kazhdan-Lusztig combinatorics, simple transitive
$2$-representations corresponding to tricolored generalized $\mathsf{ADE}$
Dynkin diagrams.
Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations
Publication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Zhang, Xiaoting
The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex J of a fiat 2-category C and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of C associated with a diagonal H-cell in J. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types B-3 and B-4.
Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations
Publication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Zhang, Xiaoting
The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex J of a fiat 2-category C and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of C associated with a diagonal H-cell in J. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types B-3 and B-4.
Trihedral Soergel bimodules
Publication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Tubbenhauer, Daniel
The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum sl(2) representation category. It also establishes a precise relation between the simple transitive 2-representations of both monoidal cate-gories, which are indexed by bicolored ADE Dynldn diagrams.
Using the quantum Satake correspondence between affine A(2) Soergel bimodules and the semisimple quotient of the quantum sl(3)representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive 2-representations corresponding to tricolored generalized ADE Dynkin diagrams.
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Funding agency
Fundação para a Ciência e a Tecnologia
Funding programme
5876
Funding Award Number
UID/MAT/04459/2013