scholarly journals On relative Grothendieck rings

1975 ◽  
pp. 79-131 ◽  
Author(s):  
Andreas W. M. Dress
Keyword(s):  
Grothendieck Rings

scholarly journals The center of the extended Haagerup subfactor has 22 simple objects

2017 ◽  
Vol 28 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Scott Morrison ◽  
Kevin Walker
Keyword(s):  
Fusion Category ◽  
Dimension Function ◽  
Fusion Ring ◽  
Fusion Categories ◽  
Modular Data ◽  
Grothendieck Rings ◽  
Combinatorial Data

We explain a technique for discovering the number of simple objects in [Formula: see text], the center of a fusion category [Formula: see text], as well as the combinatorial data of the induction and restriction functors at the level of Grothendieck rings. The only input is the fusion ring [Formula: see text] and the dimension function [Formula: see text]. In particular, we apply this to deduce that the center of the extended Haagerup subfactor has 22 simple objects, along with their decompositions as objects in either of the fusion categories associated to the subfactor. This information has been used subsequently in [T. Gannon and S. Morrison, Modular data for the extended Haagerup subfactor (2016), arXiv:1606.07165 .] to compute the full modular data. This is the published version of arXiv:1404.3955 .

scholarly journals Abstract Grothendieck rings

1987 ◽  
Vol 53 (1) ◽  
pp. 17-25
Author(s):  
Kazimierz Szymiczek
Keyword(s):  
Grothendieck Rings

scholarly journals The Structure of the Grothendieck Rings of Wreath Product Deligne Categories and their Generalisations

2019 ◽  
Author(s):  
Christopher Ryba
Keyword(s):  
Hopf Algebra ◽  
Wreath Product ◽  
Tensor Category ◽  
Product Category ◽  
Closed Field ◽  
Grothendieck Ring ◽  
Product Categories ◽  
Witt Vectors ◽  
Grothendieck Rings ◽  
Lambda Ring

Abstract Given a tensor category $\mathcal{C}$ over an algebraically closed field of characteristic zero, we may form the wreath product category $\mathcal{W}_n(\mathcal{C})$. It was shown in [10] that the Grothendieck rings of these wreath product categories stabilise in some sense as $n \to \infty $. The resulting “limit” ring, $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$, is isomorphic to the Grothendieck ring of the wreath product Deligne category $S_t(\mathcal{C})$ as defined by [9] (although it is also related to $FI_G$-modules). This ring only depends on the Grothendieck ring $\mathcal{G}(\mathcal{C})$. Given a ring $R$ that is free as a $\mathbb{Z}$-module, we construct a ring $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ that specialises to $\mathcal{G}_\infty ^{\mathbb{Z}}(\mathcal{C})$ when $R = \mathcal{G}(\mathcal{C})$. We give a description of $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ using generators very similar to the basic hooks of [5]. We also show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is a $\lambda $-ring wherever $R$ is and that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is (unconditionally) a Hopf algebra. Finally, we show that $\mathcal{G}_\infty ^{\mathbb{Z}}(R)$ is isomorphic to the Hopf algebra of distributions on the formal neighbourhood of the identity in $(W\otimes _{\mathbb{Z}} R)^\times $, where $W$ is the ring of Big Witt Vectors.

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