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Stable Grothendieck rings of wreath product categories

  • Christopher RybaEmail author
Article
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Abstract

Let k be an algebraically closed field of characteristic zero, and let \({\mathcal {C}} = {\mathcal {R}} -\hbox {mod}\) be the category of finite-dimensional modules over a fixed Hopf algebra over k. One may form the wreath product categories \( {\mathcal {W}}_{n}({\mathcal {C}}) = ( {\mathcal {R}} \wr S_n)-\hbox {mod}\) whose Grothendieck groups inherit the structure of a ring. Fixing distinguished generating sets (called basic hooks) of the Grothendieck rings, the classification of the simple objects in \( {\mathcal {W}}_{n}({\mathcal {C}}) \) allows one to demonstrate stability of structure constants in the Grothendieck rings (appropriately understood), and hence define a limiting Grothendieck ring. This ring is the Grothendieck ring of the wreath product Deligne category \(S_t({\mathcal {C}})\). We give a presentation of the ring and an expression for the distinguished basis arising from simple objects in the wreath product categories as polynomials in basic hooks. We discuss some applications when \( {\mathcal {R}} \) is the group algebra of a finite group, and some results about stable Kronecker coefficients. Finally, we explain how to generalise to the setting where \({\mathcal {C}}\) is a tensor category.

Keywords

Wreath products Grothendieck rings Deligne categories 

Notes

Acknowledgements

The author would like to thank Pavel Etingof for useful conversations and for comments on an earlier version of this paper, as well as the referees for their feedback and suggestions.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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