Stable Grothendieck rings of wreath product categories

Abstract

Abstract Let k be an algebraically closed field of characteristic zero, and let

$${\mathcal {C}} = {\mathcal {R}} -\hbox {mod}$$

C = R - mod

be the category of finite-dimensional modules over a fixed Hopf algebra over k. One may form the wreath product categories

$$ {\mathcal {W}}_{n}({\mathcal {C}}) = ( {\mathcal {R}} \wr S_n)-\hbox {mod}$$

W n

( C )

=

( R ≀

S n

)

- 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

$$ {\mathcal {W}}_{n}({\mathcal {C}}) $$

W n

( 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

$$S_t({\mathcal {C}})$$

S t

( 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

$$ {\mathcal {R}} $$

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

$${\mathcal {C}}$$

C

is a tensor category.