Stable Grothendieck rings of wreath product categories
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10801_2018_856_ReferencePDF.pdf
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Author(s)
Ryba, Christopher
Date Issued
November 29, 2018
Publisher
Springer US
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s10801-018-0856-9
