TY - UNPB

T1 - Deligne categories and representations of the finite general linear group, part 1: universal property

AU - Entova Aizenbud, Inna

AU - Heidersdorf, Thorsten

PY - 2024

Y1 - 2024

N2 - We study the Deligne interpolation categories Rep––––(GLt(Fq)) for t∈C, first introduced by F. Knop. These categories interpolate the categories of finite dimensional complex representations of the finite general linear group GLn(Fq). We describe the morphism spaces in this category via generators and relations. We show that the generating object of this category (an analogue of the representation CFnq of GLn(Fq)) carries the structure of a Frobenius algebra with a compatible Fq-linear structure; we call such objects Fq-linear Frobenius spaces, and show that Rep––––(GLt(Fq)) is the universal symmetric monoidal category generated by such an Fq-linear Frobenius space of categorical dimension t. In the second part of the paper, we prove a similar universal property for a category of representations of GL∞(Fq).

AB - We study the Deligne interpolation categories Rep––––(GLt(Fq)) for t∈C, first introduced by F. Knop. These categories interpolate the categories of finite dimensional complex representations of the finite general linear group GLn(Fq). We describe the morphism spaces in this category via generators and relations. We show that the generating object of this category (an analogue of the representation CFnq of GLn(Fq)) carries the structure of a Frobenius algebra with a compatible Fq-linear structure; we call such objects Fq-linear Frobenius spaces, and show that Rep––––(GLt(Fq)) is the universal symmetric monoidal category generated by such an Fq-linear Frobenius space of categorical dimension t. In the second part of the paper, we prove a similar universal property for a category of representations of GL∞(Fq).

M3 - Preprint

T3 - Transformation Groups

BT - Deligne categories and representations of the finite general linear group, part 1: universal property

PB - Springer

ER -