Arboreal tensor categories
Nate Harman, Ilia Nekrasov, Andrew Snowden

TL;DR
This paper introduces new symmetric tensor categories derived from tree combinatorics, including a discrete family and a complex parameterized family, expanding the landscape of pre-Tannakian categories with superexponential growth.
Contribution
It constructs novel tensor categories based on trees, proves key properties like measure classification and semi-simplicity, and demonstrates a unique family not obtainable by existing interpolation methods.
Findings
Defined the discrete family (n)nd continuous family (t)ategories.
Established measure classification for trees and proved semi-simplicity.
Showed (t)ategories are the first of their kind with superexponential growth not from interpolation.
Abstract
We introduce some new symmetric tensor categories based on the combinatorics of trees: a discrete family , for an integer, and a continuous family , for a complex number. The construction is based on the general oligomorphic theory of Harman--Snowden, but relies on two non-trivial results we establish. The first determines the measures for the class of trees, and the second is a semi-simplicity theorem. These categories have some notable properties: for instance, is the first example of a 1-parameter family of pre-Tannakian categories of superexponential growth that cannot be obtained by interpolating categories of moderate growth.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Algebraic structures and combinatorial models · Finite Group Theory Research
