Categories of two-colored pair partitions part I: categories indexed by cyclic groups

We classify certain categories of partitions of finite sets subject to specific rules on the coloring of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one black point when rotated to one line; however, crossings are al...

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Bibliographic Details
Published in:The Ramanujan journal Vol. 53; no. 1; pp. 181 - 208
Main Authors: Mang, Alexander, Weber, Moritz
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2020
Springer Nature B.V
Subjects:
Categories
Combinatorial analysis
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Partitions (mathematics)
Quantum group
Half-liberation
Tensor category
20G42 (Secondary)
Category of partitions
05A18 (Primary)
Two-colored partition
Unitary easy quantum group
Partition of a set
Unitary group
ISSN:1382-4090, 1572-9303
Online Access:Get full text
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Summary:We classify certain categories of partitions of finite sets subject to specific rules on the coloring of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one black point when rotated to one line; however, crossings are allowed. There are two families of such categories, the first of which is indexed by cyclic groups and is covered in the present article; the second family will be the content of a follow-up article. Via a Tannaka–Krein result, the categories in the two families correspond to easy quantum groups interpolating the classical unitary group U n and Wang’s free unitary quantum group U n + . In fact, they are all half-liberated in some sense and our results imply that there are many more half-liberation procedures than previously expected. However, we focus on a purely combinatorial approach leaving quantum group aspects aside.
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ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-019-00149-w