Categories of two-colored pair partitions part I: categories indexed by cyclic groups
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Categories of two-colored pair partitions part I: categories indexed by cyclic groups Alexander Mang1
· Moritz Weber1
Received: 23 October 2018 / Accepted: 7 February 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract 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 Un and Wang’s free unitary quantum group Un+ . 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. Keywords Quantum group · Unitary easy quantum group · Unitary group · Half-liberation · Tensor category · Two-colored partition · Partition of a set · Category of partitions Mathematics Subject Classification 05A18 (Primary) · 20G42 (Secondary)
The second author was supported by the ERC Advanced Grant NCDFP, held by Roland Speicher, by the SFB-TRR 195, and by the DFG project Quantenautomorphismen von Graphen. This work was part of the first author’s Bachelor’s thesis.
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Alexander Mang [email protected] Moritz Weber [email protected]
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Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany
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A. Mang, M. Weber
Introduction This article is part of the classification program begun in [11], having its roots in [3]. Our base objects are partitions of finite sets into disjoint subsets, the blocks. In addition, the points are colored either black or white. We represent partitions pictorially as diagrams using strings representing the blocks; see also [9], [10]. If a set of partitions is closed under certain natural operations like horizontal or vertical concatenation or reflection at some axes, we call it a category of partitions. Categories of partitions play a crucial role in Banica and Speicher’s approach ([3,14,15]) to compact quantum groups in Woronowicz’s sense ([16–19]). Although our investigations employ purely combinatorial means, let us briefly mention how they relate to the half-liberation procedures of Banica and Speicher. The half-liberated orthogonal quantum group On∗ , introduced by Banica and Speicher in [3], represents a midway point between the free orthogonal quantum group On+ , constructed by Wang in [13], and the classical orthogonal group On over the complex numbers. It is defined by replacing the commutation relations ab = ba by the halfcommutation relations abc = cba. On the combinatorial side, these h
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