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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2016, Issue 720

# On the representation theory of partition (easy) quantum groups

Amaury Freslon
/ Moritz Weber
Published Online: 2014-07-09 | DOI: https://doi.org/10.1515/crelle-2014-0049

## Abstract

Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S. L. Woronowicz. In the case of easy quantum groups (also called partition quantum groups), the intertwiner spaces are given by the combinatorics of partitions, see the initial work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group ${S}_{N}^{+}$, the free orthogonal quantum group ${O}_{N}^{+}$ as well as the hyperoctahedral quantum group ${H}_{N}^{+}$. We then extend our work to unitary easy quantum groups and link it with a “freeness conjecture” of T. Banica and R. Vergnioux.

## References

• [1]

Banica T., Théorie des représentations du groupe quantique compact libre $O\left(n\right)$, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 3, 241–244. Google Scholar

• [2]

Banica T., Le groupe quantique compact libre $U\left(n\right)$, Comm. Math. Phys. 190 (1997), no. 1, 143–172. Google Scholar

• [3]

Banica T., Symmetries of a generic coaction, Math. Ann. 314 (1999), no. 4, 763–780. Google Scholar

• [4]

Banica T., Quantum groups and Fuss–Catalan algebras, Comm. Math. Phys. 226 (2002), no. 1, 221–232. Google Scholar

• [5]

Banica T., Bichon J. and Collins B., The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345–384. Google Scholar

• [6]

Banica T., Curran S. and Speicher R., Classification results for easy quantum groups, Pacific J. Math. 247 (2010), no. 1, 1–26. Google Scholar

• [7]

Banica T. and Speicher R., Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), no. 4, 1461–1501. Google Scholar

• [8]

Banica T. and Vergnioux R., Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), no. 3, 327–359. Google Scholar

• [9]

Bisch D. and Jones V., Algebras associated to intermediate subfactors, Invent. Math. 128 (1997), no. 1, 89–157. Google Scholar

• [10]

Brauer R., On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857–872. Google Scholar

• [11]

Copeland A., Schmidt F. and Simion R., On two determinants with interesting factorizations, Discrete Math. 256 (2002), no. 1, 449–458. Google Scholar

• [12]

Lehrer G. and Zhang R., The second fundamental theorem of invariant theory for the orthogonal group, Ann. of Math. (2) 176 (2012), 2031–2054. Google Scholar

• [13]

Maes A. and Van Daele A., Notes on compact quantum groups, preprint 1998, http://arxiv.org/abs/math/9803122.

• [14]

Raum S., Isomorphisms and fusion rules of orthogonal free quantum groups and their free complexifications, Proc. Amer. Math. Soc 140 (2012), 3207–3218. Google Scholar

• [15]

Raum S. and Weber M., A connection between easy quantum groups, varieties of groups and reflection groups, preprint 2012, http://arxiv.org/abs/1212.4742.

• [16]

Raum S. and Weber M., Easy quantum groups and quantum subgroups of a semi-direct product quantum group, preprint 2013, http://arxiv.org/abs/1311.7630.

• [17]

Raum S. and Weber M., The full classification of orthogonal easy quantum groups, preprint 2013, http://arxiv.org/abs/1312.3857.

• [18]

Takesaki M., Theory of operator algebras I, Encyclopaedia Math. Sci. 124, Springer-Verlag, Berlin 2002. Google Scholar

• [19]

Tarrago P. and Weber M., Unitary easy quantum groups, in preparation. Google Scholar

• [20]

Tutte W., The matrix of chromatic joins, J. Combin. Theory Ser. B 57 (1993), no. 2, 269–288. Google Scholar

• [21]

Van Daele A. and Wang S., Universal quantum groups, Internat. J. Math. 7 (1996), 255–264. Google Scholar

• [22]

Wang S., Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), no. 3, 671–692. Google Scholar

• [23]

Wang S., Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195–211.

• [24]

Weber M., On the classification of easy quantum groups – The nonhyperoctahedral and the half-liberated case, Adv. Math. 245 (2013), no. 1, 500–533. Google Scholar

• [25]

Woronowicz S., Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), no. 1, 35–76. Google Scholar

• [26]

Woronowicz S., Compact quantum groups, Quantum symmetries/Symétries quantiques (Les Houches 1995), North-Holland, Amsterdam (1998), 845–884. Google Scholar

Revised: 2014-03-06

Published Online: 2014-07-09

Published in Print: 2016-11-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 720, Pages 155–197, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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