Transformation Groups Springer Science+Business Media New York (2016)
ON THE PARTITION APPROACH TO
SCHUR–WEYL DUALITY AND
FREE QUANTUM GROUPS
Postfach 151159, 66041
Abstract. We give a general deﬁnition of classical and quantum groups whose repre-
sentation theory is “determined by partitions” and study their structure. This encom-
passes many examples of classical groups for which Schur–Weyl duality is described with
diagram algebras as well as generalizations of P. Deligne’s interpolated categories of rep-
resentations. Our setting is inspired by many previous works on easy quantum groups
and appears to be well suited to the study of free fusion semirings. We classify free fusion
semirings and prove that they can always be realized through our construction, thus solv-
ing several open questions. This suggests a general decomposition result for free quantum
groups which in turn gives information on the compact groups whose Schur–Weyl dual-
ity is implemented by partitions. The paper also contains an appendix by A. Chirvasitu
proving simplicity results for the reduced C*-algebras of some free quantum groups.
Partitions of ﬁnite sets are a priori very simple set-theoretic objects. However,
their combinatorics appears to be quite rich and plays a role in many diﬀerent ar-
eas of mathematics. For example, R. Brauer introduced in  algebras generated
by partitions in pairs to study the invariants of tensor representations of the or-
thogonal and symplectic groups. His ideas were later developed by several authors
to describe Schur–Weyl duality for other classes of groups like complex reﬂection
groups or wreath products (see Section 4). More precisely, several classes of al-
gebras generated by partitions were introduced and it was proved that they are
isomorphic to centralizer algebras for certain tensor representations of the involved
groups. These ideas were reinterpreted and extended in a more categorical setting
by F. Knop and M. Mori using the idea of interpolated categories of representations
introduced by P. Deligne for symmetric groups in . In a diﬀerent context, it
was discovered by R. Speicher that passing from the combinatorics of all partitions
Partially supported by the ERC advanced grant “Noncommutative distributions in
free probability” held by R. Speicher.
Received August 5, 2015. Accepted January 8, 2016.
Corresponding Author: A. Freslon, e-mail: Amaury.Freslon@math.u-psud.fr.
Vol. 22, No.
, 2017, pp.