Summary: THE HOPF ALGEBRA OF UNIFORM BLOCK PERMUTATIONS
MARCELO AGUIAR AND ROSA C. ORELLANA
Abstract. We introduce the Hopf algebra of uniform block permutations and show that
it is self-dual, free, and cofree. These results are closely related to the fact that uniform
block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf
algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric
functions in non-commuting variables of Gebhard, Rosas, and Sagan. These two embeddings
correspond to the factorization of a uniform block permutation as a product of an invertible
element and an idempotent one.
A uniform block permutation of [n] is a certain type of bijection between two set partitions
of [n]. When the blocks of both partitions are singletons, a uniform block permutation is
simply a permutation of [n]. Let Pn be the set of uniform block permutations of [n] and Sn
the subset of permutations of [n]. The set Pn is a monoid in which the invertible elements
are precisely the elements of Sn. These notions are reviewed in Section 1.
This paper introduces and studies a graded Hopf algebra based on the set of uniform block
permutations of [n] for all n 0, by analogy with the graded Hopf algebra of permutations
of Malvenuto and Reutenauer .
Let V be a complex vector space. Classical Schur-Weyl duality states that the symmetric
group algebra can be recovered from the diagonal action of GL(V ) on V n