 
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 selfdual, 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 noncommuting 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.
Introduction
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 [21].
Let V be a complex vector space. Classical SchurWeyl duality states that the symmetric
group algebra can be recovered from the diagonal action of GL(V ) on V n
