The Hopf Algebra of Uniform Block Permutations

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. 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 [17]. 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: if dimV ≥ n then (1) CSn ∼= EndGL(V )(V ⊗n) . Malvenuto and Reutenauer deduce from here the existence of a multiplication among permutations as follows. Given σ ∈ Sp and τ ∈ Sq, view them as linear endomorphisms of the tensor algebra T (V ) := ⊕