Narrow Results

We show that there exist two natural endomorphism algebras for shuffle bialgebras such as Sh(X), where X is a graded set. One of these endomorphism algebras is a natural extension of the Malvenuto–Reutenauer Hopf algebra and is defined using graded permutations. The other one, the dendriform descent algebra, is a subalgebra of the first defined by mimicking the definition of the descent algebras by convolution from the graded projections in the tensor algebra. We study these algebras for their own, show that they carry bidendriform structures and establish freeness properties, study their generators, dimensions, bases, and also feature their relations to the internal structure of shuffle algebras. As an application of these ideas, we give a new proof of Chapoton's rigidity theorem for shuffle bialgebras.

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Schürmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behavior of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Schürmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of non-crossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Schürmann’s aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively, Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature.

We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature $\beta$ in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd charge. Our methods provide a unified framework extending the de Bruijn integral identities from classical $\beta$-ensembles ($\beta =1,2,4$) to multicomponent ensembles, as well as to iterated integrals of more general determinantal integrands.