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Register a new userMultiplicative structures of the immaculate basis of non-commutative symmetric functions
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We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the Schur basis for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutati
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The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.
We consider the group (G,*) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where * denotes the convolution operation. We introduce a larger group (Gtilde,*) of unitized functions from the same incidence algebra, which satisfy a weaker condition of being semi-multiplicative. The natural action of Gtilde on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of Gtilde in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bozejko and Wysoczanski. It is known that the group G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that Gtilde can also be identified as group of characters of a Hopf algebra T, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion of G in Gtilde turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.
We apply down operators in the affine nilCoxeter algebra to yield explicit combinatorial expansions for certain families of non-commutative k-Schur functions. This yields a combinatorial interpretation for a new family of k-Littlewood-Richardson coefficients.
In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains.
A non-commutative, planar, Hopf algebra of rooted trees was proposed in L. Foissy, Bull. Sci. Math. 126 (2002) 193-239. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators).
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