Related papers: The extra basis in noncommuting variables
We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full…
We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages…
The aim of this work is to outline in some detail the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green's functions. The key object is a…
The parabolic Kazhdan-Lusztig polynomials for Grassmannians can be computed by counting Dyck partitions. We "lift" this combinatorial formula to the corresponding category of singular Soergel bimodules to obtain bases of the Hom spaces…
The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic analogues of the by now classical ``square'' of…
We consider the group $(\mathcal{G},*)$ of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where ``$*$'' denotes the convolution operation. We introduce a larger group $(\widetilde{\mathcal{G}},*)$ of…
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric…
Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the…
We introduce a general class of combinatorial objects, which we call \emph{multi-complexes}, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of…
Let us fix a positive integer $\nu>1$. For each positive integer $n>1$, we consider a normal supercharacter theory $\mathcal{S}_n$ of $G_n$, where $G_n$ is the direct-product of $n-1$ copies of the cyclic group of order $\nu$. Then we endow…
In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain…
We discuss a formulation of harmonic superspace approach for noncommuative N=2 supersymmetric field theories paying main attention on new features arising because of noncommutativity. We begin with the known notions of the harmonic…
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…
Many identities involving symmetric functions can be proved through bijective manipulations of tableaux. In this paper, we prove identities involving polysymmetric functions through bijections and sign-reversing involutions. In their paper…
The algebra of quasisymmetric functions QSym and the shuffle algebra of compositions Sh are isomorphic as graded Hopf algebras (in characteristic zero), and isomorphisms between them can be specified via shuffle bases of QSym. We use the…
The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization…
We briefly review Boltzmann-Gibbs and nonextensive statistical mechanics as well as their connections with Fokker-Planck equations and with existing central limit theorems. We then provide some hints that might pave the road to the proof of…
The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…
We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing…