Decomposition of the product of a monomial and a Demazure atom

International audience We prove that the product of a monomial and a Demazure atom is a positive sum of Demazure atoms combinatorially. This result proves one particular case in a conjecture which provides an approach to a combinatorial proof of Schubert positivity property.

Free-Boolean independence with amalgamation

In this paper, we develop the notion of free-Boolean independence in an amalgamated setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence with amalgamation. We also provide a characterization of free-Boolean independence by conditions in terms of mixed moments. In addition, we study free-Boolean independence over a [Formula: see text]-algebra and prove a positivity property. A central limit law for our free-Boolean independence with amalgamation is also studied.

The comultiplication of modified quantum affine 𝔰𝔩n

Let [Formula: see text] be the modified quantum affine [Formula: see text] and let [Formula: see text] be the positive part of quantum affine [Formula: see text]. Let [Formula: see text] be the canonical basis of [Formula: see text] and let [Formula: see text] be the canonical basis of [Formula: see text]. In this paper, we use the theory of affine quantum Schur algebras to prove that the structure constants for the comultiplication with respect to [Formula: see text] are determined by the structure constants for the comultiplication with respect to [Formula: see text] for [Formula: see text]. In particular, from the positivity property for the comultiplication of [Formula: see text], we obtain the positivity property for the comultiplication of [Formula: see text], which is conjectured by Lusztig [Introduction to Quantum Groups, Progress in Mathematics, Vol. 110 (Birkhäuser, Boston, 1993), 25.4.2].

Arithmetic of character varieties of free groups

Via counting over finite fields, we derive explicit formulas for the [Formula: see text]-polynomials and Euler characteristics of [Formula: see text]- and [Formula: see text]-character varieties of free groups. We prove a positivity property for these polynomials and relate them to the number of subgroups of finite index.

A total positivity property of the Marchenko-Pastur Law

A property of the Marchenko-Pastur measure related to total positivity is presented. The theoretical results are applied to the accurate computation of the roots of the corresponding orthogonal polynomials, an important issue in the construction of Gaussian quadrature formulas.

IN PRAISE OF ORDER UNITS

We show that the ordered rings naturally associated to compact convex polyhedra with interior satisfy a positivity property known as order unit cancellation, and obtain other general positivity results as well.