In this paper, the Schur convexity is generalized to Schur f -convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When f : R+ →R is defined by f (x) = (xm−1)/m if m = 0 and f (x) = lnx if m = 0 , the necessary and sufficient conditions for f -convexity (is called Schur m -power convexity) of Daróczy means are given, which improve, generalize and unify Shi et...

For integers a1, . . . , an > 0 and k > 1, let Lk+2(a1, . . . , an) denote the set of permutations of {1, . . . , a1 + · · · + an} whose descent set is contained in {a1, a1 + a2, . . . , a1+ · · ·+an−1}, and which avoids the pattern 12 . . . (k+2). We exhibit some bijections between such sets, most notably showing that #Lk+2(a1, . . . , an) is symmetric in the ai and is in fact Schur-concave. T...

We extend the family of classical Schur algebras in type A, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general lin...

We introduce non-commutative analogs of k-Schur functions and prove that their images by the noncommutative nabla operator H is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual (q, t)-Catalan polynomials by computing the image of certain commutative k-Schur functions by the commutative nabla operator∇. In some particular cases, we giv...

In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F -multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F -multiplicity free quasisy...

The double Schur functions form a distinguished basis of the ring Λ(x ||a) which is a multiparameter generalization of the ring of symmetric functions Λ(x). The canonical comultiplication on Λ(x) is extended to Λ(x ||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus pr...

We establish several properties of an algorithm defined by Mason and Remmel (2010) which inserts a positive integer into a row-strict composition tableau. These properties lead to a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. Résumé. Nous établissons ...

Several representations of the generalized Drazin inverse of an anti-triangular block matrix in Banach algebra are given in terms of the generalized Banachiewicz--Schur form.  

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type A by a Schur function can be understood from the multiplication in the space of dual k-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the r-Bruhat order given by Bergeron-Sottil...