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 ...

In this note, we announce some results about the Schur multipliers of the known finite simple groups. Proofs will appear elsewhere. We shall conclude with a summary of current knowledge on the subject. Basic properties of multipliers and covering groups of finite groups are discussed in [6]. Notation for groups of Lie type is standard [3], [8]. G' denotes the commutator subgroup of the group G,...

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.