Recent work of A. Postnikov shows that cylindric skew Schur functions, which are a generalisation of skew Schur functions, have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining this motivation, we study cylindric skew Schur functions from the point of view of...

By the properties of Schur-convex function, Schur geometrically convex function and Schur harmonically convex function, Schur-convexity, Schur geometric and Schur harmonic convexities of the dual form for a class of symmetric functions are simply proved. As an application, several inequalities are obtained, some of which extend the known ones. Mathematics subject classification (2010): 26D15, 0...

An important problem in algebraic combinatorics is finding expansions of products of symmetric functions as sums of symmetric functions. Schur functions form a well-known basis for the ring of symmetric functions. The Littlewood-Richardson rule was introduced to expand the product of two Schur functions as a positive sum of Schur functions. Remmel and Whitney introduced an algorithmic way to fi...

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of the Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity for a composite function of the complete symmetric function.

We introduce a quasisymmetric generalization of Berele and Regev’s hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. In this paper we examine the combinatorics of the quasisymmetric hook Schur functions, providing analogues of the RobinsonSchensted-Knuth algorithm and a generalized Cauchy Identity. Résumé. Nous...

Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions ...

We show that the two cuspidal unipotent characters of a finite Chevalley group E7(q) have Schur index 2, provided that q is an even power of a (sufficiently large) prime number p such that p ≡ 1 mod 4. The proof uses a refinement of Kawanaka’s generalized Gelfand–Graev representations and some explicit computations with the CHEVIE computer algebra system.

Let K be an abelian extension of the rationals. Let S(K) be the Schur group of K and let CC(K) be the subgroup of S(K) generated by classes containing cyclic cyclotomic algebras. We characterize when CC(K) has finite index in S(K) in terms of the relative position of K in the lattice of cyclotomic extensions of the rationals.

We characterize the maximum r-local index of a Schur algebra over an abelian number field K in terms of global information determined by the field K, for r an arbitrary rational prime. This completes and unifies previous results of Janusz in [Jan] and Pendergrass