We propose a new combinatorial description of the product of two Schur functions. In the particular case of the square of a Schur function SI, it allows to discriminate in a very natural way between the symmetric and antisymmetric parts of the square. In other words, it describes at the same time the expansion on the basis of Schur functions of the plethysms S2(SI) and A2(SI) . More generally o...

h our previous paper [l], we observed that generalized Vandermonde determinants of the form vn;~(ll,. , G) = \q” / I 1 5 i, k 5 n, where the 2, are distinct points belonging to an interval [a, b] of the real line, the index n stands for the order, the sequence p consists of ordered integers 0 5 pi < ~2 < ... < pLn, can be factored as a product of the classical Vandermonde determinant and a Schu...

Many results involving Schur functions have analogues involving k-Schur functions. Standard strong marked tableaux play a role for k-Schur functions similar to the role standard Young tableaux play for Schur functions. We discuss results and conjectures toward an analogue of the hook-length formula.

This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten inv...

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satissed in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood-Richardson and Murnaghan-Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.

Abstract We wonder if there is a way to make all Schur functions in representations equal. This impossible for fixed value of time variables, but can be achieved averages. It appears that the corresponding measure just Gaussian times, which are independent. The generating function number Young diagrams does not straightforwardly appear as product, reproduced non-trivial way.

We prove Okounkov’s conjecture, a conjecture of Fomin-FultonLi-Poon, and a special case of Lascoux-Leclerc-Thibon’s conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. We include an alternative derivation of this result directly from Haiman’s work on Schur positive immanants. Our results imply an intriguing log-concavity propert...