We present a generalization of the classical Schur modules of GL(N) exhibiting the same interplay among algebra, geometry, and combinatorics. A generalized Young diagram D is an arbitrary finite subset of N × N. For each D, we define the Schur module SD of GL(N). We introduce a projective variety FD and a line bundle LD, and describe the Schur module in terms of sections of LD. For diagrams wit...

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. An alternative proof of this result is provided. We also give an intriguing log-concavity property of Schur functions. 1. Schur positivity conjectures The ring of...

In [20] the connections between the Schur algorithm, the Wall’s continued fractions and the orthogonal polynomials are revisited and used to establish some nice convergence properties of the sequence of Schur functions associated with a Schur function. In this report, we generalize some of Krushchev’s results to the case of a multipoint Schur algorithm, that is a Schur algorithm where all the i...

Two measures of how near an arbitrary function between groups is to being a homomorphism are considered. These have properties similar to conjugates and commutators. The authors show that there is a rich theory based on these structures, and that this theory can be used to unify disparate approaches such as group cohomology and the transfer and to prove theorems. The proof of the Schur-Zassenha...

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon’s theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof...

In [I] robust stability properties of Schur polynomials of the form f(z) = an-i zi were analyzed and a theorem analogous to Kharitonov's weak theorem [2] was derived, where all the corner points of a polyhedron are needed for stability. In this paper the analog of the strong Kharitonov theorem is derived for discrete systems. It is shown that only a relatively small number of comers are needed....

An identity is derived expressing Schur functions as sums over products of pairs $Q$-functions, generalizing previously known special cases. This shown to follow from their representations vacuum expectation values (VEV's) either charged or neutral fermionic creation and annihilation operators, Wick's theorem a factorization for VEV's two mutually anticommuting sets operators.