Littlewood-Richardson polynomials

By Alexander I. Molev. 2007

84 citations recorded by the source corpus.

We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author. The new rule provides a formula for these polynomials which is manifestly positive in the sense of W. Graham. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by A. Knutson and T. Tao by using combinatorics of puzzles while the stability property was not apparent from that formula. We also use the Littlewood-Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by A. Okounkov and G. Olshanski.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/2c6f44538973f8ebacb2ccbadd78a7cdcb7881bb

Citations in the corpus

Skew quasisymmetric Schur functions and noncommutative Schur functions (2010, unknown): “Meanwhile, regarding relations they satisfy, in [33] it was shown that c mλ,mμ ≥ cλμ, and further polynomiality properties were established in [14, 31].”
Introduction to Equivariant Cohomology in Algebraic Geometry (IMPANGA 2010) (2011, unknown): “e further applications. Generalized Schubert calculus. In the case of the Grassmannian, there are combinatorial formulas for the equivariant coeﬃcients cν λµ(t), due to Knutson and Tao [KT] and Molev [Mo]. However, the question remains open for other homogeneous spaces G/P, even in the case of ordin”