Littlewood-Richardson polynomials. (English) Zbl 1169.05050
Summary: 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 [A.I. Molev and B.E. Sagan,”A Littlewood-Richardson rule for factorial Schur functions,” Trans. Am. Math. Soc. 351, No.11, 4429–4443 (1999; Zbl 0972.05053)]. The new rule provides a formula for these polynomials which is positive in the sense of W. Graham [”Positivity in equivariant Schubert calculus,” Duke Math. J. 109, No.3, 599–614 (2001; Zbl 1069.14055)]. 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 [”Puzzles and (equivariant) cohomology of Grassmannians,” Duke Math. J. 119, No.2, 221–260 (2003).] 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 [”Shifted Schur functions,” St. Petersbg. Math. J. 9, No.2, 239–300 (1998); translation from Algebra Anal. 9, No.2, 73–146 (1997; Zbl 0894.05053).].
Keywords:
Littlewood-Richardson rule; double symmetric functions; equivariant Schubert classes; Grassmannians; quantum immanants; Schur functions; Littlewood-Richardson polynomials; combinatorics of puzzles; Casimir elements; general linear Lie algebraReferences:
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