Cell transfer and monomial positivity
Thomas Lam1
and Pavlo Pylyavskyy2
1Harvard University Department of Mathematics Cambridge MA 02138 USA
2M.I.T. Department of Mathematics Cambridge MA 02139 USA
2M.I.T. Department of Mathematics Cambridge MA 02139 USA
DOI: 10.1007/s10801-006-0054-z
Abstract
We give combinatorial proofs that certain families of differences of products of Schur functions are monomial-positive. We show in addition that such monomial-positivity is to be expected of a large class of generating functions with combinatorial definitions similar to Schur functions. These generating functions are defined on posets with labelled Hasse diagrams and include for example generating functions of Stanley's ( P,ω )-partitions.
Pages: 209–224
Keywords: keywords symmetric functions; monomial positivity; P-partitions
Full Text: PDF
References
1. F. Bergeron and P. McNamara, Some positive differences of products of Schur functions, preprint, 2004; math.CO/0412289.
2. F. Bergeron, R. Biagioli, and M. Rosas, “Inequalities between Littlewood-Richardson Coefficients,” J. Comb. Th. Ser A, to appear; math.CO/0403541.
3. N. Bergeron and F. Sottile, “Skew Schubert functions and the Pieri formula for flag manifolds, with Nantel Bergeron,” Trans. Amer. Math. Soc. 354, (2002), 651-673.
4. S. Fomin, W. Fulton, C.-K. Li, and Y.-T. Poon, “Eigenvalues, singular values, and Littlewood- Richardson coefficients,” Amer. J. Math. 127 (2005), 101-127.
5. I. Gessel and C. Krattenthaler, “Cylindric Partitions,” Trans. Amer. Math. Soc. 349 (1997), 429-479.
6. T. Lam, “Affine Stanley Symmetric Functions,” Amer. J. Math., to appear; math.CO/0501335.
7. T. Lam and P. Pylyavskyy, “P-partition products and fundamental quasi-symmetric function positivity,” preprint, 2006; math.CO/0609249.
8. T. Lam, A. Postnikov, and P. Pylyavskyy, “Schur positivity and Schur log-concavity,” Amer. J. Math., to appear; math.CO/0502446.
9. I. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, 1995.
10. P. McNamara, “Cylindric Skew Schur Functions,” Adv. Math. 205(1) (2006), 275-312.
11. A. Okounkov, “Log-Concavity of multiplicities with Applications to Characters of U (\infty ),” Adv. Math. 127(2) (1997), 258-282.
12. A. Postnikov, “Affine approach to quantum Schubert calculus,” Duke Math. J. 128(3) (2005), 473-509.
13. R. Stanley, Enumerative Combinatorics, Vol 2, Cambridge, 1999.
2. F. Bergeron, R. Biagioli, and M. Rosas, “Inequalities between Littlewood-Richardson Coefficients,” J. Comb. Th. Ser A, to appear; math.CO/0403541.
3. N. Bergeron and F. Sottile, “Skew Schubert functions and the Pieri formula for flag manifolds, with Nantel Bergeron,” Trans. Amer. Math. Soc. 354, (2002), 651-673.
4. S. Fomin, W. Fulton, C.-K. Li, and Y.-T. Poon, “Eigenvalues, singular values, and Littlewood- Richardson coefficients,” Amer. J. Math. 127 (2005), 101-127.
5. I. Gessel and C. Krattenthaler, “Cylindric Partitions,” Trans. Amer. Math. Soc. 349 (1997), 429-479.
6. T. Lam, “Affine Stanley Symmetric Functions,” Amer. J. Math., to appear; math.CO/0501335.
7. T. Lam and P. Pylyavskyy, “P-partition products and fundamental quasi-symmetric function positivity,” preprint, 2006; math.CO/0609249.
8. T. Lam, A. Postnikov, and P. Pylyavskyy, “Schur positivity and Schur log-concavity,” Amer. J. Math., to appear; math.CO/0502446.
9. I. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, 1995.
10. P. McNamara, “Cylindric Skew Schur Functions,” Adv. Math. 205(1) (2006), 275-312.
11. A. Okounkov, “Log-Concavity of multiplicities with Applications to Characters of U (\infty ),” Adv. Math. 127(2) (1997), 258-282.
12. A. Postnikov, “Affine approach to quantum Schubert calculus,” Duke Math. J. 128(3) (2005), 473-509.
13. R. Stanley, Enumerative Combinatorics, Vol 2, Cambridge, 1999.