Combinatorial Rules for Three Bases of Polynomials
Séminaire lotharingien de combinatoire, Tome 74 (2015-2018)
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We present combinatorial rules (one theorem and two conjectures) concerning three bases of Z[x1,x2,...]. First, we prove a "splitting" rule for the basis of Key polynomials [Demazure, Bull. Sci. Math. 98 (1974), 163-172], thereby establishing a new positivity theorem about them. Second, we introduce an extension of Kohnert's [Bayreuth. Math. Schriften 38 (1990), 1-97] "moves" to conjecture the first combinatorial rule for a certain deformation [Lascoux, in: Physics and Combinatorics, World Scientific Publishing, 2001, pp. 164-179] of the Key polynomials. Third, we use the same extension to conjecture a new rule for the Grothendieck polynomials [Lascoux and Schützenberger, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 629-633].
@article{SLC_2015-2018_74_a0,
author = {Colleen Ross and Alexander Yong},
title = {Combinatorial {Rules} for {Three} {Bases} of {Polynomials}},
journal = {S\'eminaire lotharingien de combinatoire},
publisher = {mathdoc},
volume = {74},
year = {2015-2018},
url = {http://geodesic.mathdoc.fr/item/SLC_2015-2018_74_a0/}
}
Colleen Ross; Alexander Yong. Combinatorial Rules for Three Bases of Polynomials. Séminaire lotharingien de combinatoire, Tome 74 (2015-2018). http://geodesic.mathdoc.fr/item/SLC_2015-2018_74_a0/