Séminaire Lotharingien de Combinatoire, B74a (2015), 11 pp.

Colleen Ross and Alexander Yong

Combinatorial Rules for Three Bases of Polynomials

Abstract. 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].

Received: November 6, 2013. Revised: April 10, 2015. Accepted: July 21, 2015.

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