Linear relations on LLT polynomials and their k-Schur positivity for k=2

2018/07/11 by Lee, Seung Jin · 1 citation
#05E05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.03951

Abstract

LLT polynomials are q-analogues of product of Schur functions that are known to be Schur-positive by Grojnowski and Haiman. However, there is no known combinatorial formula for the coefficients in the Schur expansion. Finding such a formula also provides Schur positivity of Macdonald polynomials. On the other hand, Haiman and Hugland conjectured that LLT polynomials for skew partitions lying on k adjacent diagonals are k-Schur positive, which is much stronger than Schur positivity. In this paper, we prove the conjecture for k=2 by analyzing unicellular LLT polynomials. We first present a linearity theorem for unicellular LLT polynomials for k=2. By analyzing linear relations between LLT polynomials with known results on LLT polynomials for rectangles, we provide the 2-Schur positivity of the unicellular LLT polynomials as well as LLT polynomials appearing in Haiman-Hugland conjecture for k=2.

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