Linear relations on LLT polynomials and their k-Schur positivity for $$k=2$$ k =
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Linear relations on LLT polynomials and their k-Schur positivity for k = 2 Seung Jin Lee1 Received: 23 October 2018 / Accepted: 14 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract LLT polynomials are q-analogs of products 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 Haglund 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–Haglund conjecture for k = 2. Keywords LLT polynomial · k-Schur function · Haiman–Haglund conjecture
1 Introduction LLT polynomials are certain family of symmetric functions indexed by d-tuple of skew partitions, introduced by Lascoux et al. [12] in the study of quantum affine algebras and unipotent varieties. Later, Haglund et al. [7] proved that Macdonald polynomials are positive sums of LLT polynomial indexed by d-tuple of ribbons. Grojnowski and Haiman [6] proved that LLT polynomials are Schur positive using Kazhdan– Lusztig theory. However, their proof does not provide a manifestly positive formula so finding combinatorial formulas for expansions of Macdonald polynomials and LLT
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Seung Jin Lee [email protected] Department of Mathematical Sciences, Research institute of Mathematics, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul 151-747, Republic of Korea
123
Journal of Algebraic Combinatorics
polynomials remains a wide open problem. The best known result is the formula for d = 3 due to Blasiak [3]. See [3] for more history about LLT polynomials. In his 2006 ICM talk, Haiman announced a conjecture made by Haiman and Haglund stating that the involution image of LLT polynomials indexed by d-tuple of skew partition that lies in k-adjacent diagonals is k-Schur positive (Conjecture 2.1), which is much stronger than Schur positivity. The motivation of our paper stems from their conjecture, as our second main theorem is the proof of the conjecture for k = 2. The first main theorem (Theorem 4.1) shows that unicellular LLT polynomials with k = 2 are positive sums of 2-Schur functions where the exponents of q change linearly as the index set of unicellular LLT polynomials change, providing a very nice formula. We prove this by showing that there exists a linear recurrence relation between unicellular LLT polynomials (Theorem 3.5). Note that the linear relation does not assume k = 2, which may be useful to study LLT poly
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