In mathematics, an LLT polynomial is one of a family of

symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...

s introduced by

Alain Lascoux Alain Lascoux (17 October 1944 – 20 October 2013) was a French mathematician at the University of Marne la Vallée and Nankai University. His research was primarily in algebraic combinatorics, particularly Hecke algebras and Young tableaux. ...

, Bernard Leclerc, and Jean-Yves Thibon (1997) as ''q''-analogues of products of Schur functions. J. Haglund,

M. Haiman Mark David Haiman is a mathematician at the University of California at Berkeley who proved the Macdonald positivity conjecture for Macdonald polynomials. He received his Ph.D in 1984 in the Massachusetts Institute of Technology under the direc ...

, N. Loehr (2005) showed how to expand

Macdonald polynomial In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald origin ...

s in terms of LLT polynomials. Ian Grojnowski and

Mark Haiman Mark David Haiman is a mathematician at the University of California at Berkeley who proved the Macdonald positivity conjecture for Macdonald polynomials. He received his Ph.D in 1984 in the Massachusetts Institute of Technology under the directio ...

(2007, preprint) proved a positivity conjecture for LLT polynomials that combined with the previous result implies the

Macdonald positivity conjecture In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald orig ...

for

Macdonald polynomials In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald origin ...

, and extended the definition of LLT polynomials to arbitrary finite root systems.

References

*I. Grojnowski, M. Haiman, ''Affine algebras and positivity'' (preprint availabl
here
*J. Haglund, M. Haiman, N. Loeh
A Combinatorial Formula for Macdonald Polynomials
J. Amer. Math. Soc. 18 (2005), no. 3, 735–761 *Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibo
Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties
{{MathSciNet, id=1434225 J. Math. Phys. 38 (1997), no. 2, 1041–1068. Symmetric functions Algebraic geometry Algebraic combinatorics Q-analogs Polynomials