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

Symmetric function

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...

s introduced by Alain Lascoux

Alain Lascoux

Alain Lascoux is a French mathematician at the University of Marne la Vallée and Nankai University. His research fields include algebraic combinatorics, particuarly Hecke algebra and Young tableau....

, Bernard Leclerc, and Jean-Yves Thibon (1997) as q-analogues

Q-analog

Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1...

 of products of Schur functions.

J. Haglund, M. Haiman, N. Loehr (2005) showed how to expand Macdonald polynomial

Macdonald polynomial

In mathematics, Macdonald polynomials Pλ are a family of orthogonal polynomials in several variables, introduced by...

s in therms of LLT polynomials. Ian Grojnowski

Ian Grojnowski

Ian Grojnowski is a mathematician working at the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge. Grojnowski was the first recipient of the Fröhlich Prize of the London Mathematical Society in 2004 for his work in representation theory and algebraic...

 and Mark Haiman

Mark Haiman

Mark David Haiman is a mathematician at the University of California at Berkeley who proved theMacdonald positivity conjecture for Macdonald polynomials.-References:* Haiman's *Mark Haiman J. Amer. Math. Soc. 14 , 941–1006...

 (preprint) proved a positivity conjecture for LLT polynomials that combined with the previous result implies the Macdonald positivity conjecture for Macdonald polynomials, and extended the definition of LLT polynomials to arbitrary finite root systems.