nLab Macdonald Constant-Term Conjecture

There are several Macdonald conjectures, including Macdonald conjecture on plane partitions, Macdonald positivity conjecture (see wikipedia: Macdonald polynomials)

Useful disambiguation is at wikipedia: Macdonald conjecture (disambiguation)

Idea

Freeman Dyson introduced a conjecture about a constant term of certain Laurent polynomial in $n$ variables.

Macdonald conjecture is its generalization to any root system, including affine root systems.

It has been proved by Cherednik in 1995; for that purpose he (earlier) introduced double affine Hecke algebras.

Literature

• Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math.(2) 141:1 (1995) 191–216.

See at the beginning of the talk.

• Yuri Berest, Representation homology and strong Macdonald conjectures (slides pdf)

• Wolfram’s mathworld: Macdonald’s Constant-Term Conjecture

• S. L. Cacciatori, F. Dalla Piazza, A. Scotti, Compact Lie groups: Euler constructions and generalized Dyson conjecture, Trans. Amer. Math. Soc. 369 (2017) 4709–4724 doi

category: combinatorics, algebra