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In mathematics, the '''''n''! conjecture''' is the [[conjecture]] that the [[dimension]] of a certain [[bi-graded]] [[Module (mathematics)|module]] of [[diagonal harmonics]] is ''n''!. It was made by [[A. M. Garsia]] and [[M. Haiman]] and later proved by [[M. Haiman]]. It implies [[Ian G. Macdonald|Macdonald]]'s [[Macdonald_polynomial#The_Macdonald_positivity_conjecture|positivity conjecture]] about the [[Macdonald polynomial]]s.
 
==Formulation and background==
The Macdonald polynomials <math> P_\lambda \, </math> are a two-parameter family of [[orthogonal polynomials]] indexed by a positive weight λ of a [[root system]], introduced by [[Ian G. Macdonald]] (1987). They generalize several other families of orthogonal polynomials, such as [[Jack polynomial]]s and [[Hall–Littlewood polynomial]]s. They are known to have deep relationships with [[affine Hecke algebra]]s and [[Hilbert scheme]]s, which were used to prove several conjectures made by Macdonald about them.
 
{{harvtxt|Macdonald|1988}} introduced a new basis for the space of [[symmetric function]]s, which specializes to many of the well-known bases for the symmetric functions, by suitable substitutions for the parameters ''q'' and ''t''.
 
In fact, we can obtain in this manner the [[Schur function]]s, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the [[zonal symmetric function]]s, the [[zonal spherical function]]s, and the elementary and monomial symmetric functions.
 
The so-called (''q'',''t'')-[[Kostka polynomial]]s are the coefficients of a resulting [[transition matrix]]. Macdonald conjectured that they are polynomials in ''q'' and ''t'', with non-negative integer coefficients.
 
It was [[Adriano Garsia]]'s idea to construct an appropriate [[module (algebra)|module]] in order to prove positivity (as was done in his previous joint work with [[Claudio Procesi|Procesi]] on Schur positivity of [[Kostka–Foulkes polynomials]]).
 
In an attempt to prove Macdonald's conjecture, {{harvtxt|Garsia|Haiman|1993}} introduced the bi-graded module  <math>H_\mu \, </math> of [[diagonal harmonic]]s and conjectured that the (modified) Macdonald polynomials are the Frobenius image of the character generating function of ''H''<sub>&mu;</sub>, under the diagonal action of the [[symmetric group]].
 
The proof of Macdonald's conjecture was then reduced to the ''n''! conjecture; i.e., to prove that the dimension of ''H''<sub>&mu;</sub> is&nbsp;''n''!. In 2001, Haiman proved that the dimension is indeed ''n''! (see [4]). 
 
This breakthrough led to the discovery of many hidden connections and new aspects of [[symmetric group representation theory]], as well as combinatorial objects (e.g., insertion tableaux, Haglund's inversion numbers, and the role of parking functions in representation theory).
 
==References==
*{{cite journal
|first1=A. M.
|last1=Garsia
|first2=C.
|last2=Procesi
|title=On certain graded ''S''<sub>''n''</sub>-modules and the q-Kostka polynomials
|journal=Adv. Math.
|volume=94
|issue=1
|year=1992
|pages=82&ndash;138
|doi=10.1016/0001-8708(92)90034-I
}}
*{{cite journal
|first1= A. M.
|last1=Garsia
|first2=M.
|last2=Haiman
|title=A graded representation model for the Macdonald polynomials
|journal=Proc. Nat. Acad. Sci.
|volume=90
|issue=8
|year=1993
|url=http://www.pnas.org/content/90/8/3607.abstract
|pages=3607&ndash;3610
}}
*{{cite journal
|first1=A. M.
|last1=Garsia
|first2=M.
|last2=Haiman
|title=Orbit Harmonics and Graded Representations, Research Monograph
}} to appear as part of the collection published by the Lab. de. Comb. et Informatique Mathématique, edited by S. Brlek, U. du Qu&eacute;bec á Montr&eacute;al.
*{{cite journal
|first1=M.
|last1=Haiman
|title=Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
|journal=[[J. Amer. Math. Soc.]]
|volume=14
|year=2001
|pages=941&ndash;1006
|doi=10.1090/S0894-0347-01-00373-3
}}
*{{cite journal
|first1= I. G.
|last1=Macdonald
|title=A new class of symmetric functions
|publisher=Publ. I.R.M.A. Strasbourg
|volume=20
|journal=[[Séminaire Lotharingien de Combinatoire]]
|year=1988
|url=http://www.emis.ams.org/journals/SLC/opapers/s20macdonald.html
|pages=131&ndash;171
}}
 
==External links==
*[http://archive.numdam.org/ARCHIVE/SB/SB_2001-2002__44_/SB_2001-2002__44__103_0/SB_2001-2002__44__103_0.pdf Bourbaki seminar (Procesi), PDF]
*[http://bergeron.math.uqam.ca/conjecture_fact.html ''n''! conjecture] by François Bergeron
*[http://www.math.ucsd.edu/~garsia/schur/ ''n''! homepage] of Garsia
*http://www.maths.ed.ac.uk/~igordon/pubs/grenoble3.pdf
*http://mathworld.wolfram.com/n!Theorem.html
 
[[Category:Algebraic combinatorics]]
[[Category:Algebraic geometry]]
[[Category:Orthogonal polynomials]]
[[Category:Representation theory]]
[[Category:Conjectures]]

Revision as of 20:36, 26 February 2014

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