In mathematics, the ''n''! conjecture is the

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

that the

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

of a certain bi-graded

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

diagonal harmonic In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

s is ''n''!. It was made by

A. M. Garsia Adriano Mario Garsia (born 20 August 1928) is a Tunisian-born Italian American mathematician who works in analysis, combinatorics, representation theory, and algebraic geometry. He is a student of Charles Loewner and has published work on rep ...

and

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 ...

and later proved by

. It implies

Macdonald Macdonald, MacDonald or McDonald may refer to: Organisations * McDonald's, a chain of fast food restaurants * McDonald & Co., a former investment firm * MacDonald Motorsports, a NASCAR team * Macdonald Realty, a Canadian real estate brokerage f ...

's positivity conjecture about the

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 ...

Formulation and background

The Macdonald polynomials

P_\lambda

are a two-parameter family of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the cl ...

indexed by a positive weight λ of a

root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representat ...

, introduced by

Ian G. Macdonald Ian Grant Macdonald (born 11 October 1928 in London, England) is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combi ...

(1987). They generalize several other families of orthogonal polynomials, such as

Jack polynomial In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by ...

s and Hall–Littlewood polynomials. They are known to have deep relationships with

affine Hecke algebra In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials. Definition Let V be a Euclidean space of a finite dimension and \S ...

s and

Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a d ...

s, which were used to prove several conjectures made by Macdonald about them. introduced a new basis for the space 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, 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 functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the

zonal symmetric function Zonal can refer to: * Zonal and meridional, directions on a globe, west–east and north–south, respectively. * Zonal and poloidal, directions in a toroidal magnetically confined plasma * Zonal polynomial, a symmetric multivariate polynomial * Z ...

s, the

zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vec ...

s, and the elementary and monomial symmetric functions. The so-called ''q'',''t''-

Kostka polynomial In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory. The two-variable K ...

s are the coefficients of a resulting transition matrix. Macdonald conjectured that they are polynomials in ''q'' and ''t'', with non-negative

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

coefficients. It was

Adriano Garsia Adriano Mario Garsia (born 20 August 1928) is a Tunisian-born Italian American mathematician who works in analysis, combinatorics, representation theory, and algebraic geometry. He is a student of Charles Loewner and has published work on represe ...

's idea to construct an appropriate

in order to prove positivity (as was done in his previous joint work with Procesi on Schur positivity of

Kostka–Foulkes polynomials In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory. The two-variable Kost ...

). In an attempt to prove Macdonald's conjecture, introduced the bi-graded module

H_\mu

of diagonal harmonics and conjectured that the (modified) Macdonald polynomials are the Frobenius image of the character generating function of ''H''_μ, under the diagonal action of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

. The proof of Macdonald's conjecture was then reduced to the ''n''! conjecture; i.e., to prove that the dimension of ''H''_μ is ''n''!. In 2001, Haiman proved that the dimension is indeed ''n''! (see . This breakthrough led to the discovery of many hidden connections and new aspects of

symmetric group representation theory In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from s ...

, as well as combinatorial objects (e.g., insertion tableaux, Haglund's inversion numbers, and the role of parking functions in

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

References

* * * to appear as part of the collection published by the Lab. de. Comb. et Informatique Mathématique, edited by S. Brlek, U. du Québec á Montréal. * *{{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 The ''Séminaire Lotharingien de Combinatoire'' (English: ''Lotharingian Seminar of Combinatorics'') is a peer-reviewed academic journal specialising in combinatorial mathematics, named after Lotharingia. It has existed since 1980 as a regular j ...

, year=1988 , url=http://www.emis.de/journals/SLC/opapers/s20macdonald.html , pages=131–171

External links

Bourbaki seminar (Procesi), PDF
by François Bergeron
''n''! homepage
of Garsia *http://www.maths.ed.ac.uk/~igordon/pubs/grenoble3.pdf *http://mathworld.wolfram.com/n!Theorem.html Algebraic combinatorics Algebraic geometry Orthogonal polynomials Representation theory Conjectures Factorial and binomial topics