mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

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 coo ...

of a certain bi-graded module of diagonal harmonics is ''n''!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald'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 orig ...

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 In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

indexed by a positive weight λ of a

root system In mathematics, a root system is a configuration of vector space, 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 ...

, introduced by

Ian G. Macdonald Ian Grant Macdonald (11 October 1928 – 8 August 2023) was a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinator ...

(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 t ...

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

affine Hecke algebra In mathematics, an affine Hecke algebra is the associative algebra, algebra associated to an affine Weyl group, and can be used to mathematical proof, prove Macdonald's constant term conjecture for Macdonald polynomials. Definition Let V be a Eucl ...

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 ...

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 functions, 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 vect ...

s, and the elementary and monomial symmetric functions. The so-called ''q'',''t''- Kostka polynomials 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

coefficients. It was

Adriano Garsia Adriano Mario Garsia (20 August 1928 – 6 October 2024) was a Tunisian-born Italian American mathematician who worked in analysis, combinatorics, representation theory, and algebraic geometry. He was a student of Charles Loewner and published wo ...

's idea to construct an appropriate module in order to prove positivity (as was done in his previous joint work with Procesi on Schur positivity of Kostka–Foulkes polynomials). 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 grou ...

. 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, 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 algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

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 jo ...

, 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 Theorems in algebraic geometry Orthogonal polynomials Representation theory Conjectures that have been proved Factorial and binomial topics Theorems about polynomials Module theory Theorems in linear algebra