first announced a proof of the n!-conjecture at
the 1999 CMS winter
meeting in Montreal there were still gaps in this proof and he was
fortunately able to complete it later.
One announcement made by Adriano Garsia at the Winter Meeting in Montreal is that he is still offering a prize of $1000 for the first mathematician who can produce bases for the n! modules. Mark's result implies that such a basis exists but is not constructive in its nature.
A Short Explanation of the n! Theorem
|For any partition
of n we define a polynomial in 2n variables
where the are the coordinates of the cells of the partition when it is placed in the coordinate plane with the base cell at the position (0,0) and the rest of the coordinates have non-negative x and y values.
Consider the partition . The polynomial will be
|Note that because this is a determinant, when an element of the symmetric group acts on this polynomial by permuting the x and the y variables simultaneously it permutes the columns, hence the action of the symmetric group on this polynomial is:||
|Consider the linear span of all of the derivatives of this polynomial with respect to the variables and and denote this space by . This vector space is closed under permutations that act on the and simultaneously, and hence is a symmetric group module.|
The n! Theorem states that
It is a theorem by Garsia/Haiman that the dimension is less than or equal to n!
|Example: Because n! grows
so quickly it is difficult to effectivly demonstrate an example of anything
larger than 3. In this case we will choose
|Consider all of the polynomials listed
below, along with
they form a basis for
(hence the dimension = 3! = 6)
|The connection with the Macdonald polynomials comes when
one considers the graded character of this module.
Let be the character of the homogeneous component of of degree i in the x variables and degree j in the y variables of the polynomial vector space module . Consider the Frobenius image of the character of the n! module (that is, sends the irreducible representation indexed by the partition to the Schur function indexed by the partition ).
|A refined statement of the n! Conjecture is then
where are defined directly from the Macdonald polynomials.
|In our example above, if you are familiar with the representation
of the symmetric group, it is not hard to see that the Frobenius image
of the graded representation of this module is
since the polynomial
is a copy of the alternating representation (of degree 1 in x and
1 in y), the polynomials of degree 1 in x and 0 in y
(and similarly 0 in x and 1 in y) form a two dimensional
representation corresponding to the partition (2,1), and the symmetric
group acts on the constant terms with the trivial action (corresponding
to the partition (3)).