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n! Theorem


For any partition mu of n, define a polynomial in 2n variables x_1, x_2, ... and y_1, y_2, ... as

 Delta_mu=det|x_i^(p_j)y_i^(q_j)|,
(1)

where (p_j,q_j) are the coordinates of the cells of the partition when it is placed in the coordinate plane with base cell at (0,0) and such that all other coordinates are nonnegative in x and y. Denote the linear span of all derivatives of this polynomial with respect to the variables by L[partial_xpartial_yDelta_mu], where partial represents a partial derivative. This vector space is closed under permutations acting on x_i and y_i simultaneously. Then the n! theorem states that

 dimL[partial_xpartial_yDelta_mu]=n!.
(2)

The theorem was proven by M. Haiman in Dec. 1999.

For example, consider the partition mu=(2,1). Then

Delta_((2,1))=det|1 1 1; x_1 x_2 x_3; y_1 y_2 y_3|
(3)
=x_2y_3-x_3y_2-x_1y_3+y_1x_3+x_1y_2-x_2y_1.
(4)

Then the five derivatives

partial_(x_1)Delta_((2,1))=y_2-y_3
(5)
partial_(x_2)Delta_((2,1))=y_3-y_1
(6)
partial_(y_1)Delta_((2,1))=x_3-x_2
(7)
partial_(y_2)Delta_((2,1))=x_1-x_3
(8)
partial_(x_2)partial_(y_2)Delta_((2,1))=1,
(9)

together with Delta_((2,1)), 3!=6 elements in all, form a basis for L[partial_xpartial_yDelta_((2,1))].


See also

Macdonald Polynomial

Explore with Wolfram|Alpha

References

Garsia, A. M. "Manuscripts and Publications on Macdonald Polynomials and the n! Conjecture." http://schur.ucsd.edu/~garsia/.Garsia, A. M. "A Short Explanation of the n! Theorem." http://garsia.math.yorku.ca/MPWP/nfactconj/nfactconj.html.

Referenced on Wolfram|Alpha

n! Theorem

Cite this as:

Weisstein, Eric W. "n! Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/n!Theorem.html

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