Jack function
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 the Heckman–Opdam polynomials and Macdonald polynomials.
Definition
The Jack function
of integer partition
, parameter
, and
indefinitely many arguments
can be recursively defined as
follows:
- For m=1
- For m>1
where the summation is over all partitions such that the skew partition
is a horizontal strip, namely
(
must be zero or otherwise
) and
where equals
if
and
otherwise. The expressions
and
refer to the conjugate partitions of
and
, respectively. The notation
means that the product is taken over all coordinates
of boxes in the Young diagram of the partition
.
Combinatorial formula
In 1997, F. Knop and S. Sahi [1] gave a purely combinatorial formula for the Jack polynomials
in n variables:
.
The sum is taken over all admissible tableaux of shape ,
and
with
.
An admissible tableau of shape is a filling of the Young diagram
with numbers 1,2,…,n such that for any box (i,j) in the tableau,
- T(i,j) ≠ T( i',j) whenever i' > i.
- T(i,j) ≠ T( i',j-1) whenever j>1 and i' < i.
A box is critical for the tableau T if j>1 and
.
This result can be seen as a special case of the more general combinatorial formula for Macdonald polynomials.
C normalization
The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product:
This orthogonality property is unaffected by normalization. The normalization defined above is typically referred to as the J normalization. The C normalization is defined as
where
For denoted often as just
is known as the Zonal polynomial.
P normalization
The P normalization is given by the identity ,
where
and
and
denotes the arm and leg length respectively.
Therefore, for
,
is the usual Schur function.
Similar to Schur polynomials, can be expressed as a sum over Young tableaux.
However, one need to add an extra weight to each tableau that depends on the parameter
.
Thus, a formula [2] for the Jack function is given by
where the sum is taken over all tableaux of shape , and
denotes the entry in box s of T.
The weight can be defined in the following fashion:
Each tableau T of shape
can be interpreted as a sequence of partitions
where
defines the skew shape with content i in T.
Then
where
and the product is taken only over all boxes s in
such that s has a box from
in the same row, but not
in the same column.
Connection with the Schur polynomial
When the Jack function is a scalar multiple of the Schur polynomial
where
is the product of all hook lengths of .
Properties
If the partition has more parts than the number of variables, then the Jack function is 0:
Matrix argument
In some texts, especially in random matrix theory, authors have found it more convenient to use a matrix argument in the Jack function. The connection is simple. If is a matrix with eigenvalues
, then
References
- Demmel, James; Koev, Plamen (2006), "Accurate and efficient evaluation of Schur and Jack functions", Mathematics of Computation 75 (253): 223–239, doi:10.1090/S0025-5718-05-01780-1, MR 2176397.
- Jack, Henry (1970–1971), "A class of symmetric polynomials with a parameter", Proceedings of the Royal Society of Edinburgh, Section A. Mathematics 69: 1–18, MR 0289462.
- Knop, Friedrich; Sahi, Siddhartha (19 March 1997), "A recursion and a combinatorial formula for Jack polynomials", Inventiones Mathematicae 128 (1): 9–22, doi:10.1007/s002220050134
- Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), New York: Oxford University Press, ISBN 0-19-853489-2, MR 1354144
- Stanley, Richard P. (1989), "Some combinatorial properties of Jack symmetric functions", Advances in Mathematics 77 (1): 76–115, doi:10.1016/0001-8708(89)90015-7, MR 1014073.
External links
- Software for computing the Jack function by Plamen Koev and Alan Edelman.
- MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package)
- SAGE documentation for Jack Symmetric Functions
- ↑ Knop & Sahi 1997.
- ↑ Macdonald 1995, pp. 379.