In
mathematics, the Jack function is a generalization of the Jack polynomial, introduced by
Henry Jack
Henry Jack FRSE (6 July 1917 – 5 January 1978) was a Scottish mathematician at University College Dundee. The Jack polynomials are named after him. His research dealt with the development of analytic methods to evaluate certain integrals over ...
. The Jack polynomial is a
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
,
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
which generalizes the
Schur and
zonal polynomials, and is in turn generalized by the
Heckman–Opdam polynomials In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) ''P''λ(''k'') are orthogonal polynomials in several variables associated to root systems. They were introduced by .
They generalize Jack polynomials when the roots ...
and
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 ...
s.
Definition
The Jack function
of an
integer partition
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same part ...
, parameter
, and 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 In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups ...
of the partition
.
Combinatorial formula
In 1997, F. Knop and S. Sahi 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,
*
whenever
*
whenever
and
Macdonald polynomials
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 ...
.
C normalization
The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product:
:
\langle f,g\rangle = \int_ f \left (e^,\ldots,e^ \right ) \overline \prod_ \left , e^-e^ \right , ^ d\theta_1\cdots d\theta_n
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
:
C_\kappa^(x_1,\ldots,x_n) = \frac J_\kappa^(x_1,\ldots,x_n),
where
:
j_\kappa=\prod_ \left (\kappa_j'-i+\alpha \left (\kappa_i-j+1 \right ) \right ) \left (\kappa_j'-i+1+\alpha \left (\kappa_i-j \right ) \right ).
For
\alpha=2, C_\kappa^(x_1,\ldots,x_n) is often denoted by
C_\kappa(x_1,\ldots,x_n) and called the
Zonal polynomial In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials.
They appear as zonal spherical functions of the Gelfand pairs
(S_,H_n) (here, H_n i ...
.
P normalization
The ''P'' normalization is given by the identity
J_\lambda = H'_\lambda P_\lambda, where
:
H'_\lambda = \prod_ (\alpha a_\lambda(s) + l_\lambda(s) + 1)
where
a_\lambda and
l_\lambda denotes the
arm and leg length respectively. Therefore, for
\alpha=1, P_\lambda is the usual Schur function.
Similar to Schur polynomials,
P_\lambda 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
\alpha.
Thus, a formula for the Jack function
P_\lambda is given by
:
P_\lambda = \sum_ \psi_T(\alpha) \prod_ x_
where the sum is taken over all tableaux of shape
\lambda, and
T(s) denotes the entry in box ''s'' of ''T''.
The weight
\psi_T(\alpha) can be defined in the following fashion: Each tableau ''T'' of shape
\lambda can be interpreted as a sequence of partitions
:
\emptyset = \nu_1 \to \nu_2 \to \dots \to \nu_n = \lambda
where
\nu_/\nu_i defines the skew shape with content ''i'' in ''T''. Then
:
\psi_T(\alpha) = \prod_i \psi_(\alpha)
where
:
\psi_(\alpha) = \prod_ \frac \frac
and the product is taken only over all boxes ''s'' in
\lambda such that ''s'' has a box from
\lambda/\mu in the same row, but ''not'' in the same column.
Connection with the Schur polynomial
When
\alpha=1 the Jack function is a scalar multiple of the
Schur polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. ...
:
J^_\kappa(x_1,x_2,\ldots,x_n) = H_\kappa s_\kappa(x_1,x_2,\ldots,x_n),
where
:
H_\kappa=\prod_ h_\kappa(i,j)=
\prod_ (\kappa_i+\kappa_j'-i-j+1)
is the product of all hook lengths of
\kappa.
Properties
If the partition has more parts than the number of variables, then the Jack function is 0:
:
J_\kappa^(x_1,x_2,\ldots,x_m)=0, \mbox\kappa_>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
X is a matrix with eigenvalues
x_1,x_2,\ldots,x_m, then
:
J_\kappa^(X)=J_\kappa^(x_1,x_2,\ldots,x_m).
References
*.
*.
*
*
*{{citation
, last = Stanley , first = Richard P. , authorlink = Richard P. Stanley
, doi = 10.1016/0001-8708(89)90015-7
, doi-access=free
, mr = 1014073
, issue = 1
, journal =
Advances in Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes.
At the origin, the journal aimed ...
, pages = 76–115
, title = Some combinatorial properties of Jack symmetric functions
, volume = 77
, year = 1989.
External links
Software for computing the Jack functionby Plamen Koev and Alan Edelman.
*
ttp://www.sagemath.org/doc/reference/sage/combinat/sf/jack.html SAGE documentation for Jack Symmetric Functions Orthogonal polynomials
Special functions
Symmetric functions