Generalization of the 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 the Heckman–Opdam polynomials and Macdonald polynomials.
Definition
The Jack function
of an integer partition
, parameter
, and arguments
can be recursively defined as follows:
- For m=1
![{\displaystyle J_{k}^{(\alpha )}(x_{1})=x_{1}^{k}(1+\alpha )\cdots (1+(k-1)\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edf5dc9702629010ae927b7fcaa24f6b7905ca66)
- For m>1
![{\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=\sum _{\mu }J_{\mu }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m-1})x_{m}^{|\kappa /\mu |}\beta _{\kappa \mu },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53903f86d5dda1e3b25c47ad3104b90cd73c60ed)
where the summation is over all partitions
such that the skew partition
is a horizontal strip, namely
(
must be zero or otherwise
) and ![{\displaystyle \beta _{\kappa \mu }={\frac {\prod _{(i,j)\in \kappa }B_{\kappa \mu }^{\kappa }(i,j)}{\prod _{(i,j)\in \mu }B_{\kappa \mu }^{\mu }(i,j)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d9b03580d6c83647af11674cd02aae0ec29943)
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
.
In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials
in n variables:
![{\displaystyle J_{\mu }^{(\alpha )}=\sum _{T}d_{T}(\alpha )\prod _{s\in T}x_{T(s)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16d22d5d91cfa89e886831587b2128fc1a2e4407)
The sum is taken over all admissible tableaux of shape
and
![{\displaystyle d_{T}(\alpha )=\prod _{s\in T{\text{ critical}}}d_{\lambda }(\alpha )(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64b45c02fc963f2c67bd8c27844f2e03b928802a)
with
![{\displaystyle d_{\lambda }(\alpha )(s)=\alpha (a_{\lambda }(s)+1)+(l_{\lambda }(s)+1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3381f5318cef42ab9f01153bbf572b0a4359906)
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 ![{\displaystyle i'>i.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cccf69253b35324f061a24bfae23c1067dde64bc)
whenever
and ![{\displaystyle i'<i.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bddb3cce41904752339b07fd4822d2eb8d28fe)
A box
is critical for the tableau T if
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:
![{\displaystyle \langle f,g\rangle =\int _{[0,2\pi ]^{n}}f\left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right){\overline {g\left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right)}}\prod _{1\leq j<k\leq n}\left|e^{i\theta _{j}}-e^{i\theta _{k}}\right|^{\frac {2}{\alpha }}d\theta _{1}\cdots d\theta _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca582e0a32a9dc292850e71c6dd10422a1cc0029)
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
![{\displaystyle C_{\kappa }^{(\alpha )}(x_{1},\ldots ,x_{n})={\frac {\alpha ^{|\kappa |}(|\kappa |)!}{j_{\kappa }}}J_{\kappa }^{(\alpha )}(x_{1},\ldots ,x_{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6073362f14bc36e040132ca978a2429338e76008)
where
![{\displaystyle j_{\kappa }=\prod _{(i,j)\in \kappa }\left(\kappa _{j}'-i+\alpha \left(\kappa _{i}-j+1\right)\right)\left(\kappa _{j}'-i+1+\alpha \left(\kappa _{i}-j\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc567cf5beea5e41827029b11a2883057c137030)
For
is often denoted by
and called the Zonal polynomial.
P normalization
The P normalization is given by the identity
, where
![{\displaystyle H'_{\lambda }=\prod _{s\in \lambda }(\alpha a_{\lambda }(s)+l_{\lambda }(s)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5bcc3f045f1ce7796a6bbaa26e3f3ada311c64)
where
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 for the Jack function
is given by
![{\displaystyle P_{\lambda }=\sum _{T}\psi _{T}(\alpha )\prod _{s\in \lambda }x_{T(s)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a42f378bbf705b4198ff6afe9dc9aad90e0c3c77)
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
![{\displaystyle \emptyset =\nu _{1}\to \nu _{2}\to \dots \to \nu _{n}=\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95df4b31ba4b45567849b58a9904845ff6681d4f)
where
defines the skew shape with content i in T. Then
![{\displaystyle \psi _{T}(\alpha )=\prod _{i}\psi _{\nu _{i+1}/\nu _{i}}(\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30323207772218a0e0f8198486f681b43dd26f54)
where
![{\displaystyle \psi _{\lambda /\mu }(\alpha )=\prod _{s\in R_{\lambda /\mu }-C_{\lambda /\mu }}{\frac {(\alpha a_{\mu }(s)+l_{\mu }(s)+1)}{(\alpha a_{\mu }(s)+l_{\mu }(s)+\alpha )}}{\frac {(\alpha a_{\lambda }(s)+l_{\lambda }(s)+\alpha )}{(\alpha a_{\lambda }(s)+l_{\lambda }(s)+1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23fbe7bdab308ae630a9069a21be9c787f4b52d2)
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
![{\displaystyle J_{\kappa }^{(1)}(x_{1},x_{2},\ldots ,x_{n})=H_{\kappa }s_{\kappa }(x_{1},x_{2},\ldots ,x_{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d6a04caa63d81838ffdfb99f0617bdedbb207c)
where
![{\displaystyle H_{\kappa }=\prod _{(i,j)\in \kappa }h_{\kappa }(i,j)=\prod _{(i,j)\in \kappa }(\kappa _{i}+\kappa _{j}'-i-j+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b5278ecae513055c632bfe59908de4df239df8)
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:
![{\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=0,{\mbox{ if }}\kappa _{m+1}>0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef5adc2cb710db7b6cf07ec5acafbfaf1fdc02d)
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
![{\displaystyle J_{\kappa }^{(\alpha )}(X)=J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31db42494e5f64310606c134c2fac37a2047f87f)
References
- Demmel, James; Koev, Plamen (2006), "Accurate and efficient evaluation of Schur and Jack functions", Mathematics of Computation, 75 (253): 223–239, CiteSeerX 10.1.1.134.5248, 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, arXiv:q-alg/9610016, Bibcode:1997InMat.128....9K, doi:10.1007/s002220050134, S2CID 7188322
- Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), New York: Oxford University Press, ISBN 978-0-19-853489-1, 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) Archived 2010-06-20 at the Wayback Machine
- SAGE documentation for Jack Symmetric Functions