| Type: | Package |
| Title: | Jack, Zonal, Schur, and Other Symmetric Polynomials |
| Version: | 6.1.0 |
| Date: | 2024-07-29 |
| Maintainer: | Stéphane Laurent <laurent_step@outlook.fr> |
| Description: | Schur polynomials appear in combinatorics and zonal polynomials appear in random matrix theory. They are particular cases of Jack polynomials. This package allows to compute these polynomials and other symmetric multivariate polynomials: flagged Schur polynomials, factorial Schur polynomials, t-Schur polynomials, Hall-Littlewood polynomials, Macdonald polynomials, and modified Macdonald polynomials. In addition, it can compute the Kostka-Jack numbers, the Kostka-Foulkes polynomials, the Kostka-Macdonald polynomials, and the Hall polynomials. Mainly based on Demmel & Koev's paper (2006) <doi:10.1090/S0025-5718-05-01780-1> and Macdonald's book (1995) <doi:10.1093/oso/9780198534891.003.0001>. |
| License: | GPL-3 |
| URL: | https://github.com/stla/jackR |
| BugReports: | https://github.com/stla/jackR/issues |
| Depends: | qspray (≥ 3.1.0), ratioOfQsprays (≥ 1.1.0), symbolicQspray (≥ 1.1.0) |
| Imports: | DescTools, gmp, methods, multicool, mvp, partitions, RationalMatrix, Rcpp, spray, syt (≥ 0.5.0), utils |
| Suggests: | testthat |
| LinkingTo: | BH, qspray, ratioOfQsprays, Rcpp, RcppCGAL, symbolicQspray |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.3.1 |
| SystemRequirements: | C++17, gmp, mpfr |
| NeedsCompilation: | yes |
| Packaged: | 2024-07-29 17:12:59 UTC; User |
| Author: | Stéphane Laurent [aut, cre] |
| Repository: | CRAN |
| Date/Publication: | 2024-07-29 17:50:02 UTC |
Evaluation of elementary symmetric functions
Description
Evaluates an elementary symmetric function.
Usage
ESF(x, lambda)
Arguments
x |
a numeric vector or a |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A number if x is numeric, a bigq rational number
if x is a bigq vector.
Examples
x <- c(1, 2, 5/2)
lambda <- c(3, 1)
ESF(x, lambda)
library(gmp)
x <- c(as.bigq(1), as.bigq(2), as.bigq(5,2))
ESF(x, lambda)
Hall-Littlewood polynomial
Description
Hall-Littlewood polynomial of a given partition.
Usage
HallLittlewoodPol(n, lambda, which = "P")
Arguments
n |
number of variables |
lambda |
integer partition |
which |
which Hall-Littlewood polynomial, |
Value
The Hall-Littlewood polynomial in n variables of the
integer partition lambda. This is a symbolicQspray
polynomial with a unique parameter usually denoted by t and
its coefficients are polynomial in this parameter. When substituting
t with 0 in the Hall-Littlewood P-polynomials, one
obtains the Schur polynomials.
Hall polynomials
Description
Hall polynomials g^{\lambda}_{\mu,\nu}(t) for given
integer partitions \mu and \nu.
Usage
HallPolynomials(mu, nu)
Arguments
mu, nu |
integer partitions |
Value
A list of lists. Each of these lists has two elements: an integer
partition \lambda in the field lambda, and a univariate
qspray polynomial in the field polynomial, the Hall
polynomial g^{\lambda}_{\mu,\nu}(t). Every coefficient of a
Hall polynomial is an integer.
Note
This function is slow.
Examples
HallPolynomials(c(2, 1), c(1, 1))
Evaluation of Jack polynomial - C++ implementation
Description
Evaluates a Jack polynomial.
Usage
Jack(x, lambda, alpha)
Arguments
x |
values of the variables, a vector of |
lambda |
an integer partition, given as a vector of decreasing integers |
alpha |
rational number, given as a string such as
|
Value
A bigq number.
Examples
Jack(c("1", "3/2", "-2/3"), lambda = c(3, 1), alpha = "1/4")
Symmetric polynomial in terms of Jack polynomials
Description
Expression of a symmetric polynomial as a linear combination of Jack polynomials.
Usage
JackCombination(qspray, alpha, which = "J", check = TRUE)
Arguments
qspray |
a |
alpha |
Jack parameter, must be coercible to a |
which |
which Jack polynomials, |
check |
Boolean, whether to check the symmetry of |
Value
A list defining the combination. Each element of this list is a
list with two elements: coeff, which is a bigq number if
qspray is a qspray polynomial or a ratioOfQsprays if
qspray is a symbolicQspray polynomial, and the second
element of the list is lambda, an integer partition; then this list
corresponds to the term coeff * JackPol(n, lambda, alpha, which),
where n is the number of variables in the symmetric polynomial
qspray.
Jack polynomial - C++ implementation
Description
Returns a Jack polynomial.
Usage
JackPol(n, lambda, alpha, which = "J")
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
alpha |
rational number, given as a string such as
|
which |
which Jack polynomial, |
Value
A qspray multivariate polynomial.
Examples
JackPol(3, lambda = c(3, 1), alpha = "2/5")
Jack polynomial
Description
Returns the Jack polynomial.
Usage
JackPolR(n, lambda, alpha, algorithm = "DK", basis = "canonical", which = "J")
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
alpha |
parameter of the Jack polynomial, a number, possibly (and
preferably) a |
algorithm |
the algorithm used, either |
basis |
the polynomial basis for |
which |
which Jack polynomial, |
Value
A mvp multivariate polynomial (see mvp-package),
or a qspray multivariate polynomial if alpha
is a bigq rational number and algorithm = "DK", or a
character string if basis = "MSF".
Examples
JackPolR(3, lambda = c(3,1), alpha = gmp::as.bigq(2,3),
algorithm = "naive")
JackPolR(3, lambda = c(3,1), alpha = 2/3, algorithm = "DK")
JackPolR(3, lambda = c(3,1), alpha = gmp::as.bigq(2,3), algorithm = "DK")
JackPolR(3, lambda = c(3,1), alpha= gmp::as.bigq(2,3),
algorithm = "naive", basis = "MSF")
# when the Jack polynomial is a `qspray` object, you can
# evaluate it with `qspray::evalQspray`:
jack <- JackPolR(3, lambda = c(3, 1), alpha = gmp::as.bigq(2))
evalQspray(jack, c("1", "1/2", "3"))
Evaluation of Jack polynomials
Description
Evaluates a Jack polynomial.
Usage
JackR(x, lambda, alpha, algorithm = "DK")
Arguments
x |
numeric or complex vector or |
lambda |
an integer partition, given as a vector of decreasing integers |
alpha |
ordinary number or |
algorithm |
the algorithm used, either |
Value
A numeric or complex scalar or a bigq rational number.
References
I.G. Macdonald. Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995.
J. Demmel & P. Koev. Accurate and efficient evaluation of Schur and Jack functions. Mathematics of computations, vol. 75, n. 253, 223-229, 2005.
-
Jack polynomials. https://www.symmetricfunctions.com/jack.htm
See Also
Examples
lambda <- c(2,1,1)
JackR(c(1/2, 2/3, 1), lambda, alpha = 3)
# exact value:
JackR(c(gmp::as.bigq(1,2), gmp::as.bigq(2,3), gmp::as.bigq(1)), lambda,
alpha = gmp::as.bigq(3))
Jack polynomial with symbolic Jack parameter
Description
Returns the Jack polynomial with a symbolic Jack parameter.
Usage
JackSymPol(n, lambda, which = "J")
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
which |
which Jack polynomial, |
Value
A symbolicQspray object.
Examples
JackSymPol(3, lambda = c(3, 1))
Kostka-Foulkes polynomial
Description
Kostka-Foulkes polynomial for two given partitions.
Usage
KostaFoulkesPolynomial(lambda, mu)
Arguments
lambda, mu |
integer partitions; in order for the Kostka-Foulkes
polynomial to be non-zero, a necessary condition is that |
Value
The Kostka-Foulkes polynomial associated to lambda and
mu. This is a univariate qspray polynomial whose value
at 1 is the Kostka number associated to lambda and
mu.
Kostka-Jack numbers with a given Jack parameter
Description
Kostka numbers with Jack parameter, or Kostka-Jack numbers, for partitions of a given weight and a given Jack parameter.
Usage
KostkaJackNumbers(n, alpha = "1")
Arguments
n |
positive integer, the weight of the partitions |
alpha |
the Jack parameter, a |
Details
The Kostka-Jack number K_{\lambda,\mu}(\alpha) is the
coefficient of the monomial symmetric polynomial m_\mu in the
expression of the P-Jack polynomial P_\lambda(\alpha) as a
linear combination of monomial symmetric polynomials. For \alpha=1
it is the ordinary Kostka number.
Value
The matrix of the Kostka-Jack numbers K_{\lambda,\mu}(\alpha)
given as character strings representing integers or fractions.
The row names of this matrix encode the partitions \lambda and
the column names encode the partitions \mu
See Also
KostkaJackNumbersWithGivenLambda,
symbolicKostkaJackNumbers,
skewKostkaJackNumbers.
Examples
KostkaJackNumbers(4)
Kostka-Jack numbers with a given partition \lambda
Description
Kostka numbers with Jack parameter, or Kostka-Jack numbers
K_{\lambda,\mu}(\alpha) for a given Jack parameter \alpha
and a given integer partition \lambda.
Usage
KostkaJackNumbersWithGivenLambda(lambda, alpha, output = "vector")
Arguments
lambda |
integer partition |
alpha |
the Jack parameter, a |
output |
the format of the output, either |
Details
The Kostka-Jack number K_{\lambda,\mu}(\alpha) is the
coefficient of the monomial symmetric polynomial m_\mu in the
expression of the P-Jack polynomial P_\lambda(\alpha) as a
linear combination of monomial symmetric polynomials. For \alpha=1
it is the ordinary Kostka number.
Value
If output="vector", this function returns a named vector.
This vector is made of the non-zero (i.e. positive) Kostka-Jack numbers
K_{\lambda,\mu}(\alpha) given as character strings and its names
encode the partitions \mu.
If ouput="list", this function returns a list of lists.
Each of these lists has two
elements. The first one is named mu and is an integer
partition, and the second one is named value and is a bigq
rational number, the Kostka-Jack number K_{\lambda,\mu}(\alpha).
See Also
KostkaJackNumbers,
symbolicKostkaJackNumbersWithGivenLambda.
Examples
KostkaJackNumbersWithGivenLambda(c(3, 2), alpha = "2")
Littlewood-Richardson rule for multiplication
Description
Expression of the product of two Schur polynomials as a linear combination of Schur polynomials.
Usage
LRmult(mu, nu, output = "dataframe")
Arguments
mu, nu |
integer partitions, given as vectors of decreasing integers |
output |
the type of the output, |
Value
This computes the expression of the product of the two Schur
polynomials associated to mu and nu as a linear combination
of Schur polynomials. If output="dataframe", the output is a
dataframe with two columns: the column coeff gives the coefficients
of this linear combination, these are positive integers, and the column
lambda gives the partitions defining the Schur polynomials of this
linear combination as character strings, e.g. the partition
c(4, 3, 1) is encoded by the character string "[4, 3, 1]".
If output="list", the output is a list
of lists with two elements. Each of these lists with two elements
corresponds to a term of the linear combination: the first element,
named coeff, is the coefficient, namely the Littlewood-Richardson
coefficient c^{\lambda}_{\mu,\nu}, where \lambda is the
integer partition given in the second element of the list, named
lambda, which defines the Schur polynomial of the
linear combination.
Examples
library(jack)
mu <- c(2, 1)
nu <- c(3, 2, 1)
LR <- LRmult(mu, nu, output = "list")
LRterms <- lapply(LR, function(lr) {
lr[["coeff"]] * SchurPol(3, lr[["lambda"]])
})
smu_times_snu <- Reduce(`+`, LRterms)
smu_times_snu == SchurPol(3, mu) * SchurPol(3, nu) # should be TRUE
Littlewood-Richardson rule for skew Schur polynomial
Description
Expression of a skew Schur polynomial as a linear combination of Schur polynomials.
Usage
LRskew(lambda, mu, output = "dataframe")
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
output |
the type of the output, |
Value
This computes the expression of the skew Schur polynomial
associated to the skew partition defined by lambda and mu
as a linear combination of Schur polynomials. Every coefficient of this
linear combination is a positive integer, a so-called
Littlewood-Richardson coefficient.
If output="dataframe",
the output is a dataframe with two columns: the column coeff gives
the coefficients of this linear combination, and the column nu
gives the partitions defining the Schur polynomials of this linear
combination as character strings, e.g. the partition c(4, 3, 1) is
given by "[4, 3, 1]". If output="list", the output is a list
of lists with two elements. Each of these lists with two elements
corresponds to a term of the linear combination: the first element, named
coeff, is the coefficient, namely the Littlewood-Richardson
coefficient c^{\lambda}_{\mu,\nu}, where \nu is the integer
partition given in the second element of the list, named
nu, which defines the Schur polynomial of the linear
combination.
Examples
library(jack)
LRskew(lambda = c(4, 2, 1), mu = c(3, 1))
Evaluation of monomial symmetric functions
Description
Evaluates a monomial symmetric function.
Usage
MSF(x, lambda)
Arguments
x |
a numeric vector or a |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A number if x is numeric, a bigq rational number
if x is a bigq vector.
Examples
x <- c(1, 2, 5/2)
lambda <- c(3, 1)
MSF(x, lambda)
library(gmp)
x <- c(as.bigq(1), as.bigq(2), as.bigq(5,2))
MSF(x, lambda)
Macdonald polynomial
Description
Returns the Macdonald polynomial associated to the given integer partition.
Usage
MacdonaldPol(n, lambda, which = "P")
Arguments
n |
number of variables, a positive integer |
lambda |
integer partition |
which |
which Macdonald polynomial, |
Value
A symbolicQspray multivariate polynomial, the
Macdonald polynomial associated to the integer partition
lambda. It has two parameters usually denoted by q
and t. Substituting q with 0 yields the
Hall-Littlewood polynomials.
Evaluation of Schur polynomial - C++ implementation
Description
Evaluates a Schur polynomial. The Schur polynomials are the
Jack P-polynomials with Jack parameter \alpha=1.
Usage
Schur(x, lambda)
Arguments
x |
values of the variables, a vector of |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A bigq number.
Examples
Schur(c("1", "3/2", "-2/3"), lambda = c(3, 1))
Symmetric polynomial in terms of the Schur polynomials
Description
Expression of a symmetric polynomial as a linear combination of some Schur polynomials.
Usage
SchurCombination(qspray, check = TRUE)
Arguments
qspray |
a |
check |
Boolean, whether to check the symmetry of |
Value
A list defining the combination. Each element of this list is a
list with two elements: coeff, a bigq number, and
lambda, an integer partition; then this list corresponds to the
term coeff * SchurPol(n, lambda), where n is the number of
variables in the symmetric polynomial.
See Also
Schur polynomial - C++ implementation
Description
Returns a Schur polynomial. The Schur polynomials are the
Jack P-polynomials with Jack parameter \alpha=1.
Usage
SchurPol(n, lambda)
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A qspray multivariate polynomial.
Examples
( schur <- SchurPol(3, lambda = c(3, 1)) )
schur == JackPol(3, lambda = c(3, 1), alpha = "1", which = "P")
Schur polynomial
Description
Returns the Schur polynomial.
Usage
SchurPolR(n, lambda, algorithm = "DK", basis = "canonical", exact = TRUE)
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
algorithm |
the algorithm used, either |
basis |
the polynomial basis for |
exact |
logical, whether to use exact arithmetic |
Value
A mvp multivariate polynomial (see mvp-package),
or a qspray multivariate polynomial if
exact = TRUE and algorithm = "DK", or a
character string if basis = "MSF".
Examples
SchurPolR(3, lambda = c(3,1), algorithm = "naive")
SchurPolR(3, lambda = c(3,1), algorithm = "DK")
SchurPolR(3, lambda = c(3,1), algorithm = "DK", exact = FALSE)
SchurPolR(3, lambda = c(3,1), algorithm = "naive", basis = "MSF")
Evaluation of Schur polynomials
Description
Evaluates a Schur polynomial.
Usage
SchurR(x, lambda, algorithm = "DK")
Arguments
x |
numeric or complex vector or bigq vector |
lambda |
an integer partition, given as a vector of decreasing integers |
algorithm |
the algorithm used, either |
Value
A numeric or complex scalar or a bigq rational number.
References
J. Demmel & P. Koev. Accurate and efficient evaluation of Schur and Jack functions. Mathematics of computations, vol. 75, n. 253, 223-229, 2005.
See Also
Examples
x <- c(2,3,4)
SchurR(x, c(2,1,1))
prod(x) * sum(x)
Skew factorial Schur polynomial
Description
Computes the skew factorial Schur polynomial associated to a given skew partition.
Usage
SkewFactorialSchurPol(n, lambda, mu, a, i0)
Arguments
n |
number of variables |
lambda, mu |
integer partitions defining the skew partition:
|
a |
vector of |
i0 |
positive integer, the index of |
Value
A qspray polynomial.
References
I.G. Macdonald. Schur functions: theme and variations. Publ. IRMA Strasbourg, 1992.
Examples
# for a=c(0, 0, ...), the skew factorial Schur polynomial is the
# skew Schur polynomial; let's check
n <- 4
lambda <- c(3, 3, 2, 2); mu <- c(2, 2)
a <- rep(0, 9)
i0 <- 3
skewFactorialSchurPoly <- SkewFactorialSchurPol(n, lambda, mu, a, i0)
skewSchurPoly <- SkewSchurPol(n, lambda, mu)
skewFactorialSchurPoly == skewSchurPoly # should be TRUE
Skew Hall-Littlewood polynomial
Description
Returns the skew Hall-Littlewood polynomial associated to the given skew partition.
Usage
SkewHallLittlewoodPol(n, lambda, mu, which = "P")
Arguments
n |
number of variables, a positive integer |
lambda, mu |
integer partitions defining the skew partition:
|
which |
which skew Hall-Littlewood polynomial, |
Value
A symbolicQspray multivariate polynomial, the skew
Hall-Littlewood polynomial associated to the skew partition defined by
lambda and mu. It has a single parameter usually denoted
by t and its coefficients are polynomial in this parameter.
When substituting t with 0 in the skew Hall-Littlewood
P-polynomials, one obtains the skew Schur polynomials.
Examples
n <- 3; lambda <- c(3, 2, 1); mu <- c(1, 1)
skewHLpoly <- SkewHallLittlewoodPol(n, lambda, mu)
skewSchurPoly <- SkewSchurPol(n, lambda, mu)
substituteParameters(skewHLpoly, 0) == skewSchurPoly # should be TRUE
Skew Jack polynomial
Description
Computes a skew Jack polynomial with a given Jack parameter.
Usage
SkewJackPol(n, lambda, mu, alpha, which = "J")
Arguments
n |
positive integer, the number of variables |
lambda |
outer integer partition of the skew partition |
mu |
inner integer partition of the skew partition; it must be a
subpartition of |
alpha |
the Jack parameter, any object coercible to a |
which |
which skew Jack polynomial, |
Value
A qspray polynomial.
See Also
Examples
SkewJackPol(3, c(3,1), c(2), "2")
Skew Jack polynomial with symbolic Jack parameter
Description
Computes a skew Jack polynomial with a symbolic Jack parameter.
Usage
SkewJackSymPol(n, lambda, mu, which = "J")
Arguments
n |
positive integer, the number of variables |
lambda |
outer integer partition of the skew partition |
mu |
inner integer partition of the skew partition; it must be a
subpartition of |
which |
which skew Jack polynomial, |
Value
A symbolicQspray polynomial.
Examples
SkewJackSymPol(3, c(3,1), c(2))
Skew Kostka-Foulkes polynomial
Description
Computes a skew Kostka-Foulkes polynomial.
Usage
SkewKostkaFoulkesPolynomial(lambda, mu, nu)
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
nu |
integer partition; the condition
|
Value
The skew Kostka-Foulkes polynomial associated to the skew
partitiion defined by lambda and mu and to the partition
nu. This is a univariate qspray polynomial whose value
at 1 is the skew Kostka number associated to the skew partition
defined by lambda and mu and to the partition nu.
Skew Macdonald polynomial
Description
Returns the skew Macdonald polynomial associated to the given skew partition.
Usage
SkewMacdonaldPol(n, lambda, mu, which = "P")
Arguments
n |
number of variables, a positive integer |
lambda, mu |
integer partitions defining the skew partition:
|
which |
which skew Macdonald polynomial, |
Value
A symbolicQspray multivariate polynomial, the skew
Macdonald polynomial associated to the skew partition defined by
lambda and mu. It has two parameters usually
denoted by q and t. Substituting q with 0
yields the skew Hall-Littlewood polynomials.
Skew Schur polynomial
Description
Returns the skew Schur polynomial.
Usage
SkewSchurPol(n, lambda, mu)
Arguments
n |
number of variables, a positive integer |
lambda, mu |
integer partitions defining the skew partition:
|
Details
The computation is performed with the help of the
Littlewood-Richardson rule (see LRskew).
Value
A qspray multivariate polynomial, the skew Schur polynomial
associated to the skew partition defined by lambda and mu.
Examples
SkewSchurPol(3, lambda = c(3, 2, 1), mu = c(1, 1))
Evaluation of zonal polynomial - C++ implementation
Description
Evaluates a zonal polynomial. The zonal polynomials are the
Jack C-polynomials with Jack parameter \alpha=Z.
Usage
Zonal(x, lambda)
Arguments
x |
values of the variables, a vector of |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A bigq number.
Examples
Zonal(c("1", "3/2", "-2/3"), lambda = c(3, 1))
Zonal polynomial - C++ implementation
Description
Returns a zonal polynomial. The zonal polynomials are the
Jack C-polynomials with Jack parameter \alpha=Z.
Usage
ZonalPol(n, lambda)
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A qspray multivariate polynomial.
Examples
( zonal <- ZonalPol(3, lambda = c(3, 1)) )
zonal == JackPol(3, lambda = c(3, 1), alpha = "2", which = "C")
Zonal polynomial
Description
Returns the zonal polynomial.
Usage
ZonalPolR(n, lambda, algorithm = "DK", basis = "canonical", exact = TRUE)
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
algorithm |
the algorithm used, either |
basis |
the polynomial basis for |
exact |
logical, whether to get rational coefficients |
Value
A mvp multivariate polynomial (see mvp-package),
or a qspray multivariate polynomial if
exact = TRUE and algorithm = "DK", or a
character string if basis = "MSF".
Examples
ZonalPolR(3, lambda = c(3,1), algorithm = "naive")
ZonalPolR(3, lambda = c(3,1), algorithm = "DK")
ZonalPolR(3, lambda = c(3,1), algorithm = "DK", exact = FALSE)
ZonalPolR(3, lambda = c(3,1), algorithm = "naive", basis = "MSF")
Evaluation of zonal quaternionic polynomial - C++ implementation
Description
Evaluates a zonal quaternionic polynomial. The quaternionic
zonal polynomials are the Jack C-polynomials with Jack
parameter \alpha=1/Z.
Usage
ZonalQ(x, lambda)
Arguments
x |
values of the variables, a vector of |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A bigq number.
Examples
ZonalQ(c("1", "3/2", "-2/3"), lambda = c(3, 1))
Quaternionic zonal polynomial - C++ implementation
Description
Returns a quaternionic zonal polynomial. The quaternionic
zonal polynomials are the Jack C-polynomials with Jack
parameter \alpha=1/Z.
Usage
ZonalQPol(n, lambda)
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
Value
A qspray multivariate polynomial.
Examples
( zonalQ <- ZonalQPol(3, lambda = c(3, 1)) )
zonalQ == JackPol(3, lambda = c(3, 1), alpha = "1/2", which = "C")
Quaternionic zonal polynomial
Description
Returns the quaternionic (or symplectic) zonal polynomial.
Usage
ZonalQPolR(n, lambda, algorithm = "DK", basis = "canonical", exact = TRUE)
Arguments
n |
number of variables, a positive integer |
lambda |
an integer partition, given as a vector of decreasing integers |
algorithm |
the algorithm used, either |
basis |
the polynomial basis for |
exact |
logical, whether to get rational coefficients |
Value
A mvp multivariate polynomial (see mvp-package),
or a qspray multivariate polynomial if
exact = TRUE and algorithm = "DK", or a
character string if basis = "MSF".
Examples
ZonalQPolR(3, lambda = c(3,1), algorithm = "naive")
ZonalQPolR(3, lambda = c(3,1), algorithm = "DK")
ZonalQPolR(3, lambda = c(3,1), algorithm = "DK", exact = FALSE)
ZonalQPolR(3, lambda = c(3,1), algorithm = "naive", basis = "MSF")
Evaluation of quaternionic zonal polynomials
Description
Evaluates a quaternionic (or symplectic) zonal polynomial.
Usage
ZonalQR(x, lambda, algorithm = "DK")
Arguments
x |
numeric or complex vector or bigq vector |
lambda |
an integer partition, given as a vector of decreasing integers |
algorithm |
the algorithm used, either |
Value
A numeric or complex scalar or a bigq rational number.
References
F. Li, Y. Xue. Zonal polynomials and hypergeometric functions of quaternion matrix argument. Comm. Statist. Theory Methods, 38 (8), 1184-1206, 2009
See Also
Examples
lambda <- c(2,2)
ZonalQR(c(3,1), lambda)
ZonalQR(c(gmp::as.bigq(3),gmp::as.bigq(1)), lambda)
##
x <- c(3,1)
ZonalQR(x, c(1,1)) + ZonalQR(x, 2) # sum(x)^2
ZonalQR(x, 3) + ZonalQR(x, c(2,1)) + ZonalQR(x, c(1,1,1)) # sum(x)^3
Evaluation of zonal polynomials
Description
Evaluates a zonal polynomial.
Usage
ZonalR(x, lambda, algorithm = "DK")
Arguments
x |
numeric or complex vector or bigq vector |
lambda |
an integer partition, given as a vector of decreasing integers |
algorithm |
the algorithm used, either |
Value
A numeric or complex scalar or a bigq rational number.
References
Robb Muirhead. Aspects of multivariate statistical theory. Wiley series in probability and mathematical statistics. Probability and mathematical statistics. John Wiley & Sons, New York, 1982.
Akimichi Takemura. Zonal Polynomials, volume 4 of Institute of Mathematical Statistics Lecture Notes – Monograph Series. Institute of Mathematical Statistics, Hayward, CA, 1984.
Lin Jiu & Christoph Koutschan. Calculation and Properties of Zonal Polynomials. http://koutschan.de/data/zonal/
See Also
Examples
lambda <- c(2,2)
ZonalR(c(1,1), lambda)
ZonalR(c(gmp::as.bigq(1),gmp::as.bigq(1)), lambda)
##
x <- c(3,1)
ZonalR(x, c(1,1)) + ZonalR(x, 2) # sum(x)^2
ZonalR(x, 3) + ZonalR(x, c(2,1)) + ZonalR(x, c(1,1,1)) # sum(x)^3
Factorial Schur polynomial
Description
Computes a factorial Schur polynomial.
Usage
factorialSchurPol(n, lambda, a)
Arguments
n |
number of variables |
lambda |
integer partition |
a |
vector of |
Value
A qspray polynomial.
References
I.G. Macdonald. Schur functions: theme and variations. Publ. IRMA Strasbourg, 1992.
Examples
# for a=c(0, 0, ...), the factorial Schur polynomial is the Schur polynomial
n <- 3
lambda <- c(2, 2, 2)
a <- c(0, 0, 0, 0)
factorialSchurPoly <- factorialSchurPol(n, lambda, a)
schurPoly <- SchurPol(n, lambda)
factorialSchurPoly == schurPoly # should be TRUE
Flagged Schur polynomial
Description
Computes a flagged Schur polynomial (which is not symmetric in general). See Chains in the Bruhat order for the definition.
Usage
flaggedSchurPol(lambda, a, b)
Arguments
lambda |
integer partition |
a, b |
lower bounds and upper bounds, weakly increasing vectors of
integers; |
Value
A qspray polynomial.
Examples
lambda <- c(3, 2, 2)
n <- 3
a <- c(1, 1, 1); b <- c(n, n, n)
flaggedPoly <- flaggedSchurPol(lambda, a, b)
poly <- SchurPol(n, lambda)
flaggedPoly == poly # should be TRUE
Flagged skew Schur polynomial
Description
Computes a flagged skew Schur polynomial (which is not symmetric in general). See Schur polynomials (flagged) for the definition.
Usage
flaggedSkewSchurPol(lambda, mu, a, b)
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
a, b |
lower bounds and upper bounds, weakly increasing vectors of
integers; |
Value
A qspray polynomial.
Examples
lambda <- c(3, 2, 2); mu <- c(2, 1)
n <- 3
a <- c(1, 1, 1); b <- c(n, n, n)
flaggedPoly <- flaggedSkewSchurPol(lambda, mu, a, b)
poly <- SkewSchurPol(n, lambda, mu)
flaggedPoly == poly # should be TRUE
Modified Macdonald polynomial
Description
Returns the modified Macdonald polynomial associated to a given integer partition.
Usage
modifiedMacdonaldPol(n, mu)
Arguments
n |
number of variables, a positive integer |
mu |
integer partition |
Value
A symbolicQspray multivariate polynomial, the modified
Macdonald polynomial associated to the integer partition mu.
It has two parameters and its coefficients are polynomials in these
parameters.
qt-Kostka polynomials
Description
qt-Kostka polynomials, aka Kostka-Macdonald polynomials.
Usage
qtKostkaPolynomials(mu)
Arguments
mu |
integer partition |
Value
A list. The qt-Kostka polynomials are usually denoted by
K_{\lambda, \mu}(q, t) where q and t denote the two
variables and \lambda and \mu are two integer partitions.
One obtains the Kostka-Foulkes polynomials by substituting q
with 0.
For a given partition \mu, the function returns the
polynomials K_{\lambda, \mu}(q, t) as qspray objects
for all partitions \lambda of the same weight as \mu. The
generated list is a list of lists with two elements: the integer
partition \lambda and the polynomial.
Skew qt-Kostka polynomials
Description
Skew qt-Kostka polynomials associated to a given skew partition.
Usage
qtSkewKostkaPolynomials(lambda, mu)
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
Value
A list. The skew qt-Kostka polynomials are usually denoted by
K_{\lambda/\mu, \nu}(q, t) where q and t denote the two
variables, \lambda and \mu are the two integer partitions
defining the skew partition, and \nu is an integer partition.
One obtains the skew Kostka-Foulkes polynomials by substituting q
with 0.
For given partitions \lambda and \mu, the function returns the
polynomials K_{\lambda/\mu, \nu}(q, t) as qspray objects
for all partitions \nu of the same weight as the skew partition. The
generated list is a list of lists with two elements: the integer
partition \nu and the polynomial.
Skew Kostka-Jack numbers with given Jack parameter
Description
Skew Kostka-Jack numbers associated to a given skew partition and a given Jack parameter.
Usage
skewKostkaJackNumbers(lambda, mu, alpha = NULL, output = "vector")
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
alpha |
the Jack parameter, a |
output |
the format of the output, either |
Details
The skew Kostka-Jack number K_{\lambda/\mu,\nu}(\alpha) is
the coefficient of the monomial symmetric polynomial m_\nu in the
expression of the skew P-Jack polynomial
P_{\lambda/\mu}(\alpha) as a linear combination of monomial
symmetric polynomials. For \alpha=1 it is the ordinary skew Kostka
number.
Value
If output="vector", the function returns a named vector.
This vector is made of the non-zero skew Kostka-Jack numbers
K_{\lambda/\mu,\nu}(\alpha) given as character strings and its names
encode the partitions \nu.
If ouput="list", the function returns a list. Each element of this
list is a named list with two elements: an integer partition \nu
in the field named "nu", and the corresponding skew Kostka-Jack
number K_{\lambda/\mu,\nu}(\alpha) in the field named "value".
Only the non-null skew Kostka-Jack numbers are provided by this list.
Note
The skew Kostka-Jack numbers K_{\lambda/\mu,\nu}(\alpha) are
well defined when the Jack parameter \alpha is zero, however this
function does not work with alpha=0. A possible way to get the
skew Kostka-Jack numbers K_{\lambda/\mu,\nu}(0) is to use the
function symbolicSkewKostkaJackNumbers to get the skew
Kostka-Jack numbers with a symbolic Jack parameter \alpha, and then
to substitute \alpha with 0.
See Also
symbolicSkewKostkaJackNumbers.
Examples
skewKostkaJackNumbers(c(4,2,2), c(2,2))
Symmetric polynomial in terms of symbolic Jack polynomials
Description
Expression of a symmetric polynomial as a linear combination of Jack polynomials with a symbolic Jack parameter.
Usage
symbolicJackCombination(qspray, which = "J", check = TRUE)
Arguments
qspray |
a |
which |
which Jack polynomials, |
check |
Boolean, whether to check the symmetry |
Value
A list defining the combination. Each element of this list is a
list with two elements: coeff, a bigq number, and
lambda, an integer partition; then this list corresponds to the
term coeff * JackSymPol(n, lambda, which), where n is
the number of variables in the symmetric polynomial.
Kostka-Jack numbers with symbolic Jack parameter
Description
Kostka-Jack numbers with a symbolic Jack parameter for integer partitions of a given weight.
Usage
symbolicKostkaJackNumbers(n)
Arguments
n |
positive integer, the weight of the partitions |
Value
A named list of named lists of ratioOfQsprays objects.
Denoting the Kostka-Jack numbers by K_{\lambda,\mu}(\alpha), the
names of the outer list correspond to the partitions \lambda, and
the names of the inner lists correspond to the partitions \mu.
See Also
KostkaJackNumbers,
symbolicKostkaJackNumbersWithGivenLambda.
Examples
symbolicKostkaJackNumbers(3)
Kostka-Jack numbers with symbolic Jack parameter for a
given \lambda
Description
Kostka-Jack numbers K_{\lambda,\mu}(\alpha) with a
symbolic Jack parameter \alpha for a given
integer partition \lambda.
Usage
symbolicKostkaJackNumbersWithGivenLambda(lambda)
Arguments
lambda |
integer partition |
Value
A named list of ratioOfQsprays objects. The elements of this
list are the Kostka-Jack numbers K_{\lambda,\mu}(\alpha) and
its names correspond to the partitions \mu.
See Also
KostkaJackNumbersWithGivenLambda,
symbolicKostkaJackNumbers.
Examples
symbolicKostkaJackNumbersWithGivenLambda(c(3, 1))
Skew Kostka-Jack numbers with symbolic Jack parameter
Description
Skew Kostka-Jack numbers associated to a given skew partition with a symbolic Jack parameter.
Usage
symbolicSkewKostkaJackNumbers(lambda, mu)
Arguments
lambda, mu |
integer partitions defining the skew partition:
|
Value
The function returns a list. Each element of this
list is a named list with two elements: an integer partition \nu
in the field named "nu", and the corresponding skew Kostka number
K_{\lambda/\mu,\nu}(\alpha) in the field named "value", a
ratioOfQsprays object.
Examples
symbolicSkewKostkaJackNumbers(c(4,2,2), c(2,2))
t-Schur polynomial
Description
Returns the t-Schur polynomial associated to the given partition.
Usage
tSchurPol(n, lambda)
Arguments
n |
number of variables, a positive integer |
lambda |
integer partition |
Value
A symbolicQspray multivariate polynomial, the
t-Schur polynomial associated to lambda.
It has a single parameter usually denoted by t and its
coefficients are polynomials in this parameter. Substituting
t with 0 yields the Schur polynomials.
Note
The name "t-Schur polynomial" is taken from Wheeler and Zinn-Justin's paper Hall polynomials, inverse Kostka polynomials and puzzles.
Skew t-Schur polynomial
Description
Returns the skew t-Schur polynomial associated to the given skew partition.
Usage
tSkewSchurPol(n, lambda, mu)
Arguments
n |
number of variables, a positive integer |
lambda, mu |
integer partitions defining the skew partition:
|
Value
A symbolicQspray multivariate polynomial, the skew
t-Schur polynomial associated to the skew partition defined by
lambda and mu.
It has a single parameter usually denoted by t and its
coefficients are polynomials in this parameter. Substituting
t with 0 yields the skew Schur polynomials.