SymFun 3.1: Documentationww3.haverford.edu/math/cgreene/symfun/symfun31-doc.pdfSymFun 3.1:...
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SymFun 3.1: Documentation Summer 2004: Created by Curtis Greene, Julien Colvin, Ben Fineman Summer 2009: Updates by Curtis Greene, Rengyi Xu Summer 2010: Updates by Ian Burnette Summer 2012: Updates by Curtis Greene Introduction This is a Mathematica package designed to support various manipulations of symmetric functions, with an emphasis on combinatorial applications. It is loosely modeled after John Stembridge's highly successful Maple package ("SF"), but it is somewhat less ambitious: currently it implements only the standard basis conversions and a few other operations such as the Littlewood-Richardson rule. We assume familiarity with the basic theory of symmetric functions, e.g. Macdonald' s Symmetric Functions and Hall Polynomials or Stanley' s EC2. Using the Package Find the package (symfun31.m), open it, and click the button labeled "Run Package". Alternatively, load the package using the command << symfun31.m In order for the latter method to work, the package must be in a location recognized by Mathematica's $Path variable, for example, "C:\\Program Files\\Wolfram Research\\Mathematica\\7.0\\AddOns\\ExtraPackages" Overview: Conventions, Notation, and Basic Operations Basic families of symmetric functions (elementary, homogeneous, monomial, power sum, and Schur) may be defined in several ways. The following are expressions that are "atomic", i.e., they are unevaluated by Mathemat- ica and hence treated as indeterminates: e@4D,h@5D,p@3D,m@3,2,1D,s@3,2,1D The first three combine to give monomials in the obvious way, e.g., e@3De@1D^2, h@4Dh@2Dh@1D,p@3Dp@2Dp@1D These are the standard forms for the elementary, homogenous, and power-sum symmetric function. Expressions of the form e@83,1,1<D,h@84,2,1<D,p@83,2,1<D are evaluated by Mathematica, and give equivalent e, h, and p-functions in the above monomial form. Appending “x” to the function name and including an “n” parameter yields an evaluation in n variables x[1],
Transcript of SymFun 3.1: Documentationww3.haverford.edu/math/cgreene/symfun/symfun31-doc.pdfSymFun 3.1:...
SymFun 3.1: Documentation
Summer 2004: Created by Curtis Greene, Julien Colvin, Ben FinemanSummer 2009: Updates by Curtis Greene, Rengyi XuSummer 2010: Updates by Ian BurnetteSummer 2012: Updates by Curtis Greene
IntroductionThis is a Mathematica package designed to support various manipulations of symmetric functions, with anemphasis on combinatorial applications. It is loosely modeled after John Stembridge's highly successful Maplepackage ("SF"), but it is somewhat less ambitious: currently it implements only the standard basis conversions anda few other operations such as the Littlewood-Richardson rule.
We assume familiarity with the basic theory of symmetric functions, e.g. Macdonald' s Symmetric Functions andHall Polynomials or Stanley' s EC2.
Using the PackageFind the package (symfun31.m), open it, and click the button labeled "Run Package". Alternatively, load thepackage using the command
<< symfun31.m
In order for the latter method to work, the package must be in a location recognized by Mathematica's $Pathvariable, for example,
"C:\\Program Files\\Wolfram Research\\Mathematica\\7.0\\AddOns\\ExtraPackages"
Overview: Conventions, Notation, and Basic OperationsBasic families of symmetric functions (elementary, homogeneous, monomial, power sum, and Schur) may bedefined in several ways. The following are expressions that are "atomic", i.e., they are unevaluated by Mathemat-ica and hence treated as indeterminates:
e@4D,h@5D,p@3D,m@3,2,1D,s@3,2,1D
The first three combine to give monomials in the obvious way, e.g.,
e@3De@1D^2, h@4Dh@2Dh@1D,p@3Dp@2Dp@1D
These are the standard forms for the elementary, homogenous, and power-sum symmetric function. Expressionsof the form
e@83,1,1<D, h@84,2,1<D, p@83,2,1<D
are evaluated by Mathematica, and give equivalent e, h, and p-functions in the above monomial form.
Appending “x” to the function name and including an “n” parameter yields an evaluation in n variables x[1],x[2],...,x[n], for example,
Appending “x” to the function name and including an “n” parameter yields an evaluation in n variables x[1],x[2],...,x[n], for example,
ex@83, 1, 1<, 3D
x@1D3 x@2D x@3D + 2 x@1D2 x@2D2 x@3D + x@1D x@2D3 x@3D +
2 x@1D2 x@2D x@3D2+ 2 x@1D x@2D2 x@3D2
+ x@1D x@2D x@3D3
px@83, 1, 1<, 3D
x@1D5+ 2 x@1D4 x@2D + x@1D3 x@2D2
+ x@1D2 x@2D3+ 2 x@1D x@2D4
+ x@2D5+ 2 x@1D4 x@3D +
2 x@1D3 x@2D x@3D + 2 x@1D x@2D3 x@3D + 2 x@2D4 x@3D + x@1D3 x@3D2+ x@2D3 x@3D2
+
x@1D2 x@3D3+ 2 x@1D x@2D x@3D3
+ x@2D2 x@3D3+ 2 x@1D x@3D4
+ 2 x@2D x@3D4+ x@3D5
The same result can be obtained using "conversion" commands (described fully later in this document), e.g.
EToX@e@83, 1, 1<D, 3D
x@1D3 x@2D x@3D + 2 x@1D2 x@2D2 x@3D + x@1D x@2D3 x@3D +
2 x@1D2 x@2D x@3D2+ 2 x@1D x@2D2 x@3D2
+ x@1D x@2D x@3D3
or
EToX@e@3D e@1D^2, 3D
x@1D3 x@2D x@3D + 2 x@1D2 x@2D2 x@3D + x@1D x@2D3 x@3D +
2 x@1D2 x@2D x@3D2+ 2 x@1D x@2D2 x@3D2
+ x@1D x@2D x@3D3
Conversions are available between all pairs of families, e.g.,
EToS@e@3D e@1D^2D
s@2, 2, 1D + s@3, 1, 1D + 2 s@2, 1, 1, 1D + s@1, 1, 1, 1, 1D
EToP@e@3D e@1D^2D
p@1D5
6-
1
2p@1D3 p@2D +
1
3p@1D2 p@3D
Certain other algebraic operations are computed automatically. For example, products of atomic expressionsinvolving the Schur functions are automatically evaluated using the Littlewood-Richardson rule.
s@3, 2, 1D s@2, 1D
s@3, 3, 3D + 2 s@4, 3, 2D + s@4, 4, 1D + s@5, 2, 2D + s@5, 3, 1D +
s@3, 2, 2, 2D + 2 s@3, 3, 2, 1D + 2 s@4, 2, 2, 1D + 2 s@4, 3, 1, 1D +
s@5, 2, 1, 1D + s@3, 2, 2, 1, 1D + s@3, 3, 1, 1, 1D + s@4, 2, 1, 1, 1D
Basic Commands: Entering and Manipulating Symmetric Functions� H: definitions
h@<Λ>D
Expands in terms of h[1],h[2],h[3],...
h@83, 3, 2<D
h@2D h@3D2
hx@<lambda>,<n>D
Returns h[Λ] expanded in terms of x[1],x[2],...x[n] .
2 symfun31-doc.nb
hx@82<, 3D
x@1D2+ x@1D x@2D + x@2D2
+ x@1D x@3D + x@2D x@3D + x@3D2
hx@83, 2<, 2D
x@1D5+ 2 x@1D4 x@2D + 3 x@1D3 x@2D2
+ 3 x@1D2 x@2D3+ 2 x@1D x@2D4
+ x@2D5
he@<n>D
Expands h[n] directly into the e’s. This syntax works for the "primitive" h, e, and p functions, i.e. they can beconverted directly to the others.
he@5D
e@1D5- 4 e@1D3 e@2D + 3 e@1D e@2D2
+ 3 e@1D2 e@3D - 2 e@2D e@3D - 2 e@1D e@4D + e@5D
hp@5D
p@1D5
120+
1
12p@1D3 p@2D +
1
8p@1D p@2D2
+
1
6p@1D2 p@3D +
1
6p@2D p@3D +
1
4p@1D p@4D +
p@5D5
HList@<n>D
Returns list of all h[Λ]'s of degree n, expressed as monomials in h[1],h[2],... (Useful in some technical situations.)
HList@5D
9h@5D, h@1D h@4D, h@2D h@3D, h@1D2 h@3D, h@1D h@2D2, h@1D3 h@2D, h@1D5=
HList@<n>, <k>D
Returns list of all h[Λ]'s of degree n and Length(Λ)£k, expressed as monomials in h[1],h[2],...
HList@5, 2D
8h@5D, h@1D h@4D, h@2D h@3D<
HList2@<n>, <k>D
Returns list of all h[Λ]'s of degree n and Λ[[1]]£k, expressed as monomials in h[1],h[2],...
HList2@5, 2D
9h@1D h@2D2, h@1D3 h@2D, h@1D5=
� E: definitions
e@<Λ>D
Expands in terms of e[1],e[2],e[3],...
e@83, 2, 2, 1<D
e@1D e@2D2 e@3D
ex@<Λ>, <n>D
Returns e[Λ] expanded in terms of x[1],x[2],...,x[n] . Similar construction works for e, p, and h.
ex@83, 2<, 3D
x@1D2 x@2D2 x@3D + x@1D2 x@2D x@3D2+ x@1D x@2D2 x@3D2
eh@<n>D
Returns e[n] expressed in terms of h[1],h[2],.... . Similar construction works for e, p, and h.
symfun31-doc.nb 3
eh@5D
h@1D5- 4 h@1D3 h@2D + 3 h@1D h@2D2
+ 3 h@1D2 h@3D - 2 h@2D h@3D - 2 h@1D h@4D + h@5D
ep@5D
p@1D5
120-
1
12p@1D3 p@2D +
1
8p@1D p@2D2
+
1
6p@1D2 p@3D -
1
6p@2D p@3D -
1
4p@1D p@4D +
p@5D5
EList@<n>D
Returns list of all e[lambda]'s of degree n, expressed as monomials in e[1],e[2],... (Useful in some technicalsituations.)
EList@6D
9e@6D, e@1D e@5D, e@2D e@4D, e@1D2 e@4D, e@3D2,
e@1D e@2D e@3D, e@1D3 e@3D, e@2D3, e@1D2 e@2D2, e@1D4 e@2D, e@1D6=
EList@<n>, <k>D
Returns list of all e[lambda]'s of degree n and lambda[[1]]£k, expressed as monomials in e[1],e[2],...
EList@6, 2D
9e@1D6, e@1D4 e@2D, e@1D2 e@2D2, e@2D3=
� M: definitions
m@<Λ>D
Converts m[{Λ1,Λ2,...}] notation into m[Λ1,Λ2,...]. Useful for mapping m onto a list of partitions.
m@83, 2, 1<D
m@3, 2, 1D
monomial@<exponentlist>D
Returns a single monomial of type <exponentlist>.
monomial@82, 4, 3, 1<D
x@1D2 x@2D4 x@3D3 x@4D
mx@<Λ>, <n>D
Returns m[Λ] expanded in terms of x[1],x[2],...,x[n].
mx@83, 2, 1<, 3D
x@1D3 x@2D2 x@3D + x@1D2 x@2D3 x@3D + x@1D3 x@2D x@3D2+
x@1D x@2D3 x@3D2+ x@1D2 x@2D x@3D3
+ x@1D x@2D2 x@3D3
MList@<n>D
Returns list of all m[Λ]'s of degree n. (Useful in some technical situations.)
MList@4D
8m@4D, m@3, 1D, m@2, 2D, m@2, 1, 1D, m@1, 1, 1, 1D<
MList@<n>, <k>D
Returns list of all m[Λ]'s of degree n with Length(Λ)£k
4 symfun31-doc.nb
MList@4, 3D
8m@4D, m@3, 1D, m@2, 2D, m@2, 1, 1D<
� P: definitions
p@<Λ>D
Expands in terms of p[1],p[2],p[3],...
p@83, 2, 2, 1<D
p@1D p@2D2 p@3D
px@<Λ>,<n>D
Returns p[lambda] expanded in terms of x[1],x[2],...,x[n].
px@83, 2, 2, 1<, 2D
x@1D8+ x@1D7 x@2D + 2 x@1D6 x@2D2
+ 3 x@1D5 x@2D3+
2 x@1D4 x@2D4+ 3 x@1D3 x@2D5
+ 2 x@1D2 x@2D6+ x@1D x@2D7
+ x@2D8
px@84<, 6D
x@1D4+ x@2D4
+ x@3D4+ x@4D4
+ x@5D4+ x@6D4
pe@<n>D
Returns p[n] expanded in terms of e[1],e[2],... Similar construction works for p, e, and h.
pe@4D
e@1D4- 4 e@1D2 e@2D + 2 e@2D2
+ 4 e@1D e@3D - 4 e@4D
ph@4D
-h@1D4+ 4 h@1D2 h@2D - 2 h@2D2
- 4 h@1D h@3D + 4 h@4D
PList@<n>D
Returns list of all p[Λ]'s of degree n,expressed as monomials in p[1],p[2],... (Useful in some technical situations.)
PList@5D
9p@5D, p@1D p@4D, p@2D p@3D, p@1D2 p@3D, p@1D p@2D2, p@1D3 p@2D, p@1D5=
PList@<n>, <k>D
Returns list of all p[Λ]'s of degree n, and Length[Λ]£k, expressed as monomials in p[1],p[2],...
PList@5, 2D
8p@5D, p@1D p@4D, p@2D p@3D<
� S: definitions
s@<Λ>D
Converts s[{Λ1,Λ2,...}] notation into s[Λ1,Λ2,...]. Useful for mapping m onto a list of partitions.
s@84, 2, 1, 1<D
s@4, 2, 1, 1D
sx@<Λ>,<n>D
Returns s[Λ] expanded in terms of x[1],x[2],...x[n] .
symfun31-doc.nb 5
sx@84, 2, 1<, 3D
x@1D4 x@2D2 x@3D + x@1D3 x@2D3 x@3D + x@1D2 x@2D4 x@3D + x@1D4 x@2D x@3D2+
2 x@1D3 x@2D2 x@3D2+ 2 x@1D2 x@2D3 x@3D2
+ x@1D x@2D4 x@3D2+ x@1D3 x@2D x@3D3
+
2 x@1D2 x@2D2 x@3D3+ x@1D x@2D3 x@3D3
+ x@1D2 x@2D x@3D4+ x@1D x@2D2 x@3D4
sh@<Λ>D
Returns s[Λ] in terms of a linear combination of h[1], h[2], ... . Also works if argument is a sequence, or a singleinteger.
sh@84, 2, 1<Dsh@4, 2, 1Dsh@4D
h@1D h@2D h@4D - h@3D h@4D - h@1D2 h@5D + h@1D h@6D
h@1D h@2D h@4D - h@3D h@4D - h@1D2 h@5D + h@1D h@6Dh@4D
se@<Λ>D
Returns s[Λ] in terms of a linear combination of e[1], e[2], ... Also works if argument is a sequence, or a singleinteger.
se@84, 2, 1<Dse@4, 2, 1Dse@4D
e@1D2 e@2D e@3D - e@2D2 e@3D - e@1D e@3D2-
e@1D3 e@4D + e@1D e@2D e@4D + e@3D e@4D + e@1D2 e@5D - e@1D e@6D
e@1D2 e@2D e@3D - e@2D2 e@3D - e@1D e@3D2-
e@1D3 e@4D + e@1D e@2D e@4D + e@3D e@4D + e@1D2 e@5D - e@1D e@6D
e@1D4- 3 e@1D2 e@2D + e@2D2
+ 2 e@1D e@3D - e@4D
SList@<n>D
Returns list of all s[Λ]'s of degree n. (Useful in some technical situations.)
SList@7D
8s@7D, s@6, 1D, s@5, 2D, s@5, 1, 1D, s@4, 3D, s@4, 2, 1D,
s@4, 1, 1, 1D, s@3, 3, 1D, s@3, 2, 2D, s@3, 2, 1, 1D, s@3, 1, 1, 1, 1D,
s@2, 2, 2, 1D, s@2, 2, 1, 1, 1D, s@2, 1, 1, 1, 1, 1D, s@1, 1, 1, 1, 1, 1, 1D<SList@<n>, <k>D
Returns list of all s[Λ]'s of degree n with Length(Λ)£k
SList@7, 3D
8s@7D, s@6, 1D, s@5, 2D, s@5, 1, 1D, s@4, 3D, s@4, 2, 1D, s@3, 3, 1D, s@3, 2, 2D<
Basic Commands: Partitions, Degrees, Coefficients, Etc.� Partitions
ConjugateP@<Λ>D
6 symfun31-doc.nb
Computes the conjugate of a partition Λ.
ConjugateP@84, 2, 1, 1, 1, 1, 1, 1<D
88, 2, 1, 1<
Partitions@<n>,@ Rows® r,Columns®cDD
Returns a list of partitions of n with £ r parts and largest part b c. Either, both, or none of the options is possible.(Extension of Combinatorica's Partition command.)
Partitions@7, Rows ® 3D
887<, 86, 1<, 85, 2<, 85, 1, 1<, 84, 3<, 84, 2, 1<, 83, 3, 1<, 83, 2, 2<<
Partitions@7, Columns ® 3D
883, 3, 1<, 83, 2, 2<, 83, 2, 1, 1<, 83, 1, 1, 1, 1<,
82, 2, 2, 1<, 82, 2, 1, 1, 1<, 82, 1, 1, 1, 1, 1<, 81, 1, 1, 1, 1, 1, 1<<Partitions@7, Rows ® 3, Columns ® 3D
883, 3, 1<, 83, 2, 2<<
� Degrees in a polynomial
MonDegrees@<f>,<variable list>D
Returns a sorted list of all of the total degrees of monomials (in the listed variables) appearing in f.
f = Product@1 - x@iD^2, 8i, 1, 3<D �� Expand
MonDegrees@f, 8x@1D, x@2D, x@3D<D
1 - x@1D2- x@2D2
+ x@1D2 x@2D2- x@3D2
+ x@1D2 x@3D2+ x@2D2 x@3D2
- x@1D2 x@2D2 x@3D2
86, 4, 2, 0<
MonDegrees@1 + x + y^2 + x y^3 + z^5, 8x, y, z<DMonDegrees@1 + x + y^2 + x y^3 + z^5, 8y, z<DMonDegrees@1 + x + y^2 + x y^3 + z^5, 8z<D85, 4, 2, 1, 0<
85, 3, 2, 0<
85, 0<
XDegrees@<f>D
Returns a sorted list of all of the degrees of in x (assumed) appearing in f. Useful when the number of variables isunknown.
XDegrees@x@1D x@2D^2 x@3D - 43 x@3424D + 321 x@40D^329 x@47D^32 x@412DD
8362, 4, 1<
XDegrees@1 + x@1D + x@1D x@2DD
82, 1, 0<
SDegrees@<f>D
Returns a sorted list of all of the degrees of s's appearing in f (assuming f is expressed in terms of s's).
SDegrees@6 * s@4, 3, 2, 1D + s@3, 1, 1DD
810, 5<
symfun31-doc.nb 7
SDegrees@9 + 6 s@4, 3, 2, 1D + s@3, 1, 1DDSDegrees@6 s@4, 3, 2, 1D + s@3, 1, 1D + x + 9D810, 5, 0<
810, 5, 0, x<
MDegrees@<f>D
Returns a sorted list of all of the degrees of m's appearing in f (assuming f is expressed in terms of m's).
MDegrees@5 * m@4, 2D - 4 m@1D - 7 m@2, 2, 2, 2DD
88, 6, 1<
EDegrees@<f>D
Returns a sorted list of all of the degrees of e's appearing in f (assuming f is expressed in terms of e's).
EDegrees@6 e@4D^4 e@2D e@1D^3 + e@4DD
821, 4<
HDegrees@<f>D
Returns a sorted list of all of the degrees of h's appearing in f (assuming f is expressed in terms of h's).
HDegrees@h@4D h@2D h@1D^2 - 4 h@3DD
88, 3<
PDegrees@<f>D
Returns a sorted list of all of the degrees of p's appearing in f (assuming f is expressed in terms of p's).
PDegrees@p@4D p@2D p@1D^2 - 4 p@3DD
88, 3<
� Coefficients
GetCoefficients@<f>, <variable list>D
Returns a list of the coefficients in f of the variables in <variable list>. The expression f should be a polynomial insubscripted variables, e.g., x[i]'s, h[Λ]'s, s[Λ],etc. If a monomial in f divides another monomial, only the formercoefficient is returned.
SList@5DGetCoefficients@s@3, 2D - 6 s@3, 1, 1D + 93 s@2, 1, 1, 1D - s@5D, SList@5DD8s@5D, s@4, 1D, s@3, 2D, s@3, 1, 1D, s@2, 2, 1D, s@2, 1, 1, 1D, s@1, 1, 1, 1, 1D<
8-1, 0, 1, -6, 0, 93, 0<
g = 7 e@2D^2 e@1D - e@4D e@2D e@1D - 6 e@3D^2 e@1D -
2 e@4D e@1D + 200 e@1D^5 - 3 e@7D - 81 e@2D e@1D^5 + 8 e@5D;
GetCoefficients@g, EList@5DDGetCoefficients@g, EList@7DDGetCoefficients@g, EList@3DD88, -2, 0, 0, 7, 0, 200<
8-3, 0, 0, 0, 0, -1, 0, -6, 0, 0, 0, 0, 0, -81, 0<
80, 0, 0<
GetCoefficients@g^3 �� Expand, EList@21DD
8 symfun31-doc.nb
� Symmetry tests, symmetrization
SymmetricQ@<f>,<z>,<n>D
Tests whether a polynomial in variables z[1],z[2],...,z[n] is symmetric.
SymmetricQ@Hx@1D + x@2D + x@3D + x@4D + x@5DL^10, x, 5D
True
SymmetricQ@Hx@1D + x@2D + x@3D + x@4D + x@5DL^10, x, 6D
False
Symmetrize@<f>,<x>,<n>D
Computes ÚΣ f Σ over all permutations Σ of x[1],x[2],...,x[n].
f = x@1D^2 + x@2D^2 ;
Symmetrize@f, x, 5DSymmetricQ@%, x, 5D
48 x@1D2+ 48 x@2D2
+ 48 x@3D2+ 48 x@4D2
+ 48 x@5D2
True
� Restrict number of variables
Restrict@<f>,<n>, <opts>D
Takes the symmetric function f and returns the equivalent function when the underlying x-version is restricted to nvariables.
This example takes a function in the h - basis, converts it to the s - basis, then restricts the number of variables to3, eliminating the s - terms with more than 3 parts.
f = h@1D^6 - 3 h@3D^2 + h@2D^3 - 5 h@5D h@1Dr1 = HToS@fDr1 = Restrict@r1, 3D
h@1D6+ h@2D3
- 3 h@3D2- 5 h@1D h@5D
-6 s@6D + 3 s@3, 3D + 9 s@4, 2D - s@5, 1D + 6 s@2, 2, 2D + 18 s@3, 2, 1D + 11 s@4, 1, 1D +
9 s@2, 2, 1, 1D + 10 s@3, 1, 1, 1D + 5 s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D-6 s@6D + 3 s@3, 3D + 9 s@4, 2D - s@5, 1D + 6 s@2, 2, 2D + 18 s@3, 2, 1D + 11 s@4, 1, 1D
The same result could have been obtained directly using the HToS conversion command with the option Vars ->3.
r2 = HToS@f, Vars ® 3Dr1 � r2
-6 s@6D + 3 s@3, 3D + 9 s@4, 2D - s@5, 1D + 6 s@2, 2, 2D + 18 s@3, 2, 1D + 11 s@4, 1, 1D
True
Note that the results may be complex in the case of the h - and p - bases. In the other cases, the command operatesmore simply on individual monomials.
f = h@6DRestrict@f, 4Dh@6D
-h@1D6+ 3 h@1D4 h@2D - h@2D3
- 2 h@1D3 h@3D - 2 h@1D h@2D h@3D + h@3D2+ h@1D2 h@4D + 2 h@2D h@4D
symfun31-doc.nb 9
f = p@6DRestrict@f, 4Dp@6D
-
1
24p@1D6
+
3
8p@1D4 p@2D -
3
8p@1D2 p@2D2
-
p@2D3
8-
2
3p@1D3 p@3D +
p@3D2
3+
3
4p@1D2 p@4D +
3
4p@2D p@4D
Input to the Restrict command can be a function expressed using more than one basis. The Bases option allowsthe user to specify which bases the variable-restriction gets applied to. (Admittedly, this is not likely to be veryuseful!).
fmixed = e@4D e@2D + p@3D^2 + p@5D + x@1D x@2D x@3D + x@1D x@2D x@4D +
x@1D x@3D x@4D + x@2D x@3D x@4D + 4 m@2, 2D + s@3, 1, 1, 1, 1, 1, 1, 1, 1D - h@7DRestrict@fmixed, 3DRestrict@fmixed, 3, Bases ® 8E, P, X<D
e@2D e@4D - h@7D + 4 m@2, 2D + p@3D2+ p@5D + s@3, 1, 1, 1, 1, 1, 1, 1, 1D +
x@1D x@2D x@3D + x@1D x@2D x@4D + x@1D x@3D x@4D + x@2D x@3D x@4D
-h@1D3 h@2D2+ 2 h@1D h@2D3
- 2 h@1D4 h@3D + 6 h@1D2 h@2D h@3D - 3 h@2D2 h@3D - 3 h@1D h@3D2+
4 m@2, 2D +
p@1D5
6-
5
6p@1D3 p@2D +
5
6p@1D2 p@3D +
5
6p@2D p@3D + p@3D2
+ x@1D x@2D x@3D
-h@7D + 4 m@2, 2D +
p@1D5
6-
5
6p@1D3 p@2D +
5
6p@1D2 p@3D +
5
6p@2D p@3D + p@3D2
+ s@3, 1, 1, 1, 1, 1, 1, 1, 1D + x@1D x@2D x@3D
Matrices� JTH, JTE : Jacobi-Trudi Matrices
JTH@<Λ>D
Gives Jacobi-Trudi Matrix in h-functions, whose determinant equals s[Λ].
JTH@83, 2, 1, 1<D �� MatrixForm
Det@%D �� HToS
h@3D h@4D h@5D h@6Dh@1D h@2D h@3D h@4D
0 1 h@1D h@2D0 0 1 h@1D
s@3, 2, 1, 1D
JTE@<Λ>D
Gives Jacobi-Trudi Matrix in e-functions, whose determinant equals s[Λ].
10 symfun31-doc.nb
JTE@83, 2, 1, 1<D �� MatrixForm
Det@%D �� EToS
e@4D e@5D e@6De@1D e@2D e@3D
0 1 e@1Ds@3, 2, 1, 1D
� K, KInv,KStar,KTr : Kostka matrices
K@<n>D
Returns the p(n) × p(n) Kostka matrix
K@4D �� MatrixForm
1 1 1 1 1
0 1 1 2 3
0 0 1 1 2
0 0 0 1 3
0 0 0 0 1
K@<n>, <k>D
Returns the p(n, k) × p(n, k) "restricted" Kostka matrix.
K@6, 3D �� MatrixForm
1 1 1 1 1 1 1
0 1 1 2 1 2 2
0 0 1 1 1 2 3
0 0 0 1 0 1 1
0 0 0 0 1 1 1
0 0 0 0 0 1 2
0 0 0 0 0 0 1
KInv@<n>D
Returns the p(n) × p(n) inverse Kostka matrix
KInv@4D �� MatrixForm
1 -1 0 1 -1
0 1 -1 -1 2
0 0 1 -1 1
0 0 0 1 -3
0 0 0 0 1
KInv@<n>, <k>D
Returns the p(n, k) × p(n, k) inverse "restricted" Kostka matrix
KInv@6, 3D �� MatrixForm
1 -1 0 1 0 0 0
0 1 -1 -1 0 1 0
0 0 1 -1 -1 0 -1
0 0 0 1 0 -1 1
0 0 0 0 1 -1 1
0 0 0 0 0 1 -2
0 0 0 0 0 0 1
KTr@<n>D
symfun31-doc.nb 11
Returns the transpose of the p(n) × p(n) Kostka matrix
KTr@4D �� MatrixForm
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 2 1 1 0
1 3 2 3 1
KTr@<n>, <k>D
Returns the transpose of the p(n, k) × p(n, k) "restricted" Kostka matrix
KTr@6, 3D �� MatrixForm
1 0 0 0 0 0 0
1 1 0 0 0 0 0
1 1 1 0 0 0 0
1 2 1 1 0 0 0
1 1 1 0 1 0 0
1 2 2 1 1 1 0
1 2 3 1 1 2 1
KStar@<n>D
Returns the inverse transpose of the p(n) × p(n) Kostka matrix
KStar@4D �� MatrixForm
1 0 0 0 0
-1 1 0 0 0
0 -1 1 0 0
1 -1 -1 1 0
-1 2 1 -3 1
KStar@<n>, <k>D
Returns the inverse transpose of the p(n, k) × p(n, k) "restricted" Kostka matrix
KStar@6, 3D �� MatrixForm
1 0 0 0 0 0 0
-1 1 0 0 0 0 0
0 -1 1 0 0 0 0
1 -1 -1 1 0 0 0
0 0 -1 0 1 0 0
0 1 0 -1 -1 1 0
0 0 -1 1 1 -2 1
12 symfun31-doc.nb
� Bigger Examples
K@8D �� MatrixForm
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 1 2 1 2 3 1 2 2 3 4 2 3 3 4 5 3 4 5 6 7
0 0 1 1 1 2 3 1 2 3 4 6 3 4 5 7 10 6 8 11 15 20
0 0 0 1 0 1 3 0 1 1 3 6 1 3 3 6 10 3 6 10 15 21
0 0 0 0 1 1 1 1 2 2 3 4 3 4 5 7 10 6 9 13 19 28
0 0 0 0 0 1 2 0 1 2 4 8 2 4 6 11 20 8 14 24 40 64
0 0 0 0 0 0 1 0 0 0 1 4 0 1 1 4 10 1 4 10 20 35
0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 3 4 3 4 6 9 14
0 0 0 0 0 0 0 0 1 1 2 3 2 4 5 9 15 7 13 23 40 70
0 0 0 0 0 0 0 0 0 1 1 2 1 1 3 5 10 6 9 16 30 56
0 0 0 0 0 0 0 0 0 0 1 3 0 1 2 6 15 3 9 21 45 90
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 5 0 1 5 15 35
0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 3 6 11 21 42
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 6 2 5 12 26 56
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 3 6 13 30 70
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 2 8 24 64
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 6 21
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 14
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 9 28
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 20
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
KInv@24, 4D �� MatrixForm �� Timing
KTr@12, 5D
KStar@6D �� MatrixForm
1 0 0 0 0 0 0 0 0 0 0
-1 1 0 0 0 0 0 0 0 0 0
0 -1 1 0 0 0 0 0 0 0 0
1 -1 -1 1 0 0 0 0 0 0 0
0 0 -1 0 1 0 0 0 0 0 0
0 1 0 -1 -1 1 0 0 0 0 0
-1 1 1 -1 1 -2 1 0 0 0 0
0 0 -1 1 1 -2 0 1 0 0 0
0 -1 1 1 0 0 -1 -1 1 0 0
1 -1 -2 1 -1 4 -1 1 -3 1 0
-1 2 2 -3 1 -6 4 -1 6 -5 1
� Chi, ChiInv : Character matrices
Chi@<n>D, Chi2@<n>D
Returns the p(n) × p(n) character matrix.
Chi@4D �� MatrixForm
1 1 1 1 1
-1 0 -1 1 3
0 -1 2 0 2
1 0 -1 -1 3
-1 1 1 -1 1
ChiInv@<n>D, ChiInv2@<n>D
symfun31-doc.nb 13
Returns the p(n) × p(n) inverse of the character matrix.
ChiInv@4D �� MatrixForm
1
4-
1
40
1
4-
1
41
30 -
1
30
1
31
8-
1
8
1
4-
1
8
1
81
4
1
40 -
1
4-
1
41
24
1
8
1
12
1
8
1
24
� Bigger Examples
Chi@7D �� MatrixForm �� Timing
:0.094 Second,
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-1 0 -1 1 -1 0 2 0 -1 1 3 0 2 4 6
0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14
1 0 0 0 1 -1 1 0 -1 -1 3 -3 -1 5 15
0 0 -1 -1 1 0 -2 2 -1 1 -1 0 2 4 14
0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35
-1 0 0 0 0 0 0 2 2 0 2 0 -4 0 20
0 0 1 1 -1 -1 -1 0 1 1 -3 -3 1 1 21
0 0 -1 1 1 -1 1 0 1 -1 -3 3 1 -1 21
0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35
1 0 0 0 -1 -1 -1 0 -1 1 3 3 -1 -5 15
0 0 1 -1 -1 0 2 2 -1 -1 -1 0 2 -4 14
0 1 -1 -1 0 0 0 -1 2 0 2 -2 2 -6 14
-1 0 1 1 1 0 -2 0 -1 -1 3 0 2 -4 6
1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1
>
Chi2@7D �� MatrixForm �� Timing
:21.828 Second,
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-1 0 -1 1 -1 0 2 0 -1 1 3 0 2 4 6
0 -1 1 -1 0 0 0 -1 2 0 2 2 2 6 14
1 0 0 0 1 -1 1 0 -1 -1 3 -3 -1 5 15
0 0 -1 -1 1 0 -2 2 -1 1 -1 0 2 4 14
0 1 0 0 -1 1 -1 -1 -1 -1 -1 1 -1 5 35
-1 0 0 0 0 0 0 2 2 0 2 0 -4 0 20
0 0 1 1 -1 -1 -1 0 1 1 -3 -3 1 1 21
0 0 -1 1 1 -1 1 0 1 -1 -3 3 1 -1 21
0 -1 0 0 1 1 1 -1 -1 1 -1 -1 -1 -5 35
1 0 0 0 -1 -1 -1 0 -1 1 3 3 -1 -5 15
0 0 1 -1 -1 0 2 2 -1 -1 -1 0 2 -4 14
0 1 -1 -1 0 0 0 -1 2 0 2 -2 2 -6 14
-1 0 1 1 1 0 -2 0 -1 -1 3 0 2 -4 6
1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1
>
chi10 = Chi@10D
Det@chi10DApply@Times, Flatten@Partitions@10DDD9 263 795 272 057 860 957 392 207 640 004 657 152 000 000 000
9 263 795 272 057 860 957 392 207 640 004 657 152 000 000 000
This last result illustrates the remarkable result that the determinant of the character matrix of Sn equals theproduct of all of the parts in all of the partitions of n.
14 symfun31-doc.nb
This last result illustrates the remarkable result that the determinant of the character matrix of Sn equals theproduct of all of the parts in all of the partitions of n.
� J (transposition matrix), SToEMat, EToSMat, HToH2Mat, H2ToHMat (special matrices for restricted variable situations)
J@<n>D
Returns the p(n) × p(n) "transposition" matrix that takes every partition to its conjugate
J@6D �� MatrixForm
0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0
SToEMat@<n>, <k>D
Returns the p(n, k) × p(n, k) transition matrix from the S basis to the E basis in a situation where the number ofvariables is restricted to k
SToEMat@8, 3D �� MatrixForm
1 -7 15 6 -10 -20 1 12 6 -3
0 1 -5 -1 6 8 -1 -9 -3 3
0 0 1 -1 -3 2 1 3 -2 -1
0 0 0 1 0 -4 0 3 3 -2
0 0 0 0 1 -2 -1 1 2 -1
0 0 0 0 0 1 0 -2 -1 2
0 0 0 0 0 0 1 -3 1 2
0 0 0 0 0 0 0 1 -1 -1
0 0 0 0 0 0 0 0 1 -1
0 0 0 0 0 0 0 0 0 1
EToSMat@<n>, <k>D
Returns the p(n, k) × p(n, k) transition matrix from the E basis to the S basis in a situation where the number ofvariables is restricted to k
EToSMat@8, 3D �� MatrixForm
1 7 20 21 28 64 14 70 56 42
0 1 5 6 9 24 5 30 26 21
0 0 1 1 3 8 2 13 12 11
0 0 0 1 0 4 0 5 6 5
0 0 0 0 1 2 1 6 5 6
0 0 0 0 0 1 0 2 3 3
0 0 0 0 0 0 1 3 2 3
0 0 0 0 0 0 0 1 1 2
0 0 0 0 0 0 0 0 1 1
0 0 0 0 0 0 0 0 0 1
HToH2Mat@<n>, <k>D
Returns a p(n, k) × p(n, k) matrix that converts from the basis of H-restricted expressed by HList[n,k] to the basisof H-restricted expressed by HList2[n,k]
symfun31-doc.nb 15
Returns a p(n, k) × p(n, k) matrix that converts from the basis of H-restricted expressed by HList[n,k] to the basisof H-restricted expressed by HList2[n,k]
HRestrictionRules@5, 2DHToH2Mat@5, 2D �� MatrixForm
H2ToHMat@5, 2D �� MatrixForm
9h@5D ® -2 h@1D3 h@2D + 3 h@1D h@2D2, h@1D h@4D ® -h@1D5+ h@1D3 h@2D + h@1D h@2D2,
h@2D h@3D ® -h@1D3 h@2D + 2 h@1D h@2D2, h@1D2 h@3D ® -h@1D5+ 2 h@1D3 h@2D,
h@1D h@2D2® h@1D h@2D2, h@1D3 h@2D ® h@1D3 h@2D, h@1D5
® h@1D5=
3 -2 0
1 1 -1
2 -1 0
-1 0 2
-2 0 3
-3 -1 5
� Bigger examples
SToEMat@15, 3D �� MatrixForm
1 -14 78 13 -220 -132 330 495 55 -252 -840 -360 84 630 756 84 -8 -168 -
0 1 -12 -1 55 22 -120 -135 -10 126 336 108 -56 -350 -336 -28 7 126
0 0 1 -1 -10 9 36 -18 -9 -56 -28 48 35 105 -42 -21 -6 -69
0 0 0 1 0 -11 0 45 10 0 -84 -72 0 70 168 28 0 -21 -
0 0 0 0 1 -2 -8 15 2 21 -28 -29 -20 -5 90 13 5 30
0 0 0 0 0 1 0 -9 -1 0 28 16 0 -35 -63 -7 0 15
0 0 0 0 0 0 1 -3 1 -6 17 -2 10 -20 -27 4 -4 -6
0 0 0 0 0 0 0 1 -1 0 -7 6 0 15 -3 -6 0 -10
0 0 0 0 0 0 0 0 1 0 0 -8 0 0 21 7 0 0
0 0 0 0 0 0 0 0 0 1 -4 3 -4 15 -6 -3 3 -6
0 0 0 0 0 0 0 0 0 0 1 -2 0 -5 9 2 0 6
0 0 0 0 0 0 0 0 0 0 0 1 0 0 -6 -1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 -5 6 -1 -2 9
0 0 0 0 0 0 0 0 0 0 0 0 0 1 -3 1 0 -3
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -6
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
� Other matrices: PEMat, PHMat, EPMat, EHMat, HPMat, HEMat
PEMat@<n>D
Gives n × n matrix whose determinant is p[n] in terms of e[1],...,e[n]
16 symfun31-doc.nb
PEMat@5D �� MatrixForm
Det@%DEToP@%D
e@1D 1 0 0 0
2 e@2D e@1D 1 0 0
3 e@3D e@2D e@1D 1 0
4 e@4D e@3D e@2D e@1D 1
5 e@5D e@4D e@3D e@2D e@1D
e@1D5- 5 e@1D3 e@2D + 5 e@1D e@2D2
+ 5 e@1D2 e@3D - 5 e@2D e@3D - 5 e@1D e@4D + 5 e@5Dp@5D
PHMat@<n>D
Gives n × n matrix whose determinant is ±p[n] in terms of h[1],...,h[n] (The ± is implemented in the ph function.)
PHMat@5D �� MatrixForm
Det@%DHToP@%D
h@1D 1 0 0 0
2 h@2D h@1D 1 0 0
3 h@3D h@2D h@1D 1 0
4 h@4D h@3D h@2D h@1D 1
5 h@5D h@4D h@3D h@2D h@1D
h@1D5- 5 h@1D3 h@2D + 5 h@1D h@2D2
+ 5 h@1D2 h@3D - 5 h@2D h@3D - 5 h@1D h@4D + 5 h@5Dp@5D
EPMat@<n>D
Gives n × n matrix whose determinant is n!*e[n] in terms of p[1],...,p[n]
EPMat@5D �� MatrixForm
Det@%DPToE@%D
p@1D 1 0 0 0
p@2D p@1D 2 0 0
p@3D p@2D p@1D 3 0
p@4D p@3D p@2D p@1D 4
p@5D p@4D p@3D p@2D p@1D
p@1D5- 10 p@1D3 p@2D + 15 p@1D p@2D2
+ 20 p@1D2 p@3D - 20 p@2D p@3D - 30 p@1D p@4D + 24 p@5D120 e@5D
EHMat@<n>D
Gives n × n matrix whose determinant is e[n] in terms of h[1],...,h[n]
symfun31-doc.nb 17
EHMat@5D �� MatrixForm
Det@%DHToE@%D
h@1D h@2D h@3D h@4D h@5D1 h@1D h@2D h@3D h@4D0 1 h@1D h@2D h@3D0 0 1 h@1D h@2D0 0 0 1 h@1D
h@1D5- 4 h@1D3 h@2D + 3 h@1D h@2D2
+ 3 h@1D2 h@3D - 2 h@2D h@3D - 2 h@1D h@4D + h@5De@5D
HPMat@<n>D
Gives n × n matrix whose determinant is n!*h[n] in terms of p[1],...,p[n]
HPMat@5D �� MatrixForm
Det@%DPToH@%D
p@1D -1 0 0 0
p@2D p@1D -2 0 0
p@3D p@2D p@1D -3 0
p@4D p@3D p@2D p@1D -4
p@5D p@4D p@3D p@2D p@1D
p@1D5+ 10 p@1D3 p@2D + 15 p@1D p@2D2
+ 20 p@1D2 p@3D + 20 p@2D p@3D + 30 p@1D p@4D + 24 p@5D120 h@5D
HEMat@<n>D
Gives n × n matrix whose determinant is h[n] in terms of e[1],...,e[n]
HEMat@5D �� MatrixForm
Det@%DEToH@%D
e@1D e@2D e@3D e@4D e@5D1 e@1D e@2D e@3D e@4D0 1 e@1D e@2D e@3D0 0 1 e@1D e@2D0 0 0 1 e@1D
e@1D5- 4 e@1D3 e@2D + 3 e@1D e@2D2
+ 3 e@1D2 e@3D - 2 e@2D e@3D - 2 e@1D e@4D + e@5Dh@5D
Basis Conversions� XTo[HEMPSAll]
� XToM
XToM@<f>D
If f is a symmetric function expressed in terms of x[1],x[2],..., returns f expressed in terms of the m[lambda]'s.(Assumes that f is symmetric but does not test for this.)
18 symfun31-doc.nb
XToM@x@1D^3 + 3 x@1D x@2D x@3D + x@3D^3 + x@2D^3D
m@3D + 3 m@1, 1, 1D
NumberOfVars@<f>D
Returns number of x[i] variables in an expression involving x[i]'s.
NumberOfVars@x@1D x@2D + x@3DD
3
NumberOfVars@x@1D^2 + x@2D^2D
2
XToM@x@1D^2 + x@2D^2D
m@2D
XToM@Hx@1D + x@2DL Hx@1D + x@3DL Hx@2D + x@3DLD
m@2, 1D + 2 m@1, 1, 1D
� XTo[Others]
XToB@<f>D
Converts f (expressed in terms of x[i]) into basis B, where B can be H,E,P,S, or "All".
f = Product@x@iD + x@jD, 8i, 1, 3<, 8j, i + 1, 3<Df �� XToH
f �� XToE
f �� XToP
f �� XToS
f �� XToM
Hx@1D + x@2DL Hx@1D + x@3DL Hx@2D + x@3DL
h@1D h@2D - h@3D
e@1D e@2D - e@3D
p@1D3
3-
p@3D3
s@2, 1D
m@2, 1D + 2 m@1, 1, 1D
� STo[HEMPXAll]
� SToH
SToH@<f>,< opts>DOptions@SToHD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of s functions, SToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
symfun31-doc.nb 19
SToH@s@4D + 3 s@2, 2DD% �� HToS
3 h@2D2- 3 h@1D h@3D + h@4D
s@4D + 3 s@2, 2D
SToH@s@4D + 3 s@2, 2D, Vars -> 2D
2 h@1D4- 5 h@1D2 h@2D + 4 h@2D2
HToX@% - %%, 2D
0
� Bigger Examples
� SToE
SToE@<f>, <opts>DOptions@SToED=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of s functions, SToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
SToE@4 s@4D - 3 s@3, 1D + 2 s@2, 2D - s@1, 1, 1, 1DD% �� EToS
4 e@1D4- 15 e@1D2 e@2D + 9 e@2D2
+ 9 e@1D e@3D - 8 e@4D4 s@4D + 2 s@2, 2D - 3 s@3, 1D - s@1, 1, 1, 1D
SToE@4 s@4D - 3 s@3, 1D + 2 s@2, 2D - s@1, 1, 1, 1D, Vars ® 2D
4 e@1D4- 15 e@1D2 e@2D + 9 e@2D2
EToX@% - %%, 2D
0
� Bigger examples
� SToM
SToM@<f>, <opts>DOptions@SToMD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of s functions, SToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
SToM@3 s@5D - 9 s@4, 1D + s@1, 1, 1, 1, 1DD% �� MToS
3 m@5D - 6 m@3, 2D - 6 m@4, 1D - 15 m@2, 2, 1D - 15 m@3, 1, 1D - 24 m@2, 1, 1, 1D - 32 m@1, 1, 1, 1, 1D
3 s@5D - 9 s@4, 1D + s@1, 1, 1, 1, 1D
20 symfun31-doc.nb
SToM@s@3, 2, 1DD% �� MToS
2 m@2, 2, 2D + m@3, 2, 1D + 4 m@2, 2, 1, 1D +
2 m@3, 1, 1, 1D + 8 m@2, 1, 1, 1, 1D + 16 m@1, 1, 1, 1, 1, 1Ds@3, 2, 1D
SToM@s@3, 2, 1D, Vars ® 4D
2 m@2, 2, 2D + m@3, 2, 1D + 4 m@2, 2, 1, 1D + 2 m@3, 1, 1, 1D
� Bigger examples
� SToP
SToP@<f>,<opts>DOptions@SToPD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of s functions, SToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
SToP@2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1DD% �� PToS
2 p@1D + p@1D3
2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1D
SToP@s@4, 3, 2, 1DD% �� PToS
p@1D10
4725-
1
315p@1D7 p@3D -
1
27p@1D p@3D3
+
1
75p@1D5 p@5D +
1
15p@1D2 p@3D p@5D -
p@5D2
25-
1
21p@1D3 p@7D +
1
21p@3D p@7D
s@4, 3, 2, 1D
SToP@s@4, 3, 2, 1D, Vars ® 4D
p@1D10
1728-
7
288p@1D6 p@2D2
+
1
24p@1D4 p@2D3
-
1
64p@1D2 p@2D4
-
1
144p@1D7 p@3D +
5
48p@1D5 p@2D p@3D -
17
144p@1D3 p@2D2 p@3D +
1
48p@1D p@2D3 p@3D -
7
72p@1D4 p@3D2
+
1
12p@1D2 p@2D p@3D2
-
1
72p@2D2 p@3D2
-
1
27p@1D p@3D3
+
1
144p@1D6 p@4D -
1
12p@1D4 p@2D p@4D +
1
16p@1D2 p@2D2 p@4D +
11
72p@1D3 p@3D p@4D -
1
24p@1D p@2D p@3D p@4D +
1
36p@3D2 p@4D -
1
16p@1D2 p@4D2
symfun31-doc.nb 21
PToS@%DPToS@%%, Vars ® 4Ds@4, 3, 2, 1D - s@2, 2, 2, 2, 2D - s@3, 2, 2, 2, 1D + s@3, 3, 2, 1, 1D +
s@4, 2, 2, 1, 1D + s@4, 3, 1, 1, 1D - 2 s@2, 2, 2, 2, 1, 1D + s@3, 3, 1, 1, 1, 1D +
s@4, 2, 1, 1, 1, 1D - 2 s@2, 2, 2, 1, 1, 1, 1D - 2 s@2, 2, 1, 1, 1, 1, 1, 1D -
s@3, 1, 1, 1, 1, 1, 1, 1D - 2 s@2, 1, 1, 1, 1, 1, 1, 1, 1D - s@1, 1, 1, 1, 1, 1, 1, 1, 1, 1Ds@4, 3, 2, 1D
� SToX
SToX@<f>,<vars>D
If f is a symmetric function expressed as a linear combination of s functions, SToX returns the same functionexpressed in variables x[1], ..., x[n].
SToX@2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1D, 4DXToS@%D
2 x@1D + x@1D3+ 2 x@2D + 3 x@1D2 x@2D + 3 x@1D x@2D2
+ x@2D3+ 2 x@3D +
3 x@1D2 x@3D + 6 x@1D x@2D x@3D + 3 x@2D2 x@3D + 3 x@1D x@3D2+ 3 x@2D x@3D2
+
x@3D3+ 2 x@4D + 3 x@1D2 x@4D + 6 x@1D x@2D x@4D + 3 x@2D2 x@4D + 6 x@1D x@3D x@4D +
6 x@2D x@3D x@4D + 3 x@3D2 x@4D + 3 x@1D x@4D2+ 3 x@2D x@4D2
+ 3 x@3D x@4D2+ x@4D3
2 s@1D + s@3D + 2 s@2, 1D + s@1, 1, 1D
SToX@s@3, 2, 1D, 3D
x@1D3 x@2D2 x@3D + x@1D2 x@2D3 x@3D + x@1D3 x@2D x@3D2+
2 x@1D2 x@2D2 x@3D2+ x@1D x@2D3 x@3D2
+ x@1D2 x@2D x@3D3+ x@1D x@2D2 x@3D3
SToX@s@4, 3, 2, 1D, 4DXToS@%D
x@1D4 x@2D3 x@3D2 x@4D + x@1D3 x@2D4 x@3D2 x@4D +
x@1D4 x@2D2 x@3D3 x@4D + 2 x@1D3 x@2D3 x@3D3 x@4D + x@1D2 x@2D4 x@3D3 x@4D +
x@1D3 x@2D2 x@3D4 x@4D + x@1D2 x@2D3 x@3D4 x@4D + x@1D4 x@2D3 x@3D x@4D2+
x@1D3 x@2D4 x@3D x@4D2+ 2 x@1D4 x@2D2 x@3D2 x@4D2
+ 4 x@1D3 x@2D3 x@3D2 x@4D2+
2 x@1D2 x@2D4 x@3D2 x@4D2+ x@1D4 x@2D x@3D3 x@4D2
+ 4 x@1D3 x@2D2 x@3D3 x@4D2+
4 x@1D2 x@2D3 x@3D3 x@4D2+ x@1D x@2D4 x@3D3 x@4D2
+ x@1D3 x@2D x@3D4 x@4D2+
2 x@1D2 x@2D2 x@3D4 x@4D2+ x@1D x@2D3 x@3D4 x@4D2
+ x@1D4 x@2D2 x@3D x@4D3+
2 x@1D3 x@2D3 x@3D x@4D3+ x@1D2 x@2D4 x@3D x@4D3
+ x@1D4 x@2D x@3D2 x@4D3+
4 x@1D3 x@2D2 x@3D2 x@4D3+ 4 x@1D2 x@2D3 x@3D2 x@4D3
+ x@1D x@2D4 x@3D2 x@4D3+
2 x@1D3 x@2D x@3D3 x@4D3+ 4 x@1D2 x@2D2 x@3D3 x@4D3
+ 2 x@1D x@2D3 x@3D3 x@4D3+
x@1D2 x@2D x@3D4 x@4D3+ x@1D x@2D2 x@3D4 x@4D3
+ x@1D3 x@2D2 x@3D x@4D4+
x@1D2 x@2D3 x@3D x@4D4+ x@1D3 x@2D x@3D2 x@4D4
+ 2 x@1D2 x@2D2 x@3D2 x@4D4+
x@1D x@2D3 x@3D2 x@4D4+ x@1D2 x@2D x@3D3 x@4D4
+ x@1D x@2D2 x@3D3 x@4D4
s@4, 3, 2, 1D
� SToAll
SToAll@<f>, <opts>DOptions@SToAllD=8Targets®8E,H,P,M, X<, Vars®0<
If f is a symmetric function expressed as a linear combination of s functions, SToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.
22 symfun31-doc.nb
SToAll@s@3, 2, 1DD �� ColumnForm
s@3, 2, 1De@1D e@2D e@3D - e@3D2
- e@1D2 e@4D + e@1D e@5Dh@1D h@2D h@3D - h@3D2
- h@1D2 h@4D + h@1D h@5Dp@1D6
45-
1
9p@1D3 p@3D -
p@3D2
9+
1
5p@1D p@5D
2 m@2, 2, 2D + m@3, 2, 1D + 4 m@2, 2, 1, 1D + 2 m@3, 1, 1, 1D + 8 m@2, 1, 1, 1, 1D + 16 m@1, 1, 1, 1, 1,
SToAll@s@3, 2, 1D - 2 s@2, 2, 2D, Vars ® 3, Targets ® 8E, M, X<D
9-2 s@2, 2, 2D + s@3, 2, 1D, e@1D e@2D e@3D - 3 e@3D2, m@3, 2, 1D, x@1D3 x@2D2 x@3D +
x@1D2 x@2D3 x@3D + x@1D3 x@2D x@3D2+ x@1D x@2D3 x@3D2
+ x@1D2 x@2D x@3D3+ x@1D x@2D2 x@3D3=
� PTo[HEMSXAll]
� PRestrictionRules
PRestrictionRules@<deg>, <var>D
PRestrictionRules[deg, var] creates a list of rules that express p functions of degree deg in terms of other pfuntions of degree deg when the number of variables is restricted to var. The basis used for this is p[lambda]functions where lambda[[1]]£var (note that this is NOT the same basis generated by PList[deg, var]).
PRestrictionRules@4, 2DPList@4, 2D
:p@4D ® -
1
2p@1D4
+ p@1D2 p@2D +
p@2D2
2, p@1D p@3D ® -
1
2p@1D4
+
3
2p@1D2 p@2D,
p@2D2® p@2D2, p@1D2 p@2D ® p@1D2 p@2D, p@1D4
® p@1D4>
9p@4D, p@1D p@3D, p@2D2=
� PToH
PToH@<f>,< opts>DOptions@PToHD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of p functions, PToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
PToH@p@1D^2 p@4DD% �� HToP
-h@1D6+ 4 h@1D4 h@2D - 2 h@1D2 h@2D2
- 4 h@1D3 h@3D + 4 h@1D2 h@4D
p@1D2 p@4D
Restrict@%, 3DPToH@p@1D^2 p@4D, Vars ® 3D
3 h@1D6- 8 h@1D4 h@2D + 2 h@1D2 h@2D2
+ 4 h@1D3 h@3D
3 h@1D6- 8 h@1D4 h@2D + 2 h@1D2 h@2D2
+ 4 h@1D3 h@3D
symfun31-doc.nb 23
Restrict@p@1D^2 p@4D, 3DPToH@%D
p@1D6
6- p@1D4 p@2D +
1
2p@1D2 p@2D2
+
4
3p@1D3 p@3D
3 h@1D6- 8 h@1D4 h@2D + 2 h@1D2 h@2D2
+ 4 h@1D3 h@3D
� PToE
PToE@<f>,< opts>DOptions@PToED=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of p functions, PToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
PToE@p@1D^2 p@4DD% �� EToP
e@1D6- 4 e@1D4 e@2D + 2 e@1D2 e@2D2
+ 4 e@1D3 e@3D - 4 e@1D2 e@4D
p@1D2 p@4D
Restrict@%, 3DPToE@p@1D^2 p@4D, Vars ® 3D
e@1D6- 4 e@1D4 e@2D + 2 e@1D2 e@2D2
+ 4 e@1D3 e@3D
e@1D6- 4 e@1D4 e@2D + 2 e@1D2 e@2D2
+ 4 e@1D3 e@3D
� PToM
PToM@<f>,< opts>DOptions@PToMD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of p functions, PToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
PToM@p@3D - 5 p@1D^3D% �� MToP
-4 m@3D - 15 m@2, 1D - 30 m@1, 1, 1D
-5 p@1D3+ p@3D
Restrict@%, 2DPToM@p@3D - 5 p@1D^3, Vars ® 2D
-
11
2p@1D3
+
3
2p@1D p@2D
-4 m@3D - 15 m@2, 1D
24 symfun31-doc.nb
PToM@p@4D - 3 p@3D p@1D + 2 p@1D^4DRestrict@%, 3DPToM@p@4D - 3 p@3D p@1D + 2 p@1D^4, Vars ® 3D12 m@2, 2D + 5 m@3, 1D + 24 m@2, 1, 1D + 48 m@1, 1, 1, 1D
12 m@2, 2D + 5 m@3, 1D + 24 m@2, 1, 1D
12 m@2, 2D + 5 m@3, 1D + 24 m@2, 1, 1D
� PToS
PToS@<f>,<options>DOptions@PToSD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of p functions, PToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
PToS@p@1D^3 + 2 p@3DD% �� SToP
3 s@3D + 3 s@1, 1, 1D
p@1D3+ 2 p@3D
PToS@p@1D^3 + 2 p@3D, Vars ® 2D
3 s@3D
� PToX
PToX@<f>,<n>D
If f is a symmetric function expressed as a linear combination of p functions, PToX returns the same functionexpressed in variables x[1], ..., x[n].
PToX@p@2D p@1D^2, 3D% �� XToP
x@1D4+ 2 x@1D3 x@2D + 2 x@1D2 x@2D2
+ 2 x@1D x@2D3+ x@2D4
+ 2 x@1D3 x@3D +
2 x@1D2 x@2D x@3D + 2 x@1D x@2D2 x@3D + 2 x@2D3 x@3D + 2 x@1D2 x@3D2+
2 x@1D x@2D x@3D2+ 2 x@2D2 x@3D2
+ 2 x@1D x@3D3+ 2 x@2D x@3D3
+ x@3D4
p@1D2 p@2D
� PToAll
PToAll@<f>, <opts>DOptions@PToAllD=8Targets®8E,H,S,M, X<, Vars®0<
If f is a symmetric function expressed as a linear combination of p functions, PToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.
symfun31-doc.nb 25
Sum@p@iD p@jD, 8i, 3<, 8j, 3<D;
PToAll@%D
9p@1D2+ 2 p@1D p@2D + p@2D2
+ 2 p@1D p@3D + 2 p@2D p@3D + p@3D2,
e@1D2+ 2 e@1D3
+ 3 e@1D4+ 2 e@1D5
+ e@1D6- 4 e@1D e@2D - 10 e@1D2 e@2D -
10 e@1D3 e@2D - 6 e@1D4 e@2D + 4 e@2D2+ 12 e@1D e@2D2
+ 9 e@1D2 e@2D2+ 6 e@1D e@3D +
6 e@1D2 e@3D + 6 e@1D3 e@3D - 12 e@2D e@3D - 18 e@1D e@2D e@3D + 9 e@3D2,
h@1D2- 2 h@1D3
+ 3 h@1D4- 2 h@1D5
+ h@1D6+ 4 h@1D h@2D - 10 h@1D2 h@2D +
10 h@1D3 h@2D - 6 h@1D4 h@2D + 4 h@2D2- 12 h@1D h@2D2
+ 9 h@1D2 h@2D2+ 6 h@1D h@3D -
6 h@1D2 h@3D + 6 h@1D3 h@3D + 12 h@2D h@3D - 18 h@1D h@2D h@3D + 9 h@3D2,
s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D + s@1, 1D - s@3, 1D + 2 s@3, 2D + 2 s@3, 3D -
2 s@4, 1D - s@5, 1D - 2 s@1, 1, 1D - s@2, 1, 1D - 2 s@2, 2, 1D + 2 s@2, 2, 2D -
2 s@3, 2, 1D + s@4, 1, 1D + 3 s@1, 1, 1, 1D + 2 s@2, 1, 1, 1D + s@3, 1, 1, 1D -
2 s@1, 1, 1, 1, 1D - s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D, m@2D + 2 m@3D + 3 m@4D +
2 m@5D + m@6D + 2 m@1, 1D + 2 m@2, 1D + 2 m@2, 2D + 2 m@3, 1D + 2 m@3, 2D + 2 m@3, 3D=
Sum@p@iD p@jD, 8i, 3<, 8j, 3<D;
PToAll@%, Vars ® 2, Targets ® 8E, M, X<D
9p@1D2+ 2 p@1D p@2D + p@2D2
+ 2 p@1D p@3D + 2 p@2D p@3D + p@3D2,
e@1D2+ 2 e@1D3
+ 3 e@1D4+ 2 e@1D5
+ e@1D6- 4 e@1D e@2D - 10 e@1D2 e@2D -
10 e@1D3 e@2D - 6 e@1D4 e@2D + 4 e@2D2+ 12 e@1D e@2D2
+ 9 e@1D2 e@2D2,
m@2D + 2 m@3D + 3 m@4D + 2 m@5D + m@6D + 2 m@1, 1D + 2 m@2, 1D + 2 m@2, 2D + 2 m@3, 1D +
2 m@3, 2D + 2 m@3, 3D, x@1D2+ 2 x@1D3
+ 3 x@1D4+ 2 x@1D5
+ x@1D6+ 2 x@1D x@2D +
2 x@1D2 x@2D + 2 x@1D3 x@2D + x@2D2+ 2 x@1D x@2D2
+ 2 x@1D2 x@2D2+ 2 x@1D3 x@2D2
+
2 x@2D3+ 2 x@1D x@2D3
+ 2 x@1D2 x@2D3+ 2 x@1D3 x@2D3
+ 3 x@2D4+ 2 x@2D5
+ x@2D6=
� MTo[HEPSXAll]
� MToH
MToH@<f>, <opts>DOptions@MToHD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of m functions, MToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
MToH@m@1, 1, 1, 1D + 2 m@2, 2D - 3 m@3, 1D + 4 m@4DD% �� HToM
-7 h@1D4+ 26 h@1D2 h@2D - 7 h@2D2
- 31 h@1D h@3D + 23 h@4D4 m@4D + 2 m@2, 2D - 3 m@3, 1D + m@1, 1, 1, 1D
MToH@m@1, 1, 1, 1D + 2 m@2, 2D - 3 m@3, 1D + 4 m@4D, Vars ® 3D% �� HToM
16 h@1D4- 43 h@1D2 h@2D + 16 h@2D2
+ 15 h@1D h@3D4 m@4D + 2 m@2, 2D - 3 m@3, 1D + 24 m@1, 1, 1, 1D
� Bigger Examples
MToH@m@9, 1D + m@4, 3, 3D - 10 m@6, 3D + 14 m@13, 2DD �� Timing
26 symfun31-doc.nb
MToH@m@9, 1D + m@4, 3, 3D - 10 m@6, 3D + 14 m@13, 2D, Vars ® 5D �� Timing
936.037, 20 h@1D9- h@1D10
- 28 h@1D15- 180 h@1D7 h@2D + 11 h@1D8 h@2D + 238 h@1D13 h@2D +
540 h@1D5 h@2D2- 43 h@1D6 h@2D2
- 532 h@1D11 h@2D2- 590 h@1D3 h@2D3
+ 68 h@1D4 h@2D3-
700 h@1D9 h@2D3+ 150 h@1D h@2D4
- 34 h@1D2 h@2D4+ 4620 h@1D7 h@2D4
+ 2 h@2D5- 6902 h@1D5 h@2D5
+
3864 h@1D3 h@2D6- 518 h@1D h@2D7
+ 180 h@1D6 h@3D - 11 h@1D7 h@3D - 210 h@1D12 h@3D -
930 h@1D4 h@2D h@3D + 81 h@1D5 h@2D h@3D + 1008 h@1D10 h@2D h@3D + 1170 h@1D2 h@2D2 h@3D -
166 h@1D3 h@2D2 h@3D + 420 h@1D8 h@2D2 h@3D - 150 h@2D3 h@3D + 67 h@1D h@2D3 h@3D -
8638 h@1D6 h@2D3 h@3D + 14 546 h@1D4 h@2D4 h@3D - 7672 h@1D2 h@2D5 h@3D + 518 h@2D6 h@3D +
390 h@1D3 h@3D2- 40 h@1D4 h@3D2
- 476 h@1D9 h@3D2- 720 h@1D h@2D h@3D2
+ 141 h@1D2 h@2D h@3D2+
462 h@1D7 h@2D h@3D2- 33 h@2D2 h@3D2
+ 5068 h@1D5 h@2D2 h@3D2- 9912 h@1D3 h@2D3 h@3D2
+
4732 h@1D h@2D4 h@3D2+ 120 h@3D3
- 42 h@1D h@3D3- 182 h@1D6 h@3D3
- 1442 h@1D4 h@2D h@3D3+
2464 h@1D2 h@2D2 h@3D3- 924 h@2D3 h@3D3
+ 392 h@1D3 h@3D4- 154 h@1D h@2D h@3D4
-
42 h@3D5- 150 h@1D5 h@4D + 5 h@1D6 h@4D + 600 h@1D3 h@2D h@4D - 38 h@1D4 h@2D h@4D +
728 h@1D9 h@2D h@4D - 450 h@1D h@2D2 h@4D + 69 h@1D2 h@2D2 h@4D - 4914 h@1D7 h@2D2 h@4D -
10 h@2D3 h@4D + 10 920 h@1D5 h@2D3 h@4D - 8918 h@1D3 h@2D4 h@4D + 1820 h@1D h@2D5 h@4D -
450 h@1D2 h@3D h@4D + 37 h@1D3 h@3D h@4D - 364 h@1D8 h@3D h@4D + 360 h@2D h@3D h@4D -
117 h@1D h@2D h@3D h@4D + 5572 h@1D6 h@2D h@3D h@4D - 16 282 h@1D4 h@2D2 h@3D h@4D +
13 076 h@1D2 h@2D3 h@3D h@4D - 1456 h@2D4 h@3D h@4D + 45 h@3D2 h@4D - 1540 h@1D5 h@3D2 h@4D +
7896 h@1D3 h@2D h@3D2 h@4D - 5642 h@1D h@2D2 h@3D2 h@4D - 1652 h@1D2 h@3D3 h@4D +
784 h@2D h@3D3 h@4D + 90 h@1D h@4D2- 3 h@1D2 h@4D2
+ 420 h@1D7 h@4D2+ 12 h@2D h@4D2
-
3038 h@1D5 h@2D h@4D2+ 5180 h@1D3 h@2D2 h@4D2
- 1778 h@1D h@2D3 h@4D2+ 1890 h@1D4 h@3D h@4D2
-
4802 h@1D2 h@2D h@3D h@4D2+ 994 h@2D2 h@3D h@4D2
+ 1260 h@1D h@3D2 h@4D2- 588 h@1D3 h@4D3
+
476 h@1D h@2D h@4D3- 210 h@3D h@4D3
+ 150 h@1D4 h@5D - h@1D5 h@5D - 450 h@1D2 h@2D h@5D +
13 h@1D3 h@2D h@5D - 364 h@1D8 h@2D h@5D + 90 h@2D2 h@5D - 18 h@1D h@2D2 h@5D +
2366 h@1D6 h@2D2 h@5D - 4914 h@1D4 h@2D3 h@5D + 3458 h@1D2 h@2D4 h@5D - 364 h@2D5 h@5D +
360 h@1D h@3D h@5D - 15 h@1D2 h@3D h@5D + 21 h@2D h@3D h@5D - 1512 h@1D5 h@2D h@3D h@5D +
4956 h@1D3 h@2D2 h@3D h@5D - 3444 h@1D h@2D3 h@3D h@5D + 28 h@1D4 h@3D2 h@5D -
1344 h@1D2 h@2D h@3D2 h@5D + 686 h@2D2 h@3D2 h@5D + 420 h@1D h@3D3 h@5D - 90 h@4D h@5D +
3 h@1D h@4D h@5D - 420 h@1D6 h@4D h@5D + 2828 h@1D4 h@2D h@4D h@5D - 4158 h@1D2 h@2D2 h@4D h@5D +
784 h@2D3 h@4D h@5D - 1316 h@1D3 h@3D h@4D h@5D + 2632 h@1D h@2D h@3D h@4D h@5D -
630 h@3D2 h@4D h@5D + 868 h@1D2 h@4D2 h@5D - 266 h@2D h@4D2 h@5D - 182 h@1D3 h@2D h@5D2+
364 h@1D h@2D2 h@5D2- 182 h@1D2 h@3D h@5D2
+ 154 h@2D h@3D h@5D2- 210 h@1D h@4D h@5D2
-
70 h@5D3- 150 h@1D3 h@6D + h@1D4 h@6D + 360 h@1D h@2D h@6D - 15 h@1D2 h@2D h@6D + 6 h@2D2 h@6D -
270 h@3D h@6D + 24 h@1D h@3D h@6D - 12 h@4D h@6D + 90 h@1D2 h@7D + 2 h@1D3 h@7D - 90 h@2D h@7D +
12 h@1D h@2D h@7D - 21 h@3D h@7D - 90 h@1D h@8D - 9 h@1D2 h@8D + 90 h@9D + 9 h@1D h@9D=
� MToE
MToE@<f>, <opts>DOptions@MToED=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of m functions, MToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
MToE@m@5D - 5 m@4, 1D + m@3, 2D - 2 m@3, 1, 1D + m@2, 2, 1DD% �� EToM
e@1D5- 10 e@1D3 e@2D + 21 e@1D e@2D2
+ 6 e@1D2 e@3D - 26 e@2D e@3D - 6 e@1D e@4D + 20 e@5Dm@5D + m@3, 2D - 5 m@4, 1D + m@2, 2, 1D - 2 m@3, 1, 1D
symfun31-doc.nb 27
MToE@m@5D - 5 m@4, 1D + m@3, 2D - 2 m@3, 1, 1D + m@2, 2, 1D, Vars ® 3D
e@1D5- 10 e@1D3 e@2D + 21 e@1D e@2D2
+ 6 e@1D2 e@3D - 26 e@2D e@3D
� Bigger Examples
MToE@m@15D - 3 m@6, 4, 2, 2, 1DD �� Timing
961.641 Second, e@1D15- 15 e@1D13 e@2D + 90 e@1D11 e@2D2
- 275 e@1D9 e@2D3+ 450 e@1D7 e@2D4
-
378 e@1D5 e@2D5+ 140 e@1D3 e@2D6
- 15 e@1D e@2D7+ 15 e@1D12 e@3D - 165 e@1D10 e@2D e@3D +
675 e@1D8 e@2D2 e@3D - 1260 e@1D6 e@2D3 e@3D + 1050 e@1D4 e@2D4 e@3D - 315 e@1D2 e@2D5 e@3D +
15 e@2D6 e@3D + 75 e@1D9 e@3D2- 540 e@1D7 e@2D e@3D2
+ 1260 e@1D5 e@2D2 e@3D2-
1050 e@1D3 e@2D3 e@3D2+ 225 e@1D e@2D4 e@3D2
+ 140 e@1D6 e@3D3- 525 e@1D4 e@2D e@3D3
+
450 e@1D2 e@2D2 e@3D3- 50 e@2D3 e@3D3
+ 75 e@1D3 e@3D4- 75 e@1D e@2D e@3D4
+ 3 e@3D5-
15 e@1D11 e@4D + 150 e@1D9 e@2D e@4D - 540 e@1D7 e@2D2 e@4D + 840 e@1D5 e@2D3 e@4D -
525 e@1D3 e@2D4 e@4D + 90 e@1D e@2D5 e@4D - 135 e@1D8 e@3D e@4D + 840 e@1D6 e@2D e@3D e@4D -
1575 e@1D4 e@2D2 e@3D e@4D + 900 e@1D2 e@2D3 e@3D e@4D - 75 e@2D4 e@3D e@4D -
315 e@1D5 e@3D2 e@4D + 900 e@1D3 e@2D e@3D2 e@4D - 450 e@1D e@2D2 e@3D2 e@4D -
150 e@1D2 e@3D3 e@4D + 60 e@2D e@3D3 e@4D + 60 e@1D7 e@4D2- 315 e@1D5 e@2D e@4D2
+
450 e@1D3 e@2D2 e@4D2- 150 e@1D e@2D3 e@4D2
+ 225 e@1D4 e@3D e@4D2- 450 e@1D2 e@2D e@3D e@4D2
+
90 e@2D2 e@3D e@4D2+ 90 e@1D e@3D2 e@4D2
- 50 e@1D3 e@4D3+ 60 e@1D e@2D e@4D3
-
15 e@3D e@4D3+ 15 e@1D10 e@5D - 135 e@1D8 e@2D e@5D + 420 e@1D6 e@2D2 e@5D -
525 e@1D4 e@2D3 e@5D + 225 e@1D2 e@2D4 e@5D - 15 e@2D5 e@5D + 120 e@1D7 e@3D e@5D -
630 e@1D5 e@2D e@3D e@5D + 900 e@1D3 e@2D2 e@3D e@5D - 300 e@1D e@2D3 e@3D e@5D +
225 e@1D4 e@3D2 e@5D - 450 e@1D2 e@2D e@3D2 e@5D + 90 e@2D2 e@3D2 e@5D + 60 e@1D e@3D3 e@5D -
105 e@1D6 e@4D e@5D + 450 e@1D4 e@2D e@4D e@5D - 453 e@1D2 e@2D2 e@4D e@5D +
66 e@2D3 e@4D e@5D - 294 e@1D3 e@3D e@4D e@5D + 348 e@1D e@2D e@3D e@4D e@5D -
36 e@3D2 e@4D e@5D + 84 e@1D2 e@4D2 e@5D - 51 e@2D e@4D2 e@5D + 45 e@1D5 e@5D2-
147 e@1D3 e@2D e@5D2+ 84 e@1D e@2D2 e@5D2
+ 81 e@1D2 e@3D e@5D2- 33 e@2D e@3D e@5D2
-
9 e@1D e@4D e@5D2- 25 e@5D3
- 15 e@1D9 e@6D + 120 e@1D7 e@2D e@6D - 315 e@1D5 e@2D2 e@6D +
300 e@1D3 e@2D3 e@6D - 75 e@1D e@2D4 e@6D - 105 e@1D6 e@3D e@6D + 450 e@1D4 e@2D e@3D e@6D -
441 e@1D2 e@2D2 e@3D e@6D + 42 e@2D3 e@3D e@6D - 168 e@1D3 e@3D2 e@6D + 216 e@1D e@2D e@3D2 e@6D -
42 e@3D3 e@6D + 90 e@1D5 e@4D e@6D - 306 e@1D3 e@2D e@4D e@6D + 192 e@1D e@2D2 e@4D e@6D +
216 e@1D2 e@3D e@4D e@6D - 96 e@2D e@3D e@4D e@6D - 81 e@1D e@4D2 e@6D - 66 e@1D4 e@5D e@6D +
144 e@1D2 e@2D e@5D e@6D - 21 e@2D2 e@5D e@6D - 120 e@1D e@3D e@5D e@6D + 48 e@4D e@5D e@6D +
21 e@1D3 e@6D2- 27 e@1D e@2D e@6D2
+ 69 e@3D e@6D2+ 15 e@1D8 e@7D - 105 e@1D6 e@2D e@7D +
225 e@1D4 e@2D2 e@7D - 165 e@1D2 e@2D3 e@7D + 45 e@2D4 e@7D + 90 e@1D5 e@3D e@7D -
264 e@1D3 e@2D e@3D e@7D + 108 e@1D e@2D2 e@3D e@7D + 72 e@1D2 e@3D2 e@7D +
24 e@2D e@3D2 e@7D - 102 e@1D4 e@4D e@7D + 258 e@1D2 e@2D e@4D e@7D - 147 e@2D2 e@4D e@7D -
126 e@1D e@3D e@4D e@7D + 87 e@4D2 e@7D + 78 e@1D3 e@5D e@7D - 36 e@1D e@2D e@5D e@7D -
30 e@3D e@5D e@7D - 63 e@1D2 e@6D e@7D - 24 e@2D e@6D e@7D + 42 e@1D e@7D2- 15 e@1D7 e@8D +
90 e@1D5 e@2D e@8D - 135 e@1D3 e@2D2 e@8D + 30 e@1D e@2D3 e@8D - 72 e@1D4 e@3D e@8D +
102 e@1D2 e@2D e@3D e@8D + 51 e@2D2 e@3D e@8D + 12 e@1D e@3D2 e@8D + 84 e@1D3 e@4D e@8D -
48 e@1D e@2D e@4D e@8D - 90 e@3D e@4D e@8D - 117 e@1D2 e@5D e@8D - 18 e@2D e@5D e@8D +
102 e@1D e@6D e@8D + 21 e@7D e@8D + 15 e@1D6 e@9D - 132 e@1D4 e@2D e@9D + 303 e@1D2 e@2D2 e@9D -
129 e@2D3 e@9D + 114 e@1D3 e@3D e@9D - 378 e@1D e@2D e@3D e@9D + 96 e@3D2 e@9D -
153 e@1D2 e@4D e@9D + 324 e@2D e@4D e@9D + 186 e@1D e@5D e@9D - 285 e@6D e@9D + 66 e@1D5 e@10D -
288 e@1D3 e@2D e@10D + 219 e@1D e@2D2 e@10D + 315 e@1D2 e@3D e@10D - 246 e@2D e@3D e@10D -
276 e@1D e@4D e@10D + 225 e@5D e@10D - 66 e@1D4 e@11D + 153 e@1D2 e@2D e@11D +
75 e@2D2 e@11D - 180 e@1D e@3D e@11D - 3 e@4D e@11D + 165 e@1D3 e@12D - 384 e@1D e@2D e@12D +
309 e@3D e@12D - 165 e@1D2 e@13D + 57 e@2D e@13D + 399 e@1D e@14D - 525 e@15D=
28 symfun31-doc.nb
MToE@m@15D - 3 m@6, 4, 2, 2, 1D, Vars ® 6D �� Timing
926.094 Second, e@1D15- 15 e@1D13 e@2D + 90 e@1D11 e@2D2
- 275 e@1D9 e@2D3+ 450 e@1D7 e@2D4
-
378 e@1D5 e@2D5+ 140 e@1D3 e@2D6
- 15 e@1D e@2D7+ 15 e@1D12 e@3D - 165 e@1D10 e@2D e@3D +
675 e@1D8 e@2D2 e@3D - 1260 e@1D6 e@2D3 e@3D + 1050 e@1D4 e@2D4 e@3D - 315 e@1D2 e@2D5 e@3D +
15 e@2D6 e@3D + 75 e@1D9 e@3D2- 540 e@1D7 e@2D e@3D2
+ 1260 e@1D5 e@2D2 e@3D2-
1050 e@1D3 e@2D3 e@3D2+ 225 e@1D e@2D4 e@3D2
+ 140 e@1D6 e@3D3- 525 e@1D4 e@2D e@3D3
+
450 e@1D2 e@2D2 e@3D3- 50 e@2D3 e@3D3
+ 75 e@1D3 e@3D4- 75 e@1D e@2D e@3D4
+ 3 e@3D5-
15 e@1D11 e@4D + 150 e@1D9 e@2D e@4D - 540 e@1D7 e@2D2 e@4D + 840 e@1D5 e@2D3 e@4D -
525 e@1D3 e@2D4 e@4D + 90 e@1D e@2D5 e@4D - 135 e@1D8 e@3D e@4D + 840 e@1D6 e@2D e@3D e@4D -
1575 e@1D4 e@2D2 e@3D e@4D + 900 e@1D2 e@2D3 e@3D e@4D - 75 e@2D4 e@3D e@4D -
315 e@1D5 e@3D2 e@4D + 900 e@1D3 e@2D e@3D2 e@4D - 450 e@1D e@2D2 e@3D2 e@4D -
150 e@1D2 e@3D3 e@4D + 60 e@2D e@3D3 e@4D + 60 e@1D7 e@4D2- 315 e@1D5 e@2D e@4D2
+
450 e@1D3 e@2D2 e@4D2- 150 e@1D e@2D3 e@4D2
+ 225 e@1D4 e@3D e@4D2- 450 e@1D2 e@2D e@3D e@4D2
+
90 e@2D2 e@3D e@4D2+ 90 e@1D e@3D2 e@4D2
- 50 e@1D3 e@4D3+ 60 e@1D e@2D e@4D3
-
15 e@3D e@4D3+ 15 e@1D10 e@5D - 135 e@1D8 e@2D e@5D + 420 e@1D6 e@2D2 e@5D -
525 e@1D4 e@2D3 e@5D + 225 e@1D2 e@2D4 e@5D - 15 e@2D5 e@5D + 120 e@1D7 e@3D e@5D -
630 e@1D5 e@2D e@3D e@5D + 900 e@1D3 e@2D2 e@3D e@5D - 300 e@1D e@2D3 e@3D e@5D +
225 e@1D4 e@3D2 e@5D - 450 e@1D2 e@2D e@3D2 e@5D + 90 e@2D2 e@3D2 e@5D + 60 e@1D e@3D3 e@5D -
105 e@1D6 e@4D e@5D + 450 e@1D4 e@2D e@4D e@5D - 453 e@1D2 e@2D2 e@4D e@5D +
66 e@2D3 e@4D e@5D - 294 e@1D3 e@3D e@4D e@5D + 348 e@1D e@2D e@3D e@4D e@5D -
36 e@3D2 e@4D e@5D + 84 e@1D2 e@4D2 e@5D - 51 e@2D e@4D2 e@5D + 45 e@1D5 e@5D2-
147 e@1D3 e@2D e@5D2+ 84 e@1D e@2D2 e@5D2
+ 81 e@1D2 e@3D e@5D2- 33 e@2D e@3D e@5D2
-
9 e@1D e@4D e@5D2- 25 e@5D3
- 15 e@1D9 e@6D + 120 e@1D7 e@2D e@6D - 315 e@1D5 e@2D2 e@6D +
300 e@1D3 e@2D3 e@6D - 75 e@1D e@2D4 e@6D - 105 e@1D6 e@3D e@6D + 450 e@1D4 e@2D e@3D e@6D -
441 e@1D2 e@2D2 e@3D e@6D + 42 e@2D3 e@3D e@6D - 168 e@1D3 e@3D2 e@6D +
216 e@1D e@2D e@3D2 e@6D - 42 e@3D3 e@6D + 90 e@1D5 e@4D e@6D - 306 e@1D3 e@2D e@4D e@6D +
192 e@1D e@2D2 e@4D e@6D + 216 e@1D2 e@3D e@4D e@6D - 96 e@2D e@3D e@4D e@6D -
81 e@1D e@4D2 e@6D - 66 e@1D4 e@5D e@6D + 144 e@1D2 e@2D e@5D e@6D - 21 e@2D2 e@5D e@6D -
120 e@1D e@3D e@5D e@6D + 48 e@4D e@5D e@6D + 21 e@1D3 e@6D2- 27 e@1D e@2D e@6D2
+ 69 e@3D e@6D2=
� MToP
MToP@<f>, <opts>DOptions@MToPD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of m functions, MToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
MToP@m@4, 2, 1D - 3 m@5, 2D + 2 m@3, 2, 2DD% �� PToM
p@2D2 p@3D + p@1D p@2D p@4D - 2 p@3D p@4D - 6 p@2D p@5D - p@1D p@6D + 7 p@7D-3 m@5, 2D + 2 m@3, 2, 2D + m@4, 2, 1D
MToP@m@4, 2, 1D - 3 m@5, 2D + 2 m@3, 2, 2D, Vars ® 3D
p@1D7
9-
7
12p@1D5 p@2D +
2
3p@1D3 p@2D2
+
1
4p@1D p@2D3
+
25
36p@1D4 p@3D -
8
3p@1D2 p@2D p@3D -
11
12p@2D2 p@3D +
22
9p@1D p@3D2
symfun31-doc.nb 29
� MToS
MToS@<f>, <opts>DOptions@MToSD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of m functions, MToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
MToS@m@5, 2D - 6 m@3, 1, 1D + 5 m@4, 2, 1, 1D - 9 m@7DD% �� SToM
-9 s@7D - s@4, 3D + s@5, 2D + 9 s@6, 1D + 6 s@2, 2, 1D - 6 s@3, 1, 1D - s@3, 2, 2D +
s@3, 3, 1D - 10 s@5, 1, 1D + 6 s@2, 1, 1, 1D + s@2, 2, 2, 1D + 10 s@2, 2, 2, 2D -
5 s@3, 2, 2, 1D - 5 s@3, 3, 1, 1D + 10 s@4, 1, 1, 1D + 5 s@4, 2, 1, 1D - 18 s@1, 1, 1, 1, 1D -
s@2, 2, 1, 1, 1D - 5 s@2, 2, 2, 1, 1D - 10 s@3, 1, 1, 1, 1D + 10 s@3, 2, 1, 1, 1D -
15 s@4, 1, 1, 1, 1D + 11 s@2, 1, 1, 1, 1, 1D - 5 s@2, 2, 1, 1, 1, 1D + 15 s@3, 1, 1, 1, 1, 1D -
11 s@1, 1, 1, 1, 1, 1, 1D - 30 s@2, 1, 1, 1, 1, 1, 1D + 60 s@1, 1, 1, 1, 1, 1, 1, 1D-9 m@7D + m@5, 2D - 6 m@3, 1, 1D + 5 m@4, 2, 1, 1D
MToS@m@5, 2D - 6 m@3, 1, 1D + 5 m@4, 2, 1, 1D - 9 m@7D, Vars ® 3D
-9 s@7D - s@4, 3D + s@5, 2D + 9 s@6, 1D +
6 s@2, 2, 1D - 6 s@3, 1, 1D - s@3, 2, 2D + s@3, 3, 1D - 10 s@5, 1, 1D
� MToX
MToX@<f>, <n>D
If f is a symmetric function expressed as a linear combination of m functions, MToX returns the same functionexpressed in the variables x[1], ..., x[n].
MToX@m@5, 2D + m@4D, 3D% �� XToM
x@1D4+ x@1D5 x@2D2
+ x@2D4+ x@1D2 x@2D5
+
x@1D5 x@3D2+ x@2D5 x@3D2
+ x@3D4+ x@1D2 x@3D5
+ x@2D2 x@3D5
m@4D + m@5, 2D
� MToAll
MToAll@<f>, <opts>DOptions@MToAllD=8Targets®8E,P,S,H, X<, Vars®0<
If f is a symmetric function expressed as a linear combination of m functions, MToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.
MToAll@m@3, 2, 1DD
9m@3, 2, 1D, e@1D e@2D e@3D - 3 e@3D2- 3 e@1D2 e@4D + 4 e@2D e@4D + 7 e@1D e@5D - 12 e@6D,
p@1D p@2D p@3D - p@3D2- p@2D p@4D - p@1D p@5D + 2 p@6D,
-2 s@2, 2, 2D + s@3, 2, 1D - 2 s@3, 1, 1, 1D + 4 s@2, 1, 1, 1, 1D - 6 s@1, 1, 1, 1, 1, 1D,
-6 h@1D6+ 34 h@1D4 h@2D - 48 h@1D2 h@2D2
+ 8 h@2D3- 30 h@1D3 h@3D +
61 h@1D h@2D h@3D - 15 h@3D2+ 21 h@1D2 h@4D - 20 h@2D h@4D - 17 h@1D h@5D + 12 h@6D=
30 symfun31-doc.nb
MToAll@m@4, 1, 1D - 2 m@3, 2, 1D, Vars ® 3, Targets ® 8E, S, X<D
9-2 m@3, 2, 1D + m@4, 1, 1D,
e@1D3 e@3D - 5 e@1D e@2D e@3D + 9 e@3D2, 5 s@2, 2, 2D - 3 s@3, 2, 1D + s@4, 1, 1D,
x@1D4 x@2D x@3D - 2 x@1D3 x@2D2 x@3D - 2 x@1D2 x@2D3 x@3D + x@1D x@2D4 x@3D - 2 x@1D3 x@2D x@3D2-
2 x@1D x@2D3 x@3D2- 2 x@1D2 x@2D x@3D3
- 2 x@1D x@2D2 x@3D3+ x@1D x@2D x@3D4=
� ETo[HMPSXAll]
� EToH
EToH@<f>, <opts>DOptions@EToHD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of e functions, EToH returns the same functionexpressed as a linear combination of the appropriate h functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
EToH@e@3D e@2D e@1D + e@4D e@2DD% �� HToE
2 h@1D6- 7 h@1D4 h@2D + 6 h@1D2 h@2D2
- h@2D3+
3 h@1D3 h@3D - 3 h@1D h@2D h@3D - h@1D2 h@4D + h@2D h@4De@1D e@2D e@3D + e@2D e@4D
EToH@e@3D e@2D e@1D + e@4D e@2D, Vars ® 4D
2 h@1D6- 7 h@1D4 h@2D + 6 h@1D2 h@2D2
- h@2D3+
3 h@1D3 h@3D - 3 h@1D h@2D h@3D - h@1D2 h@4D + h@2D h@4DEToH@e@3D e@2D e@1D + e@4D e@2D, Vars ® 3D
h@1D6- 3 h@1D4 h@2D + 2 h@1D2 h@2D2
+ h@1D3 h@3D - h@1D h@2D h@3D
Product@1 + e@iD, 8i, 1, 3<D �� Expand
EToH@%D% �� HToE
1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D
1 + h@1D + h@1D2+ 2 h@1D3
+ h@1D4+ h@1D5
+ h@1D6- h@2D - 3 h@1D h@2D -
2 h@1D2 h@2D - 3 h@1D3 h@2D - 3 h@1D4 h@2D + 2 h@1D h@2D2+ 2 h@1D2 h@2D2
+
h@3D + h@1D h@3D + h@1D2 h@3D + h@1D3 h@3D - h@2D h@3D - h@1D h@2D h@3D1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D
� EToM
EToM@<f>, <opts>DOptions@EToMD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of e functions, EToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
symfun31-doc.nb 31
Product@1 + e@iD, 8i, 3<D �� Expand
EToM@%D% �� MToE
1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D
1 + m@1D + m@1, 1D + m@2, 1D + 4 m@1, 1, 1D + m@2, 1, 1D + m@2, 2, 1D +
3 m@2, 2, 2D + m@3, 2, 1D + 4 m@1, 1, 1, 1D + 3 m@2, 1, 1, 1D + 8 m@2, 2, 1, 1D +
3 m@3, 1, 1, 1D + 10 m@1, 1, 1, 1, 1D + 22 m@2, 1, 1, 1, 1D + 60 m@1, 1, 1, 1, 1, 1D1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D
Product@1 + e@iD, 8i, 3<D �� Expand
EToM@%, Vars ® 3D1 + e@1D + e@2D + e@1D e@2D + e@3D + e@1D e@3D + e@2D e@3D + e@1D e@2D e@3D
1 + m@1D + m@1, 1D + m@2, 1D + 4 m@1, 1, 1D + m@2, 1, 1D + m@2, 2, 1D + 3 m@2, 2, 2D + m@3, 2, 1D
� EToP
EToP@<f>, <opts>DOptions@EToPD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of e functions, EToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
EToP@e@3D e@2D e@1D - 2 e@2D^2 e@1D^2D% �� PToE
-
5
12p@1D6
+
2
3p@1D4 p@2D -
1
4p@1D2 p@2D2
+
1
6p@1D3 p@3D -
1
6p@1D p@2D p@3D
-2 e@1D2 e@2D2+ e@1D e@2D e@3D
EToP@e@3D e@2D e@1D - 2 e@2D^2 e@1D^2, Vars ® 3D
-
5
12p@1D6
+
2
3p@1D4 p@2D -
1
4p@1D2 p@2D2
+
1
6p@1D3 p@3D -
1
6p@1D p@2D p@3D
EToP@e@3D e@2D e@1D - 2 e@2D^2 e@1D^2, Vars ® 2D
-
1
2p@1D6
+ p@1D4 p@2D -
1
2p@1D2 p@2D2
� EToS
EToS@<f>, <opts>DOptions@EToSD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of e functions, EToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
32 symfun31-doc.nb
Sum@e@iD e@jD, 8i, 1, 3<, 8j, 1, 3<DEToS@%D% �� SToE
e@1D2+ 2 e@1D e@2D + e@2D2
+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2
s@2D + s@1, 1D + 2 s@2, 1D + s@2, 2D + 2 s@1, 1, 1D +
3 s@2, 1, 1D + 2 s@2, 2, 1D + s@2, 2, 2D + 3 s@1, 1, 1, 1D + 2 s@2, 1, 1, 1D +
s@2, 2, 1, 1D + 2 s@1, 1, 1, 1, 1D + s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D
e@1D2+ 2 e@1D e@2D + e@2D2
+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2
Sum@e@iD e@jD, 8i, 1, 3<, 8j, 1, 3<DEToS@%, Vars ® 3D
e@1D2+ 2 e@1D e@2D + e@2D2
+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2
s@2D + s@1, 1D + 2 s@2, 1D + s@2, 2D + 2 s@1, 1, 1D + 3 s@2, 1, 1D + 2 s@2, 2, 1D + s@2, 2, 2D
� EToX
EToX@<f>, <n>D
If f is a symmetric function expressed as a linear combination of e functions, EToX returns the same functionexpressed in the variables x[1], ..., x[n].
EToX@e@1D^2 e@2D, 4D% �� XToE
x@1D3 x@2D + 2 x@1D2 x@2D2+ x@1D x@2D3
+ x@1D3 x@3D + 5 x@1D2 x@2D x@3D +
5 x@1D x@2D2 x@3D + x@2D3 x@3D + 2 x@1D2 x@3D2+ 5 x@1D x@2D x@3D2
+ 2 x@2D2 x@3D2+
x@1D x@3D3+ x@2D x@3D3
+ x@1D3 x@4D + 5 x@1D2 x@2D x@4D + 5 x@1D x@2D2 x@4D + x@2D3 x@4D +
5 x@1D2 x@3D x@4D + 12 x@1D x@2D x@3D x@4D + 5 x@2D2 x@3D x@4D + 5 x@1D x@3D2 x@4D +
5 x@2D x@3D2 x@4D + x@3D3 x@4D + 2 x@1D2 x@4D2+ 5 x@1D x@2D x@4D2
+ 2 x@2D2 x@4D2+
5 x@1D x@3D x@4D2+ 5 x@2D x@3D x@4D2
+ 2 x@3D2 x@4D2+ x@1D x@4D3
+ x@2D x@4D3+ x@3D x@4D3
e@1D2 e@2D
� EToAll
EToAll@<f>, <opts>DOptions@EToAllD=8Targets®8H,P,S,M, X<, Vars®0<
If f is a symmetric function expressed as a linear combination of e functions, EToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.
symfun31-doc.nb 33
Sum@e@iD e@jD, 8i, 3<, 8j, 3<DEToAll@%D
e@1D2+ 2 e@1D e@2D + e@2D2
+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2
:e@1D2+ 2 e@1D e@2D + e@2D2
+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2,
h@1D2+ 2 h@1D3
+ 3 h@1D4+ 2 h@1D5
+ h@1D6- 2 h@1D h@2D - 6 h@1D2 h@2D -
6 h@1D3 h@2D - 4 h@1D4 h@2D + h@2D2+ 4 h@1D h@2D2
+ 4 h@1D2 h@2D2+ 2 h@1D h@3D +
2 h@1D2 h@3D + 2 h@1D3 h@3D - 2 h@2D h@3D - 4 h@1D h@2D h@3D + h@3D2,
p@1D2+ p@1D3
+
7 p@1D4
12+
p@1D5
6+
p@1D6
36- p@1D p@2D -
3
2p@1D2 p@2D -
2
3p@1D3 p@2D -
1
6p@1D4 p@2D +
p@2D2
4+
1
2p@1D p@2D2
+
1
4p@1D2 p@2D2
+
2
3p@1D p@3D +
1
3p@1D2 p@3D +
1
9p@1D3 p@3D -
1
3p@2D p@3D -
1
3p@1D p@2D p@3D +
p@3D2
9,
s@2D + s@1, 1D + 2 s@2, 1D + s@2, 2D + 2 s@1, 1, 1D + 3 s@2, 1, 1D + 2 s@2, 2, 1D +
s@2, 2, 2D + 3 s@1, 1, 1, 1D + 2 s@2, 1, 1, 1D + s@2, 2, 1, 1D +
2 s@1, 1, 1, 1, 1D + s@2, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1D,
m@2D + 2 m@1, 1D + 2 m@2, 1D + m@2, 2D + 6 m@1, 1, 1D + 4 m@2, 1, 1D + 2 m@2, 2, 1D +
m@2, 2, 2D + 14 m@1, 1, 1, 1D + 6 m@2, 1, 1, 1D + 2 m@2, 2, 1, 1D +
20 m@1, 1, 1, 1, 1D + 6 m@2, 1, 1, 1, 1D + 20 m@1, 1, 1, 1, 1, 1D>
Sum@e@iD e@jD, 8i, 3<, 8j, 3<DEToAll@%, Vars ® 2, Targets ® 8P, M, X<D
e@1D2+ 2 e@1D e@2D + e@2D2
+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2
:e@1D2+ 2 e@1D e@2D + e@2D2
+ 2 e@1D e@3D + 2 e@2D e@3D + e@3D2,
p@1D2+ p@1D3
+
p@1D4
4- p@1D p@2D -
1
2p@1D2 p@2D +
p@2D2
4, m@2D + 2 m@1, 1D + 2 m@2, 1D + m@2, 2D,
x@1D2+ 2 x@1D x@2D + 2 x@1D2 x@2D + x@2D2
+ 2 x@1D x@2D2+ x@1D2 x@2D2>
� HTo[EMPSXAll]
� HRestrictionRules
HRestrictionRules@<deg>, <var>D
HRestrictionRules[deg, var] creates a list of rules that express h functions of degree deg in terms of other hfuntions of degree deg when the number of variables is restricted to var. The basis used for this is h[lambda]functions where lambda[[1]]£var (note that this is NOT the same basis generated by HList[deg, var]).
HRestrictionRules@4, 2DHList@4, 2D
9h@4D ® -h@1D4+ h@1D2 h@2D + h@2D2, h@1D h@3D ® -h@1D4
+ 2 h@1D2 h@2D,
h@2D2® h@2D2, h@1D2 h@2D ® h@1D2 h@2D, h@1D4
® h@1D4=
9h@4D, h@1D h@3D, h@2D2=
34 symfun31-doc.nb
HRestrictionRules@6, 1DHList@6, 1D
9h@6D ® h@1D6, h@1D h@5D ® h@1D6, h@2D h@4D ® h@1D6,
h@1D2 h@4D ® h@1D6, h@3D2® h@1D6, h@1D h@2D h@3D ® h@1D6, h@1D3 h@3D ® h@1D6,
h@2D3® h@1D6, h@1D2 h@2D2
® h@1D6, h@1D4 h@2D ® h@1D6, h@1D6® h@1D6=
8h@6D<
� HToE
HToE@<f>, <opts>DOptions@HToED=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of h functions, HToE returns the same functionexpressed as a linear combination of the appropriate e functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
1 + Sum@h@iD, 8i, 1, 6<DHToE@%D% �� EToH
1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D
1 + e@1D + e@1D2+ e@1D3
+ e@1D4+ e@1D5
+ e@1D6- e@2D - 2 e@1D e@2D - 3 e@1D2 e@2D -
4 e@1D3 e@2D - 5 e@1D4 e@2D + e@2D2+ 3 e@1D e@2D2
+ 6 e@1D2 e@2D2- e@2D3
+ e@3D +
2 e@1D e@3D + 3 e@1D2 e@3D + 4 e@1D3 e@3D - 2 e@2D e@3D - 6 e@1D e@2D e@3D +
e@3D2- e@4D - 2 e@1D e@4D - 3 e@1D2 e@4D + 2 e@2D e@4D + e@5D + 2 e@1D e@5D - e@6D
1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D
1 + Sum@h@iD, 8i, 1, 6<DHToE@%, Vars ® 3D1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D
1 + e@1D + e@1D2+ e@1D3
+ e@1D4+ e@1D5
+ e@1D6- e@2D - 2 e@1D e@2D - 3 e@1D2 e@2D -
4 e@1D3 e@2D - 5 e@1D4 e@2D + e@2D2+ 3 e@1D e@2D2
+ 6 e@1D2 e@2D2- e@2D3
+ e@3D +
2 e@1D e@3D + 3 e@1D2 e@3D + 4 e@1D3 e@3D - 2 e@2D e@3D - 6 e@1D e@2D e@3D + e@3D2
� HToM
HToM@<f>, <opts>DOptions@HToMD=8DegreeList®8<,Vars®0<
If f is a symmetric function expressed as a linear combination of h functions, HToM returns the same functionexpressed as a linear combination of the appropriate m functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
symfun31-doc.nb 35
1 + Sum@h@iD, 8i, 1, 6<DHToM@%D% �� MToH
1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D
1 + m@1D + m@2D + m@3D + m@4D + m@5D + m@6D + m@1, 1D + m@2, 1D + m@2, 2D + m@3, 1D +
m@3, 2D + m@3, 3D + m@4, 1D + m@4, 2D + m@5, 1D + m@1, 1, 1D + m@2, 1, 1D + m@2, 2, 1D +
m@2, 2, 2D + m@3, 1, 1D + m@3, 2, 1D + m@4, 1, 1D + m@1, 1, 1, 1D + m@2, 1, 1, 1D +
m@2, 2, 1, 1D + m@3, 1, 1, 1D + m@1, 1, 1, 1, 1D + m@2, 1, 1, 1, 1D + m@1, 1, 1, 1, 1, 1D1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D
1 + Sum@h@iD, 8i, 1, 6<DHToM@%, Vars ® 3D1 + h@1D + h@2D + h@3D + h@4D + h@5D + h@6D
1 + m@1D + m@2D + m@3D + m@4D + m@5D + m@6D + m@1, 1D + m@2, 1D + m@2, 2D +
m@3, 1D + m@3, 2D + m@3, 3D + m@4, 1D + m@4, 2D + m@5, 1D + m@1, 1, 1D +
m@2, 1, 1D + m@2, 2, 1D + m@2, 2, 2D + m@3, 1, 1D + m@3, 2, 1D + m@4, 1, 1D
� HToP
HToP@<f>, <opts>DOptions@HToPD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of h functions, HToP returns the same functionexpressed as a linear combination of the appropriate p functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
HToP@h@3D h@2D h@1D - 2 h@2D^2 h@1D^2D% �� PToH
-
5
12p@1D6
-
2
3p@1D4 p@2D -
1
4p@1D2 p@2D2
+
1
6p@1D3 p@3D +
1
6p@1D p@2D p@3D
-2 h@1D2 h@2D2+ h@1D h@2D h@3D
HToP@h@3D h@2D h@1D - 2 h@2D^2 h@1D^2, Vars ® 3D
-
5
12p@1D6
-
2
3p@1D4 p@2D -
1
4p@1D2 p@2D2
+
1
6p@1D3 p@3D +
1
6p@1D p@2D p@3D
HToP@h@3D h@2D h@1D - 2 h@2D^2 h@1D^2, Vars ® 2D
-
1
2p@1D6
-
1
2p@1D4 p@2D
� HToS
HToS@<f>, <opts>DOptions@HToSD=8DegreeList®8<, Vars®0<
If f is a symmetric function expressed as a linear combination of h functions, HToS returns the same functionexpressed as a linear combination of the appropriate s functions. Setting the option DegreeList to a list of thedegrees of f saves Mathematica some computation, possibly resulting in faster output. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer.
36 symfun31-doc.nb
1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToS@%D% �� SToH
1 + h@1D2+ 2 h@1D h@2D + h@2D2
+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2
1 + s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D + s@1, 1D + 2 s@2, 1D +
s@2, 2D + 3 s@3, 1D + 2 s@3, 2D + s@3, 3D + 2 s@4, 1D + s@4, 2D + s@5, 1D
1 + h@1D2+ 2 h@1D h@2D + h@2D2
+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2
1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToS@%, Vars ® 1D
1 + h@1D2+ 2 h@1D h@2D + h@2D2
+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2
1 + s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D
� HToX
HToX@<f>, <n>D
If f is a symmetric function expressed as a linear combination of h functions, HToX returns the same functionexpressed in the variables x[1], ..., x[n].
HToX@h@1D h@2D, 3D
x@1D3+ 2 x@1D2 x@2D + 2 x@1D x@2D2
+ x@2D3+ 2 x@1D2 x@3D +
3 x@1D x@2D x@3D + 2 x@2D2 x@3D + 2 x@1D x@3D2+ 2 x@2D x@3D2
+ x@3D3
� HToAll
HToAll@<f>, <opts>DOptions@HToAllD=8Targets®8E,P,S,M, X<, Vars®0<
If f is a symmetric function expressed as a linear combination of h functions, HToAll returns the same functionexpressed as linear combinations of the appropriate functions as specified in Targets. Setting the option Vars tosome positive integer n restricts the number of variables in the problem to n, resulting in the possible elimation ofsome terms in the answer. Note that the X's will only be displayed if Vars is specified.
symfun31-doc.nb 37
1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToAll@%D
1 + h@1D2+ 2 h@1D h@2D + h@2D2
+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2
:1 + h@1D2+ 2 h@1D h@2D + h@2D2
+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2,
1 + e@1D2+ 2 e@1D3
+ 3 e@1D4+ 2 e@1D5
+ e@1D6- 2 e@1D e@2D - 6 e@1D2 e@2D -
6 e@1D3 e@2D - 4 e@1D4 e@2D + e@2D2+ 4 e@1D e@2D2
+ 4 e@1D2 e@2D2+ 2 e@1D e@3D +
2 e@1D2 e@3D + 2 e@1D3 e@3D - 2 e@2D e@3D - 4 e@1D e@2D e@3D + e@3D2,
1 + p@1D2+ p@1D3
+
7 p@1D4
12+
p@1D5
6+
p@1D6
36+ p@1D p@2D +
3
2p@1D2 p@2D +
2
3p@1D3 p@2D +
1
6p@1D4 p@2D +
p@2D2
4+
1
2p@1D p@2D2
+
1
4p@1D2 p@2D2
+
2
3p@1D p@3D +
1
3p@1D2 p@3D +
1
9p@1D3 p@3D +
1
3p@2D p@3D +
1
3p@1D p@2D p@3D +
p@3D2
9,
1 + s@2D + 2 s@3D + 3 s@4D + 2 s@5D + s@6D + s@1, 1D + 2 s@2, 1D + s@2, 2D +
3 s@3, 1D + 2 s@3, 2D + s@3, 3D + 2 s@4, 1D + s@4, 2D + s@5, 1D,
1 + m@2D + 2 m@3D + 3 m@4D + 2 m@5D + m@6D + 2 m@1, 1D + 4 m@2, 1D + 7 m@2, 2D +
6 m@3, 1D + 6 m@3, 2D + 4 m@3, 3D + 4 m@4, 1D + 3 m@4, 2D + 2 m@5, 1D + 6 m@1, 1, 1D +
10 m@2, 1, 1D + 10 m@2, 2, 1D + 7 m@2, 2, 2D + 8 m@3, 1, 1D + 6 m@3, 2, 1D +
4 m@4, 1, 1D + 14 m@1, 1, 1, 1D + 14 m@2, 1, 1, 1D + 10 m@2, 2, 1, 1D +
8 m@3, 1, 1, 1D + 20 m@1, 1, 1, 1, 1D + 14 m@2, 1, 1, 1, 1D + 20 m@1, 1, 1, 1, 1, 1D>
1 + Sum@h@iD h@jD, 8i, 3<, 8j, 3<DHToAll@%, Vars ® 2, Targets ® 8E, M, X<D
1 + h@1D2+ 2 h@1D h@2D + h@2D2
+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2
91 + h@1D2+ 2 h@1D h@2D + h@2D2
+ 2 h@1D h@3D + 2 h@2D h@3D + h@3D2,
1 + e@1D2+ 2 e@1D3
+ 3 e@1D4+ 2 e@1D5
+ e@1D6- 2 e@1D e@2D - 6 e@1D2 e@2D -
6 e@1D3 e@2D - 4 e@1D4 e@2D + e@2D2+ 4 e@1D e@2D2
+ 4 e@1D2 e@2D2,
1 + m@2D + 2 m@3D + 3 m@4D + 2 m@5D + m@6D + 2 m@1, 1D + 4 m@2, 1D + 7 m@2, 2D +
6 m@3, 1D + 6 m@3, 2D + 4 m@3, 3D + 4 m@4, 1D + 3 m@4, 2D + 2 m@5, 1D,
1 + x@1D2+ 2 x@1D3
+ 3 x@1D4+ 2 x@1D5
+ x@1D6+ 2 x@1D x@2D + 4 x@1D2 x@2D +
6 x@1D3 x@2D + 4 x@1D4 x@2D + 2 x@1D5 x@2D + x@2D2+ 4 x@1D x@2D2
+ 7 x@1D2 x@2D2+
6 x@1D3 x@2D2+ 3 x@1D4 x@2D2
+ 2 x@2D3+ 6 x@1D x@2D3
+ 6 x@1D2 x@2D3+ 4 x@1D3 x@2D3
+
3 x@2D4+ 4 x@1D x@2D4
+ 3 x@1D2 x@2D4+ 2 x@2D5
+ 2 x@1D x@2D5+ x@2D6=
Miscellaneous� LR
LRProduct@<Μ>,<Ν>D or s@<Μ>Ds@<Ν>D
Computes the expansion of s[Μ]s[Ν], expressed as a linear combination of s[Λ]'s.
s@<Μ>D<n>
Computes the expression s@ΜDn
s@<Μ>Ds@<Ν>Ds@<Ψ>D... ��Expand
Computes the product s[Μ] s[Ν] s[j]... Products of more than two terms need to be followed by "//Expand" in orderto expand completely. There is no need to type //Expand for s@ < Μ >Dn
38 symfun31-doc.nb
Computes the product s[Μ] s[Ν] s[j]... Products of more than two terms need to be followed by "//Expand" in orderto expand completely. There is no need to type //Expand for s@ < Μ >Dn
LRProduct@82, 1<, 82, 1<D
s@3, 3D + s@4, 2D + s@2, 2, 2D + 2 s@3, 2, 1D + s@4, 1, 1D + s@2, 2, 1, 1D + s@3, 1, 1, 1D
s@2, 1D s@2, 1D
s@3, 3D + s@4, 2D + s@2, 2, 2D + 2 s@3, 2, 1D + s@4, 1, 1D + s@2, 2, 1, 1D + s@3, 1, 1, 1D
s@2, 1D s@2, 1D s@2, 1Ds@2, 1D s@2, 1D s@2, 1D �� Expand
s@2, 1D Hs@3, 3D + s@4, 2D + s@2, 2, 2D + 2 s@3, 2, 1D + s@4, 1, 1D + s@2, 2, 1, 1D + s@3, 1, 1, 1DL
2 s@5, 4D + s@6, 3D + 2 s@3, 3, 3D + 8 s@4, 3, 2D + 4 s@4, 4, 1D +
4 s@5, 2, 2D + 6 s@5, 3, 1D + 2 s@6, 2, 1D + 4 s@3, 2, 2, 2D + 8 s@3, 3, 2, 1D +
9 s@4, 2, 2, 1D + 9 s@4, 3, 1, 1D + 6 s@5, 2, 1, 1D + s@6, 1, 1, 1D +
2 s@2, 2, 2, 2, 1D + 6 s@3, 2, 2, 1, 1D + 4 s@3, 3, 1, 1, 1D + 6 s@4, 2, 1, 1, 1D +
2 s@5, 1, 1, 1, 1D + s@2, 2, 2, 1, 1, 1D + 2 s@3, 2, 1, 1, 1, 1D + s@4, 1, 1, 1, 1, 1Ds@1D s@1D s@1D s@1D s@1D s@1D s@1D s@1D s@1D �� Expand �� Timing
831.45, s@9D + 42 s@5, 4D + 48 s@6, 3D + 27 s@7, 2D + 8 s@8, 1D + 42 s@3, 3, 3D + 168 s@4, 3, 2D +
84 s@4, 4, 1D + 120 s@5, 2, 2D + 162 s@5, 3, 1D + 105 s@6, 2, 1D + 28 s@7, 1, 1D +
84 s@3, 2, 2, 2D + 168 s@3, 3, 2, 1D + 216 s@4, 2, 2, 1D + 216 s@4, 3, 1, 1D +
189 s@5, 2, 1, 1D + 56 s@6, 1, 1, 1D + 42 s@2, 2, 2, 2, 1D + 162 s@3, 2, 2, 1, 1D +
120 s@3, 3, 1, 1, 1D + 189 s@4, 2, 1, 1, 1D + 70 s@5, 1, 1, 1, 1D + 48 s@2, 2, 2, 1, 1, 1D +
105 s@3, 2, 1, 1, 1, 1D + 56 s@4, 1, 1, 1, 1, 1D + 27 s@2, 2, 1, 1, 1, 1, 1D +
28 s@3, 1, 1, 1, 1, 1, 1D + 8 s@2, 1, 1, 1, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1, 1, 1, 1D<
s@1D9 �� Timing
831.372, s@9D + 42 s@5, 4D + 48 s@6, 3D + 27 s@7, 2D + 8 s@8, 1D + 42 s@3, 3, 3D + 168 s@4, 3, 2D +
84 s@4, 4, 1D + 120 s@5, 2, 2D + 162 s@5, 3, 1D + 105 s@6, 2, 1D + 28 s@7, 1, 1D +
84 s@3, 2, 2, 2D + 168 s@3, 3, 2, 1D + 216 s@4, 2, 2, 1D + 216 s@4, 3, 1, 1D +
189 s@5, 2, 1, 1D + 56 s@6, 1, 1, 1D + 42 s@2, 2, 2, 2, 1D + 162 s@3, 2, 2, 1, 1D +
120 s@3, 3, 1, 1, 1D + 189 s@4, 2, 1, 1, 1D + 70 s@5, 1, 1, 1, 1D + 48 s@2, 2, 2, 1, 1, 1D +
105 s@3, 2, 1, 1, 1, 1D + 56 s@4, 1, 1, 1, 1, 1D + 27 s@2, 2, 1, 1, 1, 1, 1D +
28 s@3, 1, 1, 1, 1, 1, 1D + 8 s@2, 1, 1, 1, 1, 1, 1, 1D + s@1, 1, 1, 1, 1, 1, 1, 1, 1D<
Finally, a "check" that the calculation is correct:
s@2, 2D s@3, 1D s@1D �� Expand
aaa = SToX@%, 7D;
bbb = HSToX@s@2, 2D, 7DL HSToX@s@3, 1D, 7DL HSToX@s@1D, 7DL �� Expand;
aaa � bbb
s@5, 4D + s@6, 3D + s@3, 3, 3D + 3 s@4, 3, 2D + s@4, 4, 1D + 2 s@5, 2, 2D +
3 s@5, 3, 1D + s@6, 2, 1D + s@3, 2, 2, 2D + 2 s@3, 3, 2, 1D + 3 s@4, 2, 2, 1D +
2 s@4, 3, 1, 1D + 2 s@5, 2, 1, 1D + s@3, 2, 2, 1, 1D + s@4, 2, 1, 1, 1DTrue
symfun31-doc.nb 39
� Bigger examples
LRProduct@84, 3, 2, 1<, 83, 2, 1<D
s@4, 4, 4, 4D + 3 s@5, 4, 4, 3D + 2 s@5, 5, 3, 3D + 3 s@5, 5, 4, 2D + s@5, 5, 5, 1D +
3 s@6, 4, 3, 3D + 3 s@6, 4, 4, 2D + 4 s@6, 5, 3, 2D + 2 s@6, 5, 4, 1D + s@6, 6, 2, 2D +
s@6, 6, 3, 1D + s@7, 3, 3, 3D + 2 s@7, 4, 3, 2D + s@7, 4, 4, 1D + s@7, 5, 2, 2D +
s@7, 5, 3, 1D + s@4, 3, 3, 3, 3D + 3 s@4, 4, 3, 3, 2D + 2 s@4, 4, 4, 2, 2D + 3 s@4, 4, 4, 3, 1D +
3 s@5, 3, 3, 3, 2D + 6 s@5, 4, 3, 2, 2D + 6 s@5, 4, 3, 3, 1D + 6 s@5, 4, 4, 2, 1D +
2 s@5, 5, 2, 2, 2D + 6 s@5, 5, 3, 2, 1D + 3 s@5, 5, 4, 1, 1D + 3 s@6, 3, 3, 2, 2D +
3 s@6, 3, 3, 3, 1D + 3 s@6, 4, 2, 2, 2D + 8 s@6, 4, 3, 2, 1D + 3 s@6, 4, 4, 1, 1D +
4 s@6, 5, 2, 2, 1D + 4 s@6, 5, 3, 1, 1D + s@6, 6, 2, 1, 1D + s@7, 3, 2, 2, 2D + 2 s@7, 3, 3, 2, 1D +
2 s@7, 4, 2, 2, 1D + 2 s@7, 4, 3, 1, 1D + s@7, 5, 2, 1, 1D + s@4, 3, 3, 2, 2, 2D +
2 s@4, 3, 3, 3, 2, 1D + s@4, 4, 2, 2, 2, 2D + 4 s@4, 4, 3, 2, 2, 1D + 3 s@4, 4, 3, 3, 1, 1D +
3 s@4, 4, 4, 2, 1, 1D + s@5, 3, 2, 2, 2, 2D + 4 s@5, 3, 3, 2, 2, 1D + 3 s@5, 3, 3, 3, 1, 1D +
4 s@5, 4, 2, 2, 2, 1D + 8 s@5, 4, 3, 2, 1, 1D + 3 s@5, 4, 4, 1, 1, 1D + 3 s@5, 5, 2, 2, 1, 1D +
3 s@5, 5, 3, 1, 1, 1D + 2 s@6, 3, 2, 2, 2, 1D + 4 s@6, 3, 3, 2, 1, 1D + 4 s@6, 4, 2, 2, 1, 1D +
4 s@6, 4, 3, 1, 1, 1D + 2 s@6, 5, 2, 1, 1, 1D + s@7, 3, 2, 2, 1, 1D + s@7, 3, 3, 1, 1, 1D +
s@7, 4, 2, 1, 1, 1D + s@4, 3, 3, 2, 2, 1, 1D + s@4, 3, 3, 3, 1, 1, 1D + s@4, 4, 2, 2, 2, 1, 1D +
2 s@4, 4, 3, 2, 1, 1, 1D + s@4, 4, 4, 1, 1, 1, 1D + s@5, 3, 2, 2, 2, 1, 1D +
2 s@5, 3, 3, 2, 1, 1, 1D + 2 s@5, 4, 2, 2, 1, 1, 1D + 2 s@5, 4, 3, 1, 1, 1, 1D +
s@5, 5, 2, 1, 1, 1, 1D + s@6, 3, 2, 2, 1, 1, 1D + s@6, 3, 3, 1, 1, 1, 1D + s@6, 4, 2, 1, 1, 1, 1D
� Pieri
Pieri@<Λ>,<k>D
Pieri[Λ, k] computes the list of partitions that are "k-Pieri" over lambda (the partitions that come about by adding kboxes to the Ferrer's diagram of Λ with no more than one box added per column). Corresponds to the special caseof the LR rule where one multiplies by s[n].
Pieri@83, 3, 3<, 6D
889, 3, 3<, 86, 3, 3, 3<, 87, 3, 3, 2<, 88, 3, 3, 1<<
LRProduct@83, 3, 3<, 86<D
s@9, 3, 3D + s@6, 3, 3, 3D + s@7, 3, 3, 2D + s@8, 3, 3, 1D
LRProduct@83, 3, 3<, 81<D
s@4, 3, 3D + s@3, 3, 3, 1D
Pieri@82, 1<, 1D
882, 2<, 83, 1<, 82, 1, 1<<
SMult@82, 1<, 1D
s@2, 2D + s@3, 1D + s@2, 1, 1D
40 symfun31-doc.nb
