The category for elements in algebras of symmetric functions is AlgSymElt.
By default, elements are printed using a lexicographical ordering on the partitions indexing the basis elements. For partitions of the same weight w this ordering is the reverse of that in Partitions(w). More generally, basis elements indexed by partitions with smaller entries print first (see Testing Order Relations). It is possible to rearrange the linear combination of basis elements so that the basis elements indexed by the longest partition print first and also so that the basis elements indexed by the partition with greatest maximal part (first entry) print first.
Retrieve or set the style in which elements of the algebra A will print. The default is the lexicographical ordering described above, "Lex". Other options are "Length" and "MaximalPart".
> M := SFAMonomial(Rationals());
> M`PrintStyle;
Lex
> P := Partitions(3);
> P;
[
[ 3 ],
[ 2, 1 ],
[ 1, 1, 1 ]
]
> f := &+[M.p : p in P];
> f;
M.[1,1,1] + M.[2,1] + M.[3]
> M`PrintStyle := "Length";
> f;
M.[3] + M.[2,1] + M.[1,1,1]
> M`PrintStyle := "MaximalPart";
> f;
M.[1,1,1] + M.[2,1] + M.[3]
The usual unary and binary ring element operations are available for symmetric functions. It is possible to combine elements of different algebras (so long as the coefficient rings are compatible) in these operations. Where the elements are written with respect to different bases, the result will be written with respect to the basis of the second operand.
Where the elements being multiplied are expressed with respect to different bases, the result will again be expressed with respect to the basis of the second operand. The algorithm used for multiplication is dependent on the bases with respect to which the elements are expressed.
The product of the symmetric functions a and b.If b is expressed with respect to a Schur basis and a is expressed with respect to a power sum, elementary or monomial basis then the algorithm based on Muirs rule [Mui60] is used. If a is expressed with respect to a homogeneous basis the algorithm based on the Pieri rule [Mac95] is used and if a is also expressed with respect to a Schur basis then the method of Schubert polynomials [LS85] is used.
Special algorithms are also used if b is expressed with respect to a monomial basis and a is expressed with respect to a homogeneous, elementary or monomial basis.
When both a and b have a homogeneous, elementary or power sum basis, the multiplication of basis elements involves the merging of the parts of the partitions. This follows from the definition of the basis elements fλ = ∏i fλi where fλ is a homogeneous, elementary or power sum basis element.
Otherwise a is coerced into the parent of b before the elements are multiplied.
The degree of the result is the sum of the degrees of both the operands.
> Q := Rationals(); > s := SFASchur(Q); > m := SFAMonomial(Q); > m.[3]*s.[2,1]; s.[2,1,1,1,1] + s.[5,1] - s.[2,2,2] - s.[3,3]To illustrate the merging of the partitions :
> E := SFAElementary(Q); > E.4*E.3*E.1; E.[4,3,1]This is the definition of e[4, 3, 1].
Plethysm is also referred to as composition of symmetric functions.
This operator computes the plethysm or composition of the symmetric functions a and b. The result is given with respect to the basis of the second operand.The degree of the result is the product of the degrees of operands which may be very large.
> Q := Rationals(); > s := SFASchur(Q); > m := SFAMonomial(Q); > m.[3]~s.[2,1]; s.[4,1,1,1,1,1] - s.[3,2,1,1,1,1] - s.[5,1,1,1,1] + s.[2,2,2,1,1,1] + s.[3,3,1,1,1] + s.[6,1,1,1] - s.[2,2,2,2,1] - s.[3,3,2,1] - s.[6,2,1] + s.[4,4,1] + s.[3,2,2,2] + s.[5,2,2] - s.[4,3,2] + 2*s.[3,3,3] + s.[6,3] - s.[5,4]
Returns true if the partitions indexing the basis elements present in the symmetric function s are all of the same weight. This implies that each term has the same degree so is the same as the polynomial expansion of s being a homogeneous polynomial.
Return true if the symmetric functions s and t are (not) the same.
Given a symmetric function s return the coefficient of the basis element Ap, where A is the parent of s and p is a sequence defining a partition. The coefficient may be zero.
Return two parallel sequences of the partitions indexing the basis elements and the coefficients of those basis elements in the symmetric function s, which is a linear combination of basis elements.
Given a symmetric function s, return the length of s, i.e., the number of basis elements having non zero coefficients in s, with respect to the current basis.
> H := SFAHomogeneous(Rationals());
> P := Partitions(4);
> f := &+[Random(1, 5)*H.p : p in P];
> f;
H.[1,1,1,1] + 4*H.[2,1,1] + 4*H.[2,2] + 5*H.[3,1] + 5*H.[4]
> s, e := Support(f);
> s, e;
[
[ 1, 1, 1, 1 ],
[ 2, 1, 1 ],
[ 2, 2 ],
[ 3, 1 ],
[ 4 ]
]
[ 1, 4, 4, 5, 5 ]
> f eq &+[e[i]*H.s[i] : i in [1 .. Length(f)]];
true
Given a symmetric function s, return the degree of s, i.e., the maximal degree of the basis elements having non zero coefficients in s, which is the maximal weight of the partitions indexing those basis elements.
A symmetric function may be seen as a polynomial in any number of variables.
Return the polynomial expansion of the symmetric function s in the polynomial ring P.
> S := SFASchur(GF(7)); > s := S.[3,1];Now we compute the multivariate polynomial we get by restricting to 5 variables.
> G<e1, e2, e3, e4, e5> := PolynomialRing(GF(7), 5);
> p := G!s;
> p;
e1^3*e2 + e1^3*e3 + e1^3*e4 + e1^3*e5 + e1^2*e2^2 + 2*e1^2*e2*e3 + 2*e1^2*e2*e4
+ 2*e1^2*e2*e5 + e1^2*e3^2 + 2*e1^2*e3*e4 + 2*e1^2*e3*e5 + e1^2*e4^2 +
2*e1^2*e4*e5 + e1^2*e5^2 + e1*e2^3 + 2*e1*e2^2*e3 + 2*e1*e2^2*e4 +
2*e1*e2^2*e5 + 2*e1*e2*e3^2 + 3*e1*e2*e3*e4 + 3*e1*e2*e3*e5 + 2*e1*e2*e4^2 +
3*e1*e2*e4*e5 + 2*e1*e2*e5^2 + e1*e3^3 + 2*e1*e3^2*e4 + 2*e1*e3^2*e5 +
2*e1*e3*e4^2 + 3*e1*e3*e4*e5 + 2*e1*e3*e5^2 + e1*e4^3 + 2*e1*e4^2*e5 +
2*e1*e4*e5^2 + e1*e5^3 + e2^3*e3 + e2^3*e4 + e2^3*e5 + e2^2*e3^2 +
2*e2^2*e3*e4 + 2*e2^2*e3*e5 + e2^2*e4^2 + 2*e2^2*e4*e5 + e2^2*e5^2 + e2*e3^3
+ 2*e2*e3^2*e4 + 2*e2*e3^2*e5 + 2*e2*e3*e4^2 + 3*e2*e3*e4*e5 + 2*e2*e3*e5^2
+ e2*e4^3 + 2*e2*e4^2*e5 + 2*e2*e4*e5^2 + e2*e5^3 + e3^3*e4 + e3^3*e5 +
e3^2*e4^2 + 2*e3^2*e4*e5 + e3^2*e5^2 + e3*e4^3 + 2*e3*e4^2*e5 + 2*e3*e4*e5^2
+ e3*e5^3 + e4^3*e5 + e4^2*e5^2 + e4*e5^3
To check the polynomial is actually symmetric, we can
use the Magma intrinsic IsSymmetric, which also
computes an expansion as a sum of elementary symmetric polynomials.
> IsSymmetric(p,G); true e1^2*e2 + 6*e1*e3 + 6*e2^2 + e4Which is identical to the result of
> E := SFAElementary(GF(7)); > E!s; E.[2,1,1] + 6*E.[2,2] + 6*E.[3,1] + E.[4]
> S := SFASchur(Rationals()); > R<a, b, c, d, e> := PolynomialRing(Rationals(), 5); > p := Polynomial(S.[3,2], R); > q := Evaluate(p, [a+1, b+1, c+1, d+1, e+1]); > x, y := IsCoercible(S, q); > y; 175 + 175*S.[1] + 70*S.[1,1] + 70*S.[2] + 35*S.[2,1] + 7*S.[2,2] + 10*S.[3] + 5*S.[3,1] + S.[3,2]
There is an automorphism which maps the elementary symmetric function to the homogeneous symmetric function.
Given any symmetric function s, compute the image of s under the Frobenius automorphism. The image will have the same parent as s. When the power sum symmetric functions are involved, it may be necessary to work with a coefficient ring which allows division by an integer.
> S := SFASchur(Integers()); > E := SFAElementary(Integers()); > h := S!E.[3,3,3]; > h; S.[1,1,1,1,1,1,1,1,1] + 2*S.[2,1,1,1,1,1,1,1] + S.[3,1,1,1,1,1,1] + 3*S.[2,2,1,1,1,1,1] + 2*S.[3,2,1,1,1,1] + 4*S.[2,2,2,1,1,1] + S.[3,3,1,1,1] + 3*S.[3,2,2,1,1] + 2*S.[2,2,2,2,1] + 2*S.[3,3,2,1] + S.[3,2,2,2] + S.[3,3,3]Now apply the Frobenius automorphism.
> f:=Frobenius(h); > f; S.[3,3,3] + 2*S.[4,3,2] + S.[4,4,1] + S.[5,2,2] + 3*S.[5,3,1] + 2*S.[5,4] + 2*S.[6,2,1] + 4*S.[6,3] + S.[7,1,1] + 3*S.[7,2] + 2*S.[8,1] + S.[9]To check whether the coefficient of the basis element indexed by a partition in one element is the same as the coefficient of the basis element indexed by the conjugate partition in the other :
> p:=Partitions(f); > for pp in p do > if Coefficient(h, ConjugatePartition(pp)) ne Coefficient(f, pp) then > print pp; > end if; > end for;
An inner product on Λ is defined by
< mλ, hλ ' > = δλ, λ '.
This definition ensures that the bases (mλ) and (hλ) are dual to each other. This is the inner product used by Magma.
Computes the inner product of the symmetric functions a and b.
> E := SFAElementary(Rationals()); > H := SFAHomogeneous(Rationals()); > p:=RandomPartition(45); > pc:=ConjugatePartition(p); > InnerProduct(E.p,H.pc); 1As it should be the result is 1.
The Schur function is the generating function of standard tableaux. Therefore, it is possible to get the corresponding tableaux.
Given a Schur function sf over the integers with positive coefficients, return the multiset of the tableaux, with maximal entry m, for which sf is the generating function.
A Schur function indexed by a single partition, i.e. a basis element, corresponds to an irreducible character of the symmetric group.
Given an element sf of an algebra of symmetric functions return a linear combination of irreducible characters of the symmetric group, whose coefficients are the coefficients of sf with respect to the Schur function basis.
> H := SFAHomogeneous(Rationals()); > E := SFAElementary(Rationals()); > p := Partitions(7); > for I in p do > a := SymmetricCharacter(H.I); > J := ConjugatePartition(I); > b := SymmetricCharacter(E.J); > i := InnerProduct(a,b); > if i ne 1 then print i; end if; > end for;And there should be no output.
It is possible to form symmetric functions whose support is a subset of the support of a given symmetric function, subject to some restrictions.
Exact: BoolElt Default: true
Return the symmetric function which is a linear combination of those basis elements of the symmetric function a with degree n. This is the restriction of a into the submodule Λn. If Exact is false then the basis elements included will have degree ≤n. This is the restriction of a into bigcupk ≤n Λk.
Exact: BoolElt Default: true
Return the symmetric function which is a linear combination of those basis elements of the symmetric function a whose indexing partitions are of length n. If Exact is false, then the indexing partitions will be of length ≤n.
Exact: BoolElt Default: true
Return the symmetric function which is a linear combination of those basis elements of the symmetric function a whose indexing partitions have maximal part n. If Exact is false, then the indexing partitions will have maximal part ≤n.