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One area of interest in the theory of symmetric functions is the
study of the change of bases between the five different bases.
The matrices of change of base between the sλ, hλ, mλ and the
eλ are all integer matrices. Only when changing
from one of these four bases to the pλ, is the matrix
over the rationals. For a discussion of their computation and interactions see
[Mac95, pages 54--58].
In Magma there are routines available to obtain all these matrices. These
routines are described below. Some interactions of these matrices are verified
in examples.
Computes the matrix for the expansion of a Schur function
indexed by a partition of weight n as a
sum of monomial symmetric functions. This matrix is also
known as the table of Kostka numbers. These are the numbers
of tableaux of each shape and content.
The content of a tableau prescribes its entries, so for example a tableau
with content [2, 1, 4, 1] has two 1's, one 2, four 3's and one 4.
The entries of the
matrix are non negative integers. The matrix is upper triangular.
Compute the base change matrix from the Schur functions to the monomial
symmetric functions for degree 5.
The entries in this matrix are what are known as the Kostka numbers.
They count the number of young tableaux on a given shape
Look on the order of
the labelling partitions, and check whether the entry 3 in
the upper right corner is right by
generating the corresponding tableaux.
> M := SchurToMonomialMatrix(5);
> M;
[1 1 1 1 1 1 1]
[0 1 1 2 2 3 4]
[0 0 1 1 2 3 5]
[0 0 0 1 1 3 6]
[0 0 0 0 1 2 5]
[0 0 0 0 0 1 4]
[0 0 0 0 0 0 1]
> Parts := Partitions(5);
> Parts;
[
[ 5 ],
[ 4, 1 ],
[ 3, 2 ],
[ 3, 1, 1 ],
[ 2, 2, 1 ],
[ 2, 1, 1, 1 ],
[ 1, 1, 1, 1, 1 ]
]
> #TableauxOnShapeWithContent(Parts[2], Parts[6]);
3
> M[2,6];
3
Computes the matrix for the expansion of a Schur function
indexed by a partition of weight n as a
sum of homogeneous symmetric functions.
The entries of the
matrix are positive and negative integers. The matrix is lower triangular.
It is the transpose of the matrix returned by MonomialToSchurMatrix(n).
Computes the matrix for the expansion of a Schur function
indexed by a partition of weight n as a
sum of power sum symmetric functions.
The entries of the matrix are rationals.
Computes the matrix for the expansion of a Schur function
indexed by a partition of weight n as a
sum of elementary symmetric functions.
The entries of the
matrix are positive and negative integers. The matrix is upper left triangular.
We verify by hand that the action of base change matrix is
the same as coercion.
> S := SFASchur(Rationals());
> P := SFAPower(Rationals());
> NumberOfPartitions(4);
5
> m := SchurToPowerSumMatrix(4);
> Partitions(4);
[
[ 4 ],
[ 3, 1 ],
[ 2, 2 ],
[ 2, 1, 1 ],
[ 1, 1, 1, 1 ]
]
> s := S.[3, 1] + 5*S.[1, 1, 1, 1] - S.[4];
> s;
5*S.[1,1,1,1] + S.[3,1] - S.[4]
> p, c := Support(s);
> c;
[ 5, 1, -1 ]
> p;
[
[ 1, 1, 1, 1 ],
[ 3, 1 ],
[ 4 ]
]
> cm := Matrix(Rationals(), 1, 5, [-1, 1, 0, 0, 5]);
> cm*m;
[-7/4 4/3 3/8 -5/4 7/24]
> P!s;
7/24*P.[1,1,1,1] - 5/4*P.[2,1,1] + 3/8*P.[2,2] + 4/3*P.[3,1] - 7/4*P.[4]
The coefficients of the coerced element are the reverse of the matrix product,
consistent with the partition in the coerced element being in reverse order
to those in Partitions(4).
Computes the matrix for the expansion of a monomial symmetric function
indexed by a partition of weight n as a sum of Schur
symmetric functions. The entries of the matrix are positive and
negative integers. The matrix is upper triangular. It is the
transpose of the matrix returned by SchurToHomogeneousMatrix(n).
Computes the matrix for the expansion of a monomial symmetric function
indexed by a partition of weight n as a
sum of homogeneous symmetric functions.
The entries are positive and negative integers.
Computes the matrix for the expansion of a monomial symmetric function
indexed by a partition of weight n as a
sum of power sum symmetric functions.
The entries are rationals. The matrix is lower triangular.
Computes the matrix for the expansion of a monomial symmetric function
indexed by a partition of weight n as a
sum of elementary symmetric functions.
The entries of the
matrix are positive and negative integers. The matrix is upper left triangular.
Computes the matrix for the expansion of a homogeneous symmetric function
indexed by a partition of weight n as a
sum of Schur symmetric functions.
The entries of the
matrix are positive integers. The matrix is lower triangular.
It is known that the matrix computed by HomogeneousToSchurMatrix is
the transpose of the matrix computed by SchurToMonomialMatrix.
> SchurToMonomialMatrix(7) eq Transpose(HomogeneousToSchurMatrix(7));
true
Computes the matrix M for the expansion of a homogeneous symmetric function
indexed by a partition of weight n as a
sum of monomial symmetric functions.
The entries of the
matrix are positive integers. The matrix has no zero entries.
The coefficient Mμ, λ in the expansion
hλ = ∑μ Mμ, λ mμ is the number of
non negative integer matrices with row sums λi and
column sums μj, see [Mac95, page 57].
The matrix converting from homogeneous basis to monomial basis is symmetric.
> IsSymmetric(HomogeneousToMonomialMatrix(7));
true
Computes the matrix for the expansion of a homogeneous symmetric function
indexed by a partition of weight n as a
sum of power sum symmetric functions.
The entries of the
matrix are positive rationals. The matrix is upper triangular.
Computes the matrix for the expansion of a homogeneous symmetric function
indexed by a partition of weight n as a
sum of elementary symmetric functions.
The entries of the
matrix are integers. The matrix is upper triangular.
It is known that the matrix compute by HomogeneousToElementaryMatrix
is the same as the matrix computed by ElementaryToHomogeneousMatrix.
> HomogeneousToElementaryMatrix(7) eq ElementaryToHomogeneousMatrix(7);
true
Computes the matrix for the expansion of a power sum symmetric function
indexed by a partition of weight n as a
sum of Schur symmetric functions.
The entries of the
matrix are positive and negative integers. This matrix is
the character table of the symmetric group.
The matrix returned by PowerSumToSchurMatrix is compared to the character
table of the appropriate symmetric group.
> PowerSumToSchurMatrix(5);
[ 1 -1 0 1 0 -1 1]
[ 1 0 -1 0 1 0 -1]
[ 1 -1 1 0 -1 1 -1]
[ 1 1 -1 0 -1 1 1]
[ 1 0 1 -2 1 0 1]
[ 1 2 1 0 -1 -2 -1]
[ 1 4 5 6 5 4 1]
> CharacterTable(Sym(5));
Character Table
---------------
-----------------------------
Class | 1 2 3 4 5 6 7
Size | 1 10 15 20 30 24 20
Order | 1 2 2 3 4 5 6
-----------------------------
p = 2 1 1 1 4 3 6 4
p = 3 1 2 3 1 5 6 2
p = 5 1 2 3 4 5 1 7
-----------------------------
X.1 + 1 1 1 1 1 1 1
X.2 + 1 -1 1 1 -1 1 -1
X.3 + 4 2 0 1 0 -1 -1
X.4 + 4 -2 0 1 0 -1 1
X.5 + 5 1 1 -1 -1 0 1
X.6 + 5 -1 1 -1 1 0 -1
X.7 + 6 0 -2 0 0 1 0
In the character table the first row is the unity character, which corresponds
to the first column of the transition matrix. The second row of the character table
is the alternating character which corresponds to the last column of the
transition matrix. The first column of the character table contains the dimensions of the
irreducible characters, this is the last row of the transition matrix.
Computes the matrix for the expansion of a power sum symmetric function
indexed by a partition of weight n as a
sum of monomial symmetric functions.
The entries of the
matrix are positive integers.
The matrix is lower triangular.
Computes the matrix for the expansion of a power sum symmetric function
indexed by a partition of weight n as a
sum of homogeneous symmetric functions.
The entries of the
matrix are integers.
The matrix is upper triangular.
Computes the matrix for the expansion of a power sum symmetric function
indexed by a partition of weight n as a
sum of elementary symmetric functions.
The entries of the
matrix are integers.
The matrix is upper triangular.
Computes the matrix for the expansion of an elementary symmetric function
indexed by a partition of weight n as a
sum of Schur symmetric functions.
The entries of the
matrix are positive integers.
Computes the matrix M for the expansion of an elementary symmetric function
indexed by a partition of weight n as a
sum of monomial symmetric functions.
The entries of the
matrix are positive integers.
The coefficient Mμ, λ in the expansion
eλ = ∑μ Mμ, λ mμ is the number of
0-1 integer matrices with row sum λi and
column sum μj, see [Mac95, page 57].
The matrix converting from elementary basis to monomial basis is symmetric.
> IsSymmetric(ElementaryToMonomialMatrix(7));
true
Computes the matrix for the expansion of a elementary symmetric function
indexed by a partition of weight n as a
sum of homogeneous symmetric functions.
Computes the matrix for the expansion of a elementary symmetric function
indexed by a partition of weight n as a
sum of power sum symmetric functions.
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