Transition Matrices#

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 \(s_\lambda\), \(h_\lambda\), \(m_\lambda\) and the \(e_\lambda\) are all integer matrices. Only when changing from one of these four bases to the \(p_\lambda\), is the matrix over the rationals. For a discussion of their computation and interactions see [Macdonald, 1995, 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.

Transition Matrices from Schur Basis#

SchurToMonomialMatrix(n): RngIntElt -> AlgMatElt#: 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.

Example: creation (ex-c92c1f)#

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

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SchurToHomogeneousMatrix(n): RngIntElt -> AlgMatElt#: 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).

SchurToPowerSumMatrix(n): RngIntElt -> AlgMatElt#: 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.

SchurToElementaryMatrix(n): RngIntElt -> AlgMatElt#: 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.

Example: trans (ex-ed1ff8)#

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]

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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).

Transition Matrices from Monomial Basis#

MonomialToSchurMatrix(n): RngIntElt -> AlgMatElt#: 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).

MonomialToHomogeneousMatrix(n): RngIntElt -> AlgMatElt#: 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.

MonomialToPowerSumMatrix(n): RngIntElt -> AlgMatElt#: 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.

MonomialToElementaryMatrix(n): RngIntElt -> AlgMatElt#: 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.

Transition Matrices from Homogeneous Basis#

HomogeneousToSchurMatrix(n): RngIntElt -> AlgMatElt#: 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.

Example: HS (ex-eb6dd0)#

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

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HomogeneousToMonomialMatrix(n): RngIntElt -> AlgMatElt#

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 \(M_{\mu, \lambda}\) in the expansion \(h_\lambda = \sum_\mu M_{\mu, \lambda} m_\mu\) is the number of non negative integer matrices with row sums \(\lambda_i\) and column sums \(\mu_j\), see [Macdonald, 1995, page 57].

Example: hm (ex-5582be)#

The matrix converting from homogeneous basis to monomial basis is symmetric.

> IsSymmetric(HomogeneousToMonomialMatrix(7));
true

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HomogeneousToPowerSumMatrix(n): RngIntElt -> AlgMatElt#: 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.

HomogeneousToElementaryMatrix(n): RngIntElt -> AlgMatElt#: 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.

Example: HE (ex-6bb56e)#

It is known that the matrix compute by HomogeneousToElementaryMatrix is the same as the matrix computed by ElementaryToHomogeneousMatrix.

> HomogeneousToElementaryMatrix(7) eq ElementaryToHomogeneousMatrix(7);
true

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Transition Matrices from Power Sum Basis#

PowerSumToSchurMatrix(n): RngIntElt -> AlgMatElt#: 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.

Example: Character Table (ex-75ee8a)#

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

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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.

PowerSumToMonomialMatrix(n): RngIntElt -> AlgMatElt#: 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.

PowerSumToHomogeneousMatrix(n): RngIntElt -> AlgMatElt#: 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.

PowerSumToElementaryMatrix(n): RngIntElt -> AlgMatElt#: 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.

Transition Matrices from Elementary Basis#

ElementaryToSchurMatrix(n): RngIntElt -> AlgMatElt#: 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.

ElementaryToMonomialMatrix(n): RngIntElt -> AlgMatElt#

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 \(M_{\mu, \lambda}\) in the expansion \(e_\lambda = \sum_\mu M_{\mu, \lambda} m_\mu\) is the number of \(0\)-\(1\) integer matrices with row sum \(\lambda_i\) and column sum \(\mu_j\), see [Macdonald, 1995, page 57].

Example: em (ex-5a0537)#

The matrix converting from elementary basis to monomial basis is symmetric.

> IsSymmetric(ElementaryToMonomialMatrix(7));
true

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ElementaryToHomogeneousMatrix(n): RngIntElt -> AlgMatElt#: 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.

ElementaryToPowerSumMatrix(n): RngIntElt -> AlgMatElt#: 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.