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 [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.
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: Kostka Numbers (ex-c92c1f)
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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
- 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: Transition Matrices Agree with Coercion (ex-ed1ff8)
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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).
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: Homogeneous-to-Schur Matrix as a Transpose (ex-eb6dd0)
It is known that the matrix computed by
HomogeneousToSchurMatrixis the transpose of the matrix computed bySchurToMonomialMatrix.> SchurToMonomialMatrix(7) eq Transpose(HomogeneousToSchurMatrix(7)); true
- 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 [Mac95, page 57].
- Example: Symmetry of the Homogeneous-to-Monomial Matrix (ex-5582be)
The matrix converting from homogeneous basis to monomial basis is symmetric.
> IsSymmetric(HomogeneousToMonomialMatrix(7)); true
- 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.
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 of the Symmetric Group (ex-75ee8a)
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The matrix returned by
PowerSumToSchurMatrixis 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.
- 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.
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 [Mac95, page 57].
- Example: Symmetry of the Elementary-to-Monomial Matrix (ex-5a0537)
The matrix converting from elementary basis to monomial basis is symmetric.
> IsSymmetric(ElementaryToMonomialMatrix(7)); true