The decomposition numbers for the algebraic group \(SL_n\) of type \(A_{n−1}\) determine the decomposition numbers for the corresponding Schur algebras, and thus also determine the decomposition numbers for symmetric groups. People working with Schur algebras and symmetric groups often prefer to use partitions to label highest weights. Although it is trivial to convert between \(SL_n\) weight notation and partition notation, for the sake of convenience, we provide a few functions that perform such conversions, and various other functions related to Schur algebras and symmetric groups.
NOTE. The SymmetricGroupDecompositionMatrix (4.2-3) function for symmetric group decomposition numbers is quite slow, so readers interested in symmetric group computations may want to look elsewhere for more effcient tools.
Weyl modules for a Schur algebra are constructed by the following.
‣ SchurAlgebraWeylModule( p, ptn ) | ( operation ) |
Returns: a Weyl module
This function creates and returns a Weyl module of highest weight defined by the given partition ptn. The length of the partition, which may be padded by zeros as necessary, defines the underlying \(GL_n\) and the Schur algebra degree.
gap> V:= SchurAlgebraWeylModule(3,[1,1,0]); Schur algebra module V[ 1, 1, 0 ]
Here we define the Weyl module for \(GL_3\) of highest weight [1, 1] in the partition notation.
‣ IsSchurAlgebraWeylModule( V ) | ( filter ) |
Returns: true or false
Returns true if and only if the given V is a Schur algebra Weyl module.
Decomposition matrices for Schur algebras and symmteric groups in positive characteristic can be computed.
‣ SchurAlgebraDecompositionMatrix( p, n, r ) | ( operation ) |
Returns: a matrix
Returns the decomposition matrix for the Schur algebra \(S(n,r)\) in characteristc p. The rows and columns of the matrix are indexed by the partitions produced by BoundedPartitions(n,r) ordered the same as in the output of that function.
gap> SchurAlgebraDecompositionMatrix(3,4,3); [ [ 1, 1, 0 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ]
Here we compute the decomposition matrix for \(S(4,3)\) in characteristic \(3\). The rows and columns of the matrix are indexed by the following partitions:
gap> BoundedPartitions(4,3); [ [ 3, 0, 0, 0 ], [ 2, 1, 0, 0 ], [ 1, 1, 1, 0 ] ]
‣ SymmetricGroupDecompositionNumbers( p, ptn ) | ( operation ) |
Returns: a list
Returns the decomposition numbers of the dual Specht module indexed by the given partition ptn in characteristic p.
gap> SymmetricGroupDecompositionNumbers(2,[2,1,1]); [ [ 2, 1, 1 ], 1, [ 1, 1, 1, 1 ], 1 ]
‣ SymmetricGroupDecompositionMatrix( p, n ) | ( operation ) |
Returns: a matrix
Returns the decomposition matrix for the symmetric group on n letters in characteristc p. The rows of the matrix are labeled by the partitions of n in the order produced by AllPartitions(n), and the columns are labeled by the p-restricted partitions of n. NOTE. GAP has a built-in Partitions function that also gives all the partitions of n, but the ordering is different.
gap> SymmetricGroupDecompositionMatrix(2,4); [ [ 0, 1 ], [ 1, 1 ], [ 1, 0 ], [ 1, 1 ], [ 0, 1 ] ] gap> AllPartitions(4); [ [ 4 ], [ 3, 1 ], [ 2, 2 ], [ 2, 1, 1 ], [ 1, 1, 1, 1 ] ] gap> pRestrictedPartitions(2,4); [ [ 2, 1, 1 ], [ 1, 1, 1, 1 ] ]
This section documents a number of functions for converting between weights and partitions (in type A) as well as other related functions.
‣ CompositionToWeight( mu ) | ( operation ) |
Returns: a list (a weight)
This converts the given composition mu into a weight by taking successive differences of its parts.
gap> CompositionToWeight([1,2,0,1]); [ -1, 2, -1 ]
‣ WeightToComposition( r, wt ) | ( operation ) |
Returns: a list (a composition) or fail
This converts the given weight wt into a composition of degree r. Without degree information, this function is ill defined. Returns fail if the operation is impossible.
gap> WeightToComposition(4,[-1, 2, -1]); [ 1, 2, 0, 1 ] gap> WeightToComposition(8,[-1, 2, -1]); [ 2, 3, 1, 2 ] gap> WeightToComposition(6,[-1, 2, -1]); fail
‣ AllPartitions( n ) | ( operation ) |
Returns: a list of partitions
Lists all the partitions of n. Note that GAP has a built-in Partitions function that also gives all the partitions of n, but with a different ordering.
gap> AllPartitions(5); [ [ 5 ], [ 4, 1 ], [ 3, 2 ], [ 3, 1, 1 ], [ 2, 2, 1 ], [ 2, 1, 1, 1 ], [ 1, 1, 1, 1, 1 ] ]
‣ BoundedPartitions( n, r, s ) | ( operation ) |
‣ BoundedPartitions( n, r ) | ( operation ) |
Returns: a list of partitions
Returns a list of n part partitions of degree r such that each part lies in the closed interval [0,s]. The second form returns a list of n part partitions of degree r. Note that BoundedPartitions(n,r) is equivalent to BoundedPartitions(n,r,r).
gap> BoundedPartitions(5,3,2); [ [ 2, 1, 0, 0, 0 ], [ 1, 1, 1, 0, 0 ] ] gap> BoundedPartitions(5,3,3); [ [ 3, 0, 0, 0, 0 ], [ 2, 1, 0, 0, 0 ], [ 1, 1, 1, 0, 0 ] ] gap> BoundedPartitions(5,3); [ [ 3, 0, 0, 0, 0 ], [ 2, 1, 0, 0, 0 ], [ 1, 1, 1, 0, 0 ] ]
‣ Conjugate( ptn ) | ( operation ) |
Returns: a list (a partition)
Returns the conjugate partition of ptn.
gap> Conjugate([4]); [ 1, 1, 1, 1 ] gap> Conjugate([2,1,1,1]); [ 4, 1 ]
‣ pRegular( p, ptn ) | ( operation ) |
Returns: true or false
Returns true if and only if the given partition ptn is p-regular.
gap> pRegular(3,[3,1,1]); true gap> pRegular(2,[3,1,1]); false
‣ pRegularPartitions( p, n ) | ( operation ) |
Returns: a list
Returns a list of all p-regular partitions of n.
gap> pRegularPartitions(3,5); [ [ 4, 1 ], [ 3, 1, 1 ], [ 5 ], [ 2, 2, 1 ], [ 3, 2 ] ] gap> pRegularPartitions(2,5); [ [ 3, 2 ], [ 4, 1 ], [ 5 ] ]
‣ pRestricted( p, ptn ) | ( operation ) |
Returns: true or false
Returns true if and only if the given partition ptn is p-restricted.
gap> pRestricted(3,[3,1,1]); true gap> pRestricted(2,[3,1]); false
‣ pRestrictedPartitions( p, n ) | ( operation ) |
Returns: a list
Returns a list of all p-restricted partitions of n.
gap> pRestrictedPartitions(3,5); [ [ 3, 2 ], [ 3, 1, 1 ], [ 2, 2, 1 ], [ 2, 1, 1, 1 ], [ 1, 1, 1, 1, 1 ] ] gap> pRestrictedPartitions(2,5); [ [ 2, 2, 1 ], [ 2, 1, 1, 1 ], [ 1, 1, 1, 1, 1 ] ]
‣ Mullineux( p, mu ) | ( operation ) |
Returns: a list
Applies the Mullineux map to the partition mu in characteristic p.
gap> Mullineux(2,[1,1,1]); p-singular! gap> Mullineux(2,[3]); [ 3 ] gap> Mullineux(3,[3]); [ 2, 1 ] gap> Mullineux(3,[2,1]); [ 3 ]
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