71.4 Schur

Schur(e)
Schur(e, p)
Schur(e, p, val [,HeckeRing])

This function behaves almost identically to the function Specht (see Specht), the only difference being that the three functions in the record S returned by Schur are called S.W, S.P, and S.F and that they correspond to the q-Weyl modules, the projective decomposable modules, and the simple modules of the q--Schur algebra respectively. Note that our labeling of these modules is non--standard, following that used by James in [J]. The standard labeling can be obtained from ours by replacing all partitions by their conjugates.

Almost all of the functions in Specht which accept a Specht record H will also accept a record S returned by Schur

In the current version of Specht the decomposition matrices of q--Schur algebras are not fully supported. The InducedDecompositionMatrix function can be applied to these matrices; however there are no additional routines available for calculating the columns corresponding to e--singular partitions. The decomposition matrices for the q--Schur algebras defined over a field of characteristic 0 for <n> le 10 are in the Specht libraries.

gap> S:=Schur(2);
Schur(e=2, W(), P(), F(), Pq())
gap> InducedDecompositionMatrix(DecompositionMatrix(S,3));
# The following projectives are missing from <d>:
#  [ 2, 2 ]
4     

 1                   # 'DecompositionMatrix'(S,4) returns the
3,1   

 1 1                 # full decomposition matrix. The point
2^2   

 . 1 .               # of this example is to emphasize the
2,1^2 

 1 1 . 1             # limitations of 'Schur'.
1^4   

 1 . . 1 1 

Note that when S is defined over a field of characteristic zero then it contains a function S.Pq for calculating canonical basis elements (see Specht Specht); currently S.Pq(mu) is implemented only for e--regular partitions. There is also a function H.Wq.

See also Specht Specht. This function requires the package ``specht'' (see RequirePackage).

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