CrystalDecompositionMatrix(H, n [,Ordering])
CrystalDecompositionMatrix(H, filename [,Ordering])
This function is similar to DecompositionMatrix
, except that it
returns a crystallized decomposition matrix. The columns of
decomposition matrices correspond to projective indecomposables; the
columns of crystallized decomposition matrices correspond to the
canonical basis elements of the Fock space (see
Specht). Consequently, the entries in these matrices are polynomials
(in v
), and by specializing (ie. setting v
equal to 1; see
Specialized), the decomposition matrices of H are obtained (see
Specht).
Crystallized decomposition matrices are defined only for Hecke algebras over a base field of characteristic zero. Unlike ``normal'' decomposition matrices, crystallized decomposition matrices cannot be induced.
gap> CrystalDecompositionMatrix(Specht(3), 6); 6
|
1 5,1
|
v 1 4,2
|
. . 1 4,1^2
|
. v . 1 3^2
|
. v . . 1 3,2,1
|
v v^2 . v v 1 3,1^3
|
. . . v^2 . v 2^3
|
v^2 . . . . v 2^2,1^2
|
. . . . . . 1 2,1^4
|
. . . . v v^2 . 1^6
|
. . . . v^2 . . gap> Specialized(last); # set 'v' equal to $1$. 6
|
1 5,1
|
1 1 4,2
|
. . 1 4,1^2
|
. 1 . 1 3^2
|
. 1 . . 1 3,2,1
|
1 1 . 1 1 1 3,1^3
|
. . . 1 . 1 2^3
|
1 . . . . 1 2^2,1^2
|
. . . . . . 1 2,1^4
|
. . . . 1 1 . 1^6
|
. . . . 1 . .
See also Specht
Specht, Schur
Schur, DecompositionMatrix
DecompositionMatrix, and Specialized
Specialized. This function
requires the package ``specht'' (see RequirePackage).
GAP 3.4.4