71.6 CrystalDecompositionMatrix

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    

See also Specht Specht, Schur Schur, DecompositionMatrix DecompositionMatrix, and Specialized Specialized. This function requires the package ``specht'' (see RequirePackage).

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