71.18 AdjustmentMatrix

AdjustmentMatrix(dp, d)

James [J] noticed, and Geck [G] proved, that the decomposition matrices dp for Hecke algebras defined over fields of positive characteristic admit a factorization dp = d * a where d is a decomposition matrix for a suitable Hecke algebra defined over a field of characteristic zero, and a is the so--called adjustment matrix. This function returns the adjustment matrix a.

gap> H:=Specht(2);; Hp:=Specht(2,2);;
gap> d:=DecompositionMatrix(H,13);; dp:=DecompositionMatrix(Hp,13);;
gap> a:=AdjustmentMatrix(dp,d);
13     

 1
12,1   

 . 1
11,2   

 1 . 1
10,3   

 . . . 1
10,2,1 

 . . . . 1
9,4    

 1 . 1 . . 1
9,3,1  

 2 . . . . . 1
8,5    

 . 1 . . . . . 1
8,4,1  

 1 . . . . . . . 1
8,3,2  

 . 2 . . . . . 1 . 1
7,6    

 1 . . . . 1 . . . . 1
7,5,1  

 . . . . . . 1 . . . . 1
7,4,2  

 1 . 1 . . 1 . . . . 1 . 1
7,3,2,1

 . . . . . . . . . . . . . 1
6,5,2  

 . 1 . . . . . 1 . 1 . . . . 1
6,4,3  

 2 . . . 1 . . . . . . . . . . 1
6,4,2,1

 . 2 . 1 . . . . . . . . . . . . 1
5,4,3,1

 4 . 2 . . . . . . . . . . . . . . 1 
gap> MatrixDecompositionMatrix(dp)=
>           MatrixDecompositionMatrix(d)*MatrixDecompositionMatrix(a);
true 

In the last line we have checked our calculation.

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

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