Cartan Matrices

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Cartan Matrices

The Cartan matrix of a root datum gives the values of the bilinear pairing < o , o >:X x Y -> R on the simple (co)roots. More precisely it is the matrix C=[< alpha_i, alpha_j^star >].

The possible Cartan matrices can be determined from the Dynkin diagrams:

In the crystallographic case, the Cartan matrix is defined over the integers as follows: C_(ii)=2; if i and j do not have an edge between them then C_(ij)= 0; if the i and j are connected by a single edge then C_(ij)= - 1; if the i and j are connected by m edges with the arrow pointing from i to j then C_(ij)= - m and C_(ji)= - 1.
In the noncrystallographic case, the Cartan matrix is defined over a cyclotomic extension of the rational field (Chapter CYCLOTOMIC FIELDS). These Cartan matrices are not completely determined by the Dynkin diagram, but they satisfy the following properties: C_(ii)=2; if the i and j do not have an edge between them then C_(ij)= 0; if i and j are connected by an edge of multiplicity m then C_(ij) and C_(ji) are real elements of the extension of Q by the mth root of unity, with C_(ij)C_(ji)=4cos^2(pi/m). We also need every nondiagonal entry of C to be nonpositive; since cyclotomic fields in Magma are not ordered, it is enough to require that the nondiagonal entries are nonpositive whenever they are rational.

The Cartan type of a root datum is entirely determined by its Cartan matrix. Cartan types can be given as strings.

CartanMatrix( t ) : MonStgElt -> AlgMatElt

Returns the Cartan matrix with the Cartan type t given as a string.

IsCartanMatrix( M ) : AlgMatElt -> BoolElt

Return true if the matrix M is a Cartan matrix.

CartanName( C ) : AlgMatElt -> List

Returns the name of the Cartan type of the Cartan matrix C. If C is not a Cartan matrix, this gives an error message listing the rows/columns where the problem was detected.

Example `RootDtm_CartanMatrices (H33E1)`

The code for interpreting a string as a Cartan type is quite flexible: any symbol other than a letter or a number is ignored; letters and numbers must alternate, except in type I where the two numbers may be separated by any nonalphanumeric character.

> C := CartanMatrix( "A_5B3 c2I2 5" );
> CartanName( C );
A5 B3 B2 I2(5)

The user can also supply a Cartan matrix.

> C := Matrix( 4, 4, [2,-1,0,0, -1,2,0,0, 0,0,2,-3, 0,0,-2,2] );
> IsCartanMatrix( C );
false

> CartanName( C );
Not a Cartan matrix at rows/columns  [ 4, 3 ]
> C[4,3] := -1;
> C;
[ 2 -1  0  0]
[-1  2  0  0]
[ 0  0  2 -3]
[ 0  0 -1  2]
> CartanName( C );
A2 G2

DynkinDiagram( t ) : List -> .

DynkinDiagram( C ) : AlgMatElt -> .

Prints the Dynkin diagram of the Cartan type t or Cartan matrix C.

IsCartanIrreducible( C ) : AlgMatElt -> BoolElt

Returns true if the matrix C is Cartan irreducible. A matrix is Cartan irreducible if it cannot be conjugated to a direct sum of matrices using a permutation matrix. This is equivalent to the Dynkin diagram being connected.

IsCrystallographic( C ) : AlgMatElt -> BoolElt

Returns true if the Cartan matrix C is crystallographic (ie. if all its entries are integral).

FundamentalGroup( t ) : AlgMatElt -> GrpAb

FundamentalGroup( C ) : AlgMatElt -> GrpAb

The fundamental group Lambda/Z Phi for the Cartan type t or Cartan matrix C. Note that the order of this group is the determinant of C.

Example `RootDtm_CartanMatrixFunctions (H33E2)`

> C := CartanMatrix( "A5 B2" );
> DynkinDiagram( C );

A5    1 - 2 - 3 - 4 - 5

B2    6 =>= 7
> IsCartanIrreducible( C );
false
> IsCrystallographic( C );
true
> #FundamentalGroup( C ) eq Determinant( C );
true

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