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Cartan Matrices

A Cartan matrix is a real valued matrix C=(c_ij)_{i, j=1}ⁿ satisfying the properties:

1.

c_ii=2;

2.

c_ij≤0 for i≠j;

3.

c_ij=0 if, and only if, c_ji=0; and

4.

if n_ij:=c_ijc_ji<4, then n_ij=4cos²(π/m_ij) for some integer m_ijge2.

Given a Cartan matrix, the corresponding Coxeter matrix M=(m_ij)_{i, j=1}ⁿ is defined by m_ii=1; m_ij as in (4) if n_ij<4; m_ij=∞ (ie, 0) if n_ij≥4. The significance of Cartan matrices is due to the following construction: Let X be a real inner-product space with basis α₁, ..., α_n. Take the unique basis α^star₁, ..., α^star_n for X such that (α_i, α_j^star)=c_ij. Let s_i be the reflection in α_i and α_i^star, i.e. s_i:V -> V is defined by vs_i= v - (v, α_i^star)α_i. Then the group generated by s₁, ..., s_n is a Coxeter group with Coxeter matrix M. In other words, a Cartan matrix specifies a faithful representation of the Coxeter group as a real reflection group. For more details on reflection groups see Chapter REFLECTION GROUPS.

IsCartanMatrix(C) : AlgMatElt -> BoolElt

    RealInjection: Any                  Default: false

Returns true if, and only if, the matrix C is a Cartan matrix.

Number field elements and cyclotomic field elements do not have a natural identification with real numbers. The RealInjection flag allows the user to provide one. If the base field of C is a number field, the flag should be an injection into the real field; if the base field is cyclotomic, the flag should be an injection into the complex field taking real values on the entries of C. If no real injection is given, conditions (2) and (4) of the definition are not checked.

CartanMatrix(M) : AlgMatElt -> AlgMatElt

CartanMatrix(G) : GrphUnd -> AlgMatElt

    Symmetric: BoolElt                  Default: false

    BaseField: MonStgElt                Default: "NumberField"

A Cartan matrix corresponding to the Coxeter matrix M or Coxeter graph G. Note that the Cartan matrix of a Coxeter system is not unique. By default this function returns the Cartan matrix with c_ij= - 4cos²(π/m_ij), c_ji= - 1 when m_ijne2 and i<j. This matrix is crystallographic whenever there exists a crystallographic Cartan matrix corresponding to M.

If the Symmetric flag is set true, the symmetric Cartan matrix with c_ij=c_ji= - 2cos(π/m_ij) is returned.

The BaseField flag determines the field over which the Cartan matrix is defined. If the matrix is crystallographic however, it is defined over the integers regardless of the value of this flag. The possible values are:

1.

"NumberField": An algebraic number field. This is the default. See Chapter NUMBER FIELDS.

2.

"Cyclotomic" or "SparseCyclotomic": A cyclotomic field with the sparse representation for elements. See Chapter CYCLOTOMIC FIELDS.

3.

"DenseCyclotomic": A cyclotomic field with the dense representation for elements. See Chapter CYCLOTOMIC FIELDS.

CartanMatrix(D) : GrphDir -> AlgMatElt

The crystallographic Cartan matrix corresponding to the Dynkin digraph D.

> C := Matrix(2,2, [ 2,-3, -1,2 ]);
> C;
> IsCartanMatrix(C);
true
> CoxeterMatrix(C);
[1 6]
[6 1]
>
> G := PathGraph(4);
> AssignLabel(G, 1,2, 4);
> AssignLabel(G, 3,4, 4);
> CartanMatrix(G);
[ 2 -2  0  0]
[-1  2 -1  0]
[ 0 -1  2 -2]
[ 0  0 -1  2]
> CartanMatrix(G : Symmetric, BaseField := "Cyclotomic");
[2 zeta(8)_8^3 - zeta(8)_8 0 0]
[zeta(8)_8^3 - zeta(8)_8 2 -1 0]
[0 -1 2 zeta(8)_8^3 - zeta(8)_8]
[0 0 zeta(8)_8^3 - zeta(8)_8 2]

IsCoxeterIsomorphic(C1, C2) : AlgMatElt, AlgMatElt -> RngIntElt

Tests if the Cartan matrices C₁ and C₂ give rise to isomorphic Coxeter systems; i.e., their Coxeter matrices are equal modulo a permutation of the underlying basis. If true, a sequence giving the permutation of the underlying basis which takes the Coxeter matrix of C₁ to the Coxeter matrix of C₂ is also returned.

IsCartanEquivalent(C1, C2) : AlgMatElt, AlgMatElt -> BoolElt

Returns true if, and only if, the crystallographic Cartan matrices C₁ and C₂ are Cartan equivalent, i.e. they are equal modulo permutation of the underlying basis. If so, a sequence giving the permutation of the underlying basis which takes C₁ to C₂ is also returned.

> C1 := Matrix(2,2, [ 2,-2, -2,2 ]);
> C2 := Matrix(2,2, [ 2,-1, -5,2 ]);
> IsCoxeterIsomorphic(C1, C2);
true [ 1, 2 ]
> IsCartanEquivalent(C1, C2);
false

NumberOfPositiveRoots(C) : AlgMatElt -> RngIntElt

NumPosRoots(C) : AlgMatElt -> RngIntElt

The number of positive roots of the root system with Cartan matrix C. See Subsection Simple and Positive Roots for the definition of positive roots.

CoxeterGroupOrder(C) : AlgMatElt -> .

CoxeterGroupFactoredOrder(C) : AlgMatElt -> .

The (factored) order of the Coxeter group with Cartan matrix C.

FundamentalGroup(C) : AlgMatElt -> GrpAb

The fundamental group of the crystallographic Cartan matrix C, i.e. Zⁿ/Γ where n is the degree of C and Γ is the lattice generated by the rows of C. The natural mapping ZⁿtoZⁿ/Γ is the second returned value.

> C := CartanMatrix(PathGraph(4));
> FundamentalGroup(C);
Abelian Group isomorphic to Z/5
Defined on 1 generator
Relations:
    5*$.1 = 0
Mapping from: Standard Lattice of rank 4 and degree 4 to Abelian Group
isomorphic to Z/5
Defined on 1 generator
Relations:
    5*$.1 = 0

IsCoxeterIrreducible(C) : AlgMatElt -> BoolElt

Returns true if, and only if, C is the Cartan matrix of an irreducible Coxeter system. If the Coxeter matrix is reducible, this function also returns a nontrivial subset I of {1, ..., n} such that m_ij=2 (i.e. c_ij=0) whenever i∈I, j∉I.

IsCrystallographic(C) : AlgMatElt -> BoolElt

Returns true if, and only if, the Cartan matrix C is crystallographic, i.e. C has integral entries.

IsSimplyLaced(C) : AlgMatElt -> BoolElt

Returns true if, and only if, the Cartan matrix C is simply laced, i.e. all the entries in its Coxeter matrix are 1, 2, or 3.

> C := Matrix(2,2, [ 2,-2, -2,2 ]);
> IsCoxeterIrreducible(C);
true
> IsCrystallographic(C);
true
> IsSimplyLaced(C);
false

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