.. _SectCartanCarMat: .. _cartanmat: Cartan Matrices =============== A *Cartan matrix* is a real valued matrix :math:`C=(c_{ij})_{i,j=1}^n` satisfying the properties: 1. :math:`c_{ii}=2`; 2. :math:`c_{ij}\le 0` for :math:`i\ne j`; 3. :math:`c_{ij}=0` if, and only if, :math:`c_{ji}=0`; and 4. if :math:`n_{ij}:=c_{ij}c_{ji}<4`, then :math:`n_{ij}=4\cos^2(\pi/m_{ij})` for some integer :math:`m_{ij}\ge2`. In Magma, Cartan matrices can be defined over the integer ring (Chapter :ref:`RngInt`), the rational field (Chapter :ref:`FldRat`), number fields (Chapter :ref:`FldNum:main`), and cyclotomic fields (Chapter :ref:`ChapFldCyc`). The real field (Chapter :ref:`ChapFldRe`) is *not* allowed since it is not infinite precision. A Cartan matrix is called *crystallographic* if all its entries are integers. Given a Cartan matrix, the corresponding Coxeter matrix :math:`M=(m_{ij})_{i,j=1}^n` is defined by :math:`m_{ii}=1`; :math:`m_{ij}` as in (4) if :math:`n_{ij}<4`; :math:`m_{ij}=\infty` (ie, :math:`0`) if :math:`n_{ij}\ge 4`. The significance of Cartan matrices is due to the following construction: Let :math:`X` be a real inner-product space with basis :math:`\alpha_1,\dots,\alpha_n`. Take the unique basis :math:`\alpha^\star_1,\dots,\alpha^\star_n` for :math:`X` such that :math:`(\alpha_i,\alpha_j^\star)=c_{ij}`. Let :math:`s_i` be the reflection in :math:`\alpha_i` and :math:`\alpha_i^\star`, i.e. :math:`s_i:V\rightarrow V` is defined by :math:`vs_i= v-(v,\alpha_i^\star)\alpha_i`. Then the group generated by :math:`s_1,\dots,s_n` is a Coxeter group with Coxeter matrix :math:`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 :ref:`ChapGrpRfl`. .. magma:function:: \name{IntrIsCart}{IsCartanMatrix}(C) :input_types: AlgMatElt :output_types: BoolElt :parameters: RealInjection : Any : \texttt{false} Returns ``true`` if, and only if, the matrix :math:`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 :math:`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 :math:`C`. If no real injection is given, conditions (2) and (4) of the definition are not checked. .. magma:function:: CartanMatrix(M) :input_types: AlgMatElt :output_types: AlgMatElt :label: CartanMatrix_AlgMatElt .. magma:function:: CartanMatrix(G) :input_types: GrphUnd :output_types: AlgMatElt :label: CartanMatrix_GrphUnd Symmetric : BoolElt : ``false`` BaseField : MonStgElt : “NumberField" A Cartan matrix corresponding to the Coxeter matrix :math:`M` or Coxeter graph :math:`G`. Note that the Cartan matrix of a Coxeter system is not unique. By default this function returns the Cartan matrix with :math:`c_{ij}=-4\cos^2(\pi/m_{ij})`, :math:`c_{ji}=-1` when :math:`m_{ij}\ne2` and :math:`i C := Matrix(2,2, [ 2,-3, -1,2 ]); > C; > IsCartanMatrix(C); true %%a> assert $1; > 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] .. magma:function:: IsCoxeterIsomorphic(C1, C2) :input_types: AlgMatElt, AlgMatElt :output_types: RngIntElt :label: IsCoxeterIsomorphic_AlgMatElt_AlgMatElt Tests if the Cartan matrices :math:`C_1` and :math:`C_2` 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 :math:`C_1` to the Coxeter matrix of :math:`C_2` is also returned. .. magma:function:: IsCartanEquivalent(C1, C2) :input_types: AlgMatElt, AlgMatElt :output_types: BoolElt :label: IsCartanEquivalent_AlgMatElt_AlgMatElt Returns ``true`` if, and only if, the crystallographic Cartan matrices :math:`C_1` and :math:`C_2` 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 :math:`C_1` to :math:`C_2` is also returned. .. magma:example:: Example: Cartan Matrix Equivalence :label: CartanMatrixEquivalence Cartan equivalence is a stronger condition than Coxeter isomorphism. .. code-block:: magma > 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 %%a> assert not $1; .. magma:function:: NumberOfPositiveRoots(C) :input_types: AlgMatElt :output_types: RngIntElt :label: NumberOfPositiveRoots_AlgMatElt .. magma:function:: NumPosRoots(C) :input_types: AlgMatElt :output_types: RngIntElt :label: NumPosRoots_AlgMatElt The number of positive roots of the root system with Cartan matrix :math:`C`. See Subsection :ref:`SubsectRSPosSimple` for the definition of positive roots. .. magma:function:: CoxeterGroupOrder(C) :input_types: AlgMatElt :output_types: . :label: CoxeterGroupOrder_AlgMatElt .. magma:function:: CoxeterGroupFactoredOrder(C) :input_types: AlgMatElt :output_types: . :label: CoxeterGroupFactoredOrder_AlgMatElt The (factored) order of the Coxeter group with Cartan matrix :math:`C`. .. magma:function:: FundamentalGroup(C) :input_types: AlgMatElt :output_types: GrpAb :label: FundamentalGroup_AlgMatElt The fundamental group of the crystallographic Cartan matrix :math:`C`, i.e. :math:`{\mathbb{Z}}^n/\Gamma` where :math:`n` is the degree of :math:`C` and :math:`\Gamma` is the lattice generated by the rows of :math:`C`. The natural mapping :math:`{\mathbb{Z}}^n\to{\mathbb{Z}}^n/\Gamma` is the second returned value. .. magma:example:: Example: Cartan Matrix Operations :label: CartanMatrixOperations .. code-block:: magma > C := CartanMatrix(PathGraph(4)); > FundamentalGroup(C); Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*$.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 .. magma:function:: IsCoxeterIrreducible(C) :input_types: AlgMatElt :output_types: BoolElt :label: IsCoxeterIrreducible_AlgMatElt Returns ``true`` if, and only if, :math:`C` is the Cartan matrix of an irreducible Coxeter system. If the Coxeter matrix is reducible, this function also returns a nontrivial subset :math:`I` of :math:`\{1,\dots,n\}` such that :math:`m_{ij}=2` (i.e. :math:`c_{ij}=0`) whenever :math:`i\in I`, :math:`j\notin I`. .. magma:function:: IsCrystallographic(C) :input_types: AlgMatElt :output_types: BoolElt :label: IsCrystallographic_AlgMatElt Returns ``true`` if, and only if, the Cartan matrix :math:`C` is crystallographic, i.e. :math:`C` has integral entries. .. magma:function:: IsSimplyLaced(C) :input_types: AlgMatElt :output_types: BoolElt :label: IsSimplyLaced_AlgMatElt Returns ``true`` if, and only if, the Cartan matrix :math:`C` is simply laced, i.e. all the entries in its Coxeter matrix are 1, 2, or 3. .. magma:example:: Example: Cartan Matrix Properties :label: CartanMatrixProperties .. code-block:: magma > C := Matrix(2,2, [ 2,-2, -2,2 ]); > IsCoxeterIrreducible(C); true %%a> assert $1; > IsCrystallographic(C); true %%a> assert $1; > IsSimplyLaced(C); false %%a> assert not $1;