# Constructing Root Systems We first describe some optional parameters that are common to many functions described in this section. ```magma RealInjection: Any Default: false ``` 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 the Cartan matrix $C$ 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 $C$ (see more in Section [Cartan Matrices](../CoxeterSystems/cartanmat.md#sectcartancarmat)). ```magma Nonreduced: SetEnum Default: {} ``` The optional argument `Nonreduced` is used to distinguish the reducedness of a root system in case the input doesn’t uniquely determine it. ```magma Symmetric: BoolElt Default: false ``` If the `Symmetric` flag is set `true`, the symmetric Cartan matrix is used. For types $I_2(m)$, $H_3$, $H_4$ the symmetric Cartan matrix is *always* used, since the root system is nonreduced otherwise. ```magma BaseField: MonStgElt Default: "NumberField" ``` The `BaseField` flag determines the field over which the Cartan matrix is defined. The possible values are: 1. `"NumberField"`: An algebraic number field. This is the default. See Chapter [Number Fields](../../GlobalFields/NumberFields/index-number-fields.md#fldnum-main). 2. `"Cyclotomic"` or `"SparseCyclotomic"`: A cyclotomic field with the sparse representation for elements. See Chapter [Cyclotomic Fields](../../GlobalFields/CyclotomicFields/index-cyclotomic-fields.md#chapfldcyc). 3. `"DenseCyclotomic"`: A cyclotomic field with the dense representation for elements. See Chapter [Cyclotomic Fields](../../GlobalFields/CyclotomicFields/index-cyclotomic-fields.md#chapfldcyc). ## `RootSystem(N): MonStgElt -> RootSys` ```magma Symmetric: BoolElt Default: false BaseField: MonStgElt Default: "NumberField" ``` The root system with Cartan name given by the string $N$. In addition to the Cartan names in Section [Finite and Affine Coxeter Groups](../CoxeterSystems/finiteaffine.md#sectcartanfinaff), we allow `"BCn"` for the irreducible nonreduced system, and `"Tn"` for the $n$-dimensional toral subsystem. Note that `"Tn"` is used for input only and does not appear in the string returned by `CartanName` when applied to the resulting root system (see example below). For descriptions of the parameters `Symmetric` and `BaseField` see the beginning of this section ## `Example: Creating Root Systems Name (ex-ff0977)` ```magma > RootSystem("H3 E6"); Root system of type H3 E6 > RootSystem("A2 T1 I2(5)"); Root system of type A2 I2(5) ``` ## `RootSystem(M): AlgMatElt -> RootSys` ## `RootSystem(G): GrphUnd -> RootSys` ```magma Nonreduced: SetEnum Default: {} Symmetric : BoolElt Default: false BaseField : MonStgElt Default: "NumberField" ``` The semisimple root system with Coxeter matrix $M$ or Coxeter graph $G$ (see Chapter [Coxeter Systems](../CoxeterSystems/index-coxeter-systems.md#chapcartan)). If the corresponding Coxeter group is infinite, an error is flagged. For descriptions of the parameters `Nonreduced`, `Symmetric`, and `BaseField` see the beginning of this section. ## `RootSystem(C): AlgMatElt -> RootSys` ```magma RealInjection: Any Default: false Nonreduced : SetEnum Default: {} ``` The semisimple root system with Cartan matrix $C$ (see Chapter [Coxeter Systems](../CoxeterSystems/index-coxeter-systems.md#chapcartan)). If the corresponding Coxeter group is infinite, an error is flagged. For descriptions of the parameters `RealInjection` and `Nonreduced` see the beginning of this section. ## `RootSystem(D): GrphDir -> RootSys` ```magma Nonreduced: SetEnum Default: {} ``` The semisimple crystallographic root system with Cartan matrix $C$, or Dynkin diagram $D$ (see Chapter [Coxeter Systems](../CoxeterSystems/index-coxeter-systems.md#chapcartan)). If the corresponding Coxeter group is infinite, an error is flagged. For a description of the parameter `Nonreduced` see the beginning of this section. ## `Example: Creating Root Systems Matrix (ex-4f54b6)` ```magma > M := SymmetricMatrix([1, 3,1, 2,3,1]); > RootSystem(M); Root system of type A3 > M := SymmetricMatrix([1, 3,1, 3,3,1]); > RootSystem(M); >> RootSystem(M); ^ Runtime error in 'RootSystem': Not a finite root system in rows/columns [ 1, 2, 3 ] ``` ## `RootSystem(A, B): Mtrx, Mtrx -> RootSys` ```magma RealInjection: Any Default: false Nonreduced : SetEnum Default: {} ``` The root system with simple roots given by the rows of the matrix $A$ and simple coroots given by the rows of the matrix $B$. The matrices $A$ and $B$ must have the following properties: 1. $A$ and $B$ must have the same number of rows and the same number of columns; they must be defined over the same ring, which must be the integers, the rational field, a number field, or a cyclotomic field; 2. the number of columns must be at least the number of rows; and 3. $AB^t$ must be the Cartan matrix of a finite Coxeter group. For descriptions of the parameters `RealInjection` and `Nonreduced` see the beginning of this section. ## `Example: G2Root System (ex-ad34e4)` The following code creates a nonsemisimple root system of type $G_2$. ```magma > A := Matrix(2,3, [1,-1,0, -1,1,-1]); > B := Matrix(2,3, [1,-1,1, 0,1,-1]); > RootSystem(A, B); Root system of type G2 ``` ## `IrreducibleRootSystem(X, n): MonStgElt, RngIntElt -> RootSys` ```magma Symmetric: BoolElt Default: false BaseField: MonStgElt Default: "NumberField" ``` The irreducible root system with Cartan name $X_n$ (or $I_2(n)$ if $X=``I"$) given by the string $X$ and integer $n$. In addition to the Cartan names in Section [Finite and Affine Coxeter Groups](../CoxeterSystems/finiteaffine.md#sectcartanfinaff), we allow `"BCn"` for the irreducible nonreduced system. For descriptions of the parameters `Symmetric` and `BaseField` see the beginning of this section. ## `StandardRootSystem(X, n): MonStgElt, RngIntElt -> RootSys` The standard root system with Cartan name $X_n$ (or $I_2(n)$ if $X=``I"$) given by the string $X$ and integer $n$, i.e. the root system whose Coxeter form is the same as the standard inner product. In addition to the Cartan names in Section [Finite and Affine Coxeter Groups](../CoxeterSystems/finiteaffine.md#sectcartanfinaff), we allow `"BCn"` for the irreducible nonreduced system. For type $A_n$, the standard root system is not semisimple. ## `Example: Irreducible Root System (ex-28a4fc)` ```magma > Rs := { IrreducibleRootSystem("I", n) : n in [3..20] }; > { R : R in Rs | IsCrystallographic(R) }; { Root system of type I2(3) , Root system of type I2(4) , Root system of type I2(6) } ``` ## `ToralRootSystem(n): RngIntElt -> RootSys` The toral root system of dimension $n$, i.e., the $n$-dimensional root system with no roots or coroots. ## `TrivialRootSystem() -> RootSys` The trivial root system of dimension $0$.