# Coxeter Systems - [Introduction](introduction.md) - [Coxeter Matrices](coxetermat.md) - [`IsCoxeterMatrix(M): AlgMatElt → BoolElt`](coxetermat.md#function-iscoxetermatrix-algmatelt) - [`CoxeterMatrix(G): GrphUnd → AlgMatElt`](coxetermat.md#function-coxetermatrix-grphund) - [`CoxeterMatrix(C): AlgMatElt → AlgMatElt`](coxetermat.md#function-coxetermatrix-algmatelt) - [`CoxeterMatrix(D): GrphDir → AlgMatElt`](coxetermat.md#function-coxetermatrix-grphdir) - [`Example: Coxeter Matrix Construction`](coxetermat.md#example-ex-8161cb) - [`IsCoxeterIsomorphic(M1, M2): AlgMatElt, AlgMatElt → BoolElt, SeqEnum`](coxetermat.md#function-iscoxeterisomorphic-algmatelt-algmatelt) - [`CoxeterGroupOrder(M): AlgMatElt → RngIntElt`](coxetermat.md#function-coxetergrouporder-algmatelt) - [`CoxeterGroupFactoredOrder(M): AlgMatElt → SeqEnum`](coxetermat.md#function-coxetergroupfactoredorder-algmatelt) - [`Example: Coxeter Matrix Operations`](coxetermat.md#example-ex-84a070) - [`IsCoxeterIrreducible(M): AlgMatElt → BoolElt`](coxetermat.md#function-iscoxeterirreducible-algmatelt) - [`IsSimplyLaced(M): AlgMatElt → BoolElt`](coxetermat.md#function-issimplylaced-algmatelt) - [`Example: Coxeter Matrix Properties`](coxetermat.md#example-ex-e19461) - [Coxeter Graphs](coxetergraph.md) - [`IsCoxeterGraph(G): GrphUnd → BoolElt`](coxetergraph.md#function-iscoxetergraph-grphund) - [`CoxeterGraph(M): AlgMatElt → GrphUnd`](coxetergraph.md#function-coxetergraph-algmatelt) - [`CoxeterGraph(C): AlgMatElt → GrphUnd`](coxetergraph.md#function-coxetergraph-algmatelt-2) - [`CoxeterGraph(D): GrphDir → GrphUnd`](coxetergraph.md#function-coxetergraph-grphdir) - [`CoxeterGroupOrder(G): GrphUnd → .`](coxetergraph.md#function-coxetergrouporder-grphund) - [`CoxeterGroupFactoredOrder(G): GrphUnd → .`](coxetergraph.md#function-coxetergroupfactoredorder-grphund) - [`Example: Coxeter Graph Operations`](coxetergraph.md#example-ex-a98e43) - [`IsSimplyLaced(G): GrphUnd → BoolElt`](coxetergraph.md#function-issimplylaced-grphund) - [`Example: Coxeter Graph Properties`](coxetergraph.md#example-ex-9639e1) - [Cartan Matrices](cartanmat.md) - [`IsCartanMatrix(C): AlgMatElt → BoolElt`](cartanmat.md#function-intriscart) - [`CartanMatrix(M): AlgMatElt → AlgMatElt`](cartanmat.md#function-cartanmatrix-algmatelt) - [`CartanMatrix(G): GrphUnd → AlgMatElt`](cartanmat.md#function-cartanmatrix-grphund) - [`CartanMatrix(D): GrphDir → AlgMatElt`](cartanmat.md#function-cartanmatrix-grphdir) - [`Example: Cartan Matrix Construction`](cartanmat.md#example-ex-0d7e25) - [`IsCoxeterIsomorphic(C1, C2): AlgMatElt, AlgMatElt → RngIntElt`](cartanmat.md#function-iscoxeterisomorphic-algmatelt-algmatelt-2) - [`IsCartanEquivalent(C1, C2): AlgMatElt, AlgMatElt → BoolElt`](cartanmat.md#function-iscartanequivalent-algmatelt-algmatelt) - [`Example: Cartan Matrix Equivalence`](cartanmat.md#example-ex-1ab47d) - [`NumberOfPositiveRoots(C): AlgMatElt → RngIntElt`](cartanmat.md#function-numberofpositiveroots-algmatelt) - [`NumPosRoots(C): AlgMatElt → RngIntElt`](cartanmat.md#function-numposroots-algmatelt) - [`CoxeterGroupOrder(C): AlgMatElt → .`](cartanmat.md#function-coxetergrouporder-algmatelt-2) - [`CoxeterGroupFactoredOrder(C): AlgMatElt → .`](cartanmat.md#function-coxetergroupfactoredorder-algmatelt-2) - [`FundamentalGroup(C): AlgMatElt → GrpAb`](cartanmat.md#function-fundamentalgroup-algmatelt) - [`Example: Cartan Matrix Operations`](cartanmat.md#example-ex-918786) - [`IsCoxeterIrreducible(C): AlgMatElt → BoolElt`](cartanmat.md#function-iscoxeterirreducible-algmatelt-2) - [`IsCrystallographic(C): AlgMatElt → BoolElt`](cartanmat.md#function-iscrystallographic-algmatelt) - [`IsSimplyLaced(C): AlgMatElt → BoolElt`](cartanmat.md#function-issimplylaced-algmatelt-2) - [`Example: Cartan Matrix Properties`](cartanmat.md#example-ex-839a84) - [Dynkin Digraphs](dynkindigraph.md) - [`IsDynkinDigraph(D): GrphDir → BoolElt`](dynkindigraph.md#function-isdynkindigraph-grphdir) - [`DynkinDigraph(C): AlgMatElt → GrphDir`](dynkindigraph.md#function-dynkindigraph-algmatelt) - [`CoxeterGroupOrder(D): GrphDir → .`](dynkindigraph.md#function-coxetergrouporder-grphdir) - [`CoxeterGroupFactoredOrder(D): GrphDir → .`](dynkindigraph.md#function-coxetergroupfactoredorder-grphdir) - [`FundamentalGroup(D): GrphDir → GrpAb`](dynkindigraph.md#function-fundamentalgroup-grphdir) - [`IsSimplyLaced(D): GrphDir → BoolElt`](dynkindigraph.md#function-issimplylaced-grphdir) - [`Example: Cartan Matrices`](dynkindigraph.md#example-ex-9a3784) - [Finite and Affine Coxeter Groups](finiteaffine.md) - [`IsCoxeterFinite(M): AlgMatElt → BoolElt`](finiteaffine.md#function-iscoxeterfinite-algmatelt) - [`IsCoxeterFinite(G): GrphUnd → BoolElt`](finiteaffine.md#function-iscoxeterfinite-grphund) - [`IsCoxeterFinite(C): AlgMatElt → BoolElt`](finiteaffine.md#function-iscoxeterfinite-algmatelt-2) - [`IsCoxeterFinite(D): GrphDir → BoolElt`](finiteaffine.md#function-iscoxeterfinite-grphdir) - [`IsCoxeterFinite(N): MonStgElt → BoolElt`](finiteaffine.md#function-iscoxeterfinite-monstgelt) - [`IsCoxeterAffine(M): AlgMatElt → BoolElt`](finiteaffine.md#function-iscoxeteraffine-algmatelt) - [`IsCoxeterAffine(G): GrphUnd → BoolElt`](finiteaffine.md#function-iscoxeteraffine-grphund) - [`IsCoxeterAffine(C): AlgMatElt → BoolElt`](finiteaffine.md#function-iscoxeteraffine-algmatelt-2) - [`IsCoxeterAffine(D): GrphDir → BoolElt`](finiteaffine.md#function-iscoxeteraffine-grphdir) - [`IsCoxeterAffine(N): MonStgElt → BoolElt`](finiteaffine.md#function-iscoxeteraffine-monstgelt) - [`Example: Testing`](finiteaffine.md#example-ex-ebe4bc) - [`CoxeterMatrix(N): MonStgElt → AlgMatElt`](finiteaffine.md#function-coxetermatrix-monstgelt) - [`CoxeterGraph(N): MonStgElt → GrpUnd`](finiteaffine.md#function-coxetergraph-monstgelt) - [`CartanMatrix(N): MonStgElt → AlgMatElt`](finiteaffine.md#function-cartanmatrix-monstgelt) - [`DynkinDigraph(N): MonStgElt → GrphDir`](finiteaffine.md#function-dynkindigraph-monstgelt) - [`Example: Matrices And Graphs`](finiteaffine.md#example-ex-f6d3c0) - [`IrreducibleCoxeterMatrix(X, n): MonStgElt, RngIntElt → AlgMatElt`](finiteaffine.md#function-irreduciblecoxetermatrix-monstgelt-rngintelt) - [`IrreducibleCoxeterGraph(X, n): MonStgElt, RngIntElt → GrpUnd`](finiteaffine.md#function-irreduciblecoxetergraph-monstgelt-rngintelt) - [`IrreducibleCartanMatrix(X, n): MonStgElt, RngIntElt → AlgMatElt`](finiteaffine.md#function-irreduciblecartanmatrix-monstgelt-rngintelt) - [`IrreducibleDynkinDigraph(X, n): MonStgElt, RngIntElt → GrphDir`](finiteaffine.md#function-irreducibledynkindigraph-monstgelt-rngintelt) - [`Example: Irreducible Coxeter`](finiteaffine.md#example-ex-fc81db) - [`IsCoxeterIsomorphic(N1, N2): MonStgElt, MonStgElt → BoolElt`](finiteaffine.md#function-iscoxeterisomorphic-monstgelt-monstgelt) - [`IsCartanEquivalent(N1, N2): MonStgElt, MonStgElt → BoolElt`](finiteaffine.md#function-iscartanequivalent-monstgelt-monstgelt) - [`Example: Isomorphism And Equivalence`](finiteaffine.md#example-ex-d3e86e) - [`IsSimplyLaced(N): MonStgElt → BoolElt`](finiteaffine.md#function-issimplylaced-monstgelt) - [`CoxeterGroupOrder(N): MonStgElt → .`](finiteaffine.md#function-coxetergrouporder-monstgelt) - [`CoxeterGroupFactoredOrder(N): MonStgElt → .`](finiteaffine.md#function-coxetergroupfactoredorder-monstgelt) - [`NumberOfPositiveRoots(N): MonStgElt → .`](finiteaffine.md#function-numberofpositiveroots-monstgelt) - [`NumPosRoots(N): MonStgElt → .`](finiteaffine.md#function-numposroots-monstgelt) - [`FundamentalGroup(N): MonStgElt → GrpAb`](finiteaffine.md#function-fundamentalgroup-monstgelt) - [`Example: Group Orders`](finiteaffine.md#example-ex-d07ed7) - [`CartanName(M): AlgMatElt → MonStgElt`](finiteaffine.md#function-cartanname-algmatelt) - [`CartanName(G): GrphUnd → MonStgElt`](finiteaffine.md#function-cartanname-grphund) - [`CartanName(C): AlgMatElt → MonStgElt`](finiteaffine.md#function-cartanname-algmatelt-2) - [`CartanName(D): GrphDir → MonStgElt`](finiteaffine.md#function-cartanname-grphdir) - [`Example: Cartan Name`](finiteaffine.md#example-ex-8583c3) - [`DynkinDiagram(M): AlgMatElt`](finiteaffine.md#function-dynkindiagram-algmatelt) - [`DynkinDiagram(G): GrphUnd`](finiteaffine.md#function-dynkindiagram-grphund) - [`DynkinDiagram(C): AlgMatElt`](finiteaffine.md#function-dynkindiagram-algmatelt-2) - [`DynkinDiagram(D): GrphDir`](finiteaffine.md#function-dynkindiagram-grphdir) - [`DynkinDiagram(N): MonStgElt`](finiteaffine.md#function-dynkindiagram-monstgelt) - [`Example: Dynkin Diagram`](finiteaffine.md#example-ex-08ba7b) - [`CoxeterDiagram(M): AlgMatElt`](finiteaffine.md#function-coxeterdiagram-algmatelt) - [`CoxeterDiagram(G): GrphUnd`](finiteaffine.md#function-coxeterdiagram-grphund) - [`CoxeterDiagram(C): AlgMatElt`](finiteaffine.md#function-coxeterdiagram-algmatelt-2) - [`CoxeterDiagram(D): GrphDir`](finiteaffine.md#function-coxeterdiagram-grphdir) - [`CoxeterDiagram(N): MonStgElt`](finiteaffine.md#function-coxeterdiagram-monstgelt) - [`Example: Coxeter Diagram`](finiteaffine.md#example-ex-f3140d) - [Hyperbolic Groups](related.md) - [`IsCoxeterHyperbolic(M): AlgMatElt → BoolElt`](related.md#function-iscoxeterhyperbolic-algmatelt) - [`IsCoxeterCompactHyperbolic(M): AlgMatElt → BoolElt`](related.md#function-iscoxetercompacthyperbolic-algmatelt) - [`IsCoxeterHyperbolic(G): GrphUnd → BoolElt`](related.md#function-iscoxeterhyperbolic-grphund) - [`IsCoxeterCompactHyperbolic(G): GrphUnd → BoolElt`](related.md#function-iscoxetercompacthyperbolic-grphund) - [`HyperbolicCoxeterMatrix(i): RngIntElt → AlgMatElt`](related.md#function-hyperboliccoxetermatrix-rngintelt) - [`HyperbolicCoxeterGraph(i): RngIntElt → GrphUnd`](related.md#function-hyperboliccoxetergraph-rngintelt) - [`Example: Hyperbolic`](related.md#example-ex-4b6e52) - [Related Structures](related-2.md) - [`RootSystem(M): AlgMatElt → RootSys`](related-2.md#function-rootsystem-algmatelt) - [`RootSystem(G): GrphUnd → RootSys`](related-2.md#function-rootsystem-grphund) - [`RootSystem(C): AlgMatElt → RootSys`](related-2.md#function-rootsystem-algmatelt-2) - [`RootSystem(D): GrphDir → RootSys`](related-2.md#function-rootsystem-grphdir) - [`RootSystem(N): MonStgElt → RootSys`](related-2.md#function-rootsystem-monstgelt) - [`RootDatum(C): AlgMatElt → RootDtm`](related-2.md#function-rootdatum-algmatelt) - [`RootDatum(M): AlgMatElt → RootDtm`](related-2.md#function-rootdatum-algmatelt-2) - [`RootDatum(G): GrphUnd → RootDtm`](related-2.md#function-rootdatum-grphund) - [`RootDatum(D): GrphDir → RootDtm`](related-2.md#function-rootdatum-grphdir) - [`RootDatum(N): MonStgElt → RootDtm`](related-2.md#function-rootdatum-monstgelt) - [`CoxeterGroup(grpcat, M): Cat, AlgMatElt → grpcat`](related-2.md#function-coxetergroup-cat-algmatelt) - [`CoxeterGroup(grpcat, G): Cat, GrphUnd → grpcat`](related-2.md#function-coxetergroup-cat-grphund) - [`CoxeterGroup(grpcat, D): Cat, GrphDir → grpcat`](related-2.md#function-coxetergroup-cat-grphdir) - [`CoxeterGroup(grpcat, N): Cat, MonStgElt → grpcat`](related-2.md#function-coxetergroup-cat-monstgelt) - [`CoxeterGroup(M): AlgMatElt → Grp`](related-2.md#function-coxetergroup-algmatelt) - [`CoxeterGroup(C): AlgMatElt → Grp`](related-2.md#function-coxetergroup-algmatelt-2) - [`CoxeterGroup(G): GrphUnd → Grp`](related-2.md#function-coxetergroup-grphund) - [`CoxeterGroup(D): GrphDir → Grp`](related-2.md#function-coxetergroup-grphdir) - [`CoxeterGroup(N): MonStgElt → Grp`](related-2.md#function-coxetergroup-monstgelt) - [`ReflectionGroup(M): AlgMatElt → GrpMat`](related-2.md#function-reflectiongroup-algmatelt) - [`ReflectionGroup(G): GrphUnd → GrpMat`](related-2.md#function-reflectiongroup-grphund) - [`ReflectionGroup(C): AlgMatElt → GrpMat`](related-2.md#function-reflectiongroup-algmatelt-2) - [`ReflectionGroup(D): GrphDir → GrpMat`](related-2.md#function-reflectiongroup-grphdir) - [`ReflectionGroup(N): MonStgElt → GrpMat`](related-2.md#function-reflectiongroup-monstgelt) - [`LieAlgebra(C, k): AlgMatElt, Rng → AlgLie`](related-2.md#function-liealgebra-algmatelt-rng) - [`LieAlgebra(D, k): GrphDir, Rng → AlgLie`](related-2.md#function-liealgebra-grphdir-rng) - [`LieAlgebra(N, k): MonStgElt, Rng → AlgLie`](related-2.md#function-liealgebra-monstgelt-rng) - [`MatrixLieAlgebra(C, k): AlgMatElt, Rng → AlgLie`](related-2.md#function-matrixliealgebra-algmatelt-rng) - [`MatrixLieAlgebra(D, k): GrphDir, Rng → AlgLie`](related-2.md#function-matrixliealgebra-grphdir-rng) - [`MatrixLieAlgebra(N, k): MonStgElt, Rng → AlgLie`](related-2.md#function-matrixliealgebra-monstgelt-rng) - [`GroupOfLieType(C, k): AlgMatElt, Rng → GrpLie`](related-2.md#function-groupoflietype-algmatelt-rng) - [`GroupOfLieType(D, k): GrphDir, Rng → GrpLie`](related-2.md#function-groupoflietype-grphdir-rng) - [`GroupOfLieType(N, k): MonStgElt, Rng → GrpLie`](related-2.md#function-groupoflietype-monstgelt-rng)