CartanType
A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.
An irreducible finite Cartan type, written with its letter and rank, as in CartanType(Tuple("A", 3)).
CartanType(...)An irreducible finite Cartan type, written with its letter and rank, as in CartanType(Tuple("A", 3)).Typed signatures
| Type | Package | Applies |
|---|---|---|
(tuple<string, integer>) -> cartan_type | combinatorics | always |
Statistics
On CartanType, called as CombinatorialStat(x, "Name").
- AdjointRepresentationDimensionSt001148The dimension of the adjoint representation of the Lie algebra: n plus twice the number of positive roots.
- AlmostPositiveRootCountSt000853The number of almost positive roots: the positive roots and the negatives of the simple roots.
- AtomicLengthOfLongestElementSt001849The atomic length of the longest element of the Weyl group: the sum of the heights of the positive roots.
- CartanMatrixThe Cartan matrix of the type in Bourbaki's numbering: the entry at row i and column j is the pairing of the simple root i with the coroot j.
- CartanMatrixDeterminantSt000821The determinant of the Cartan matrix, the order of the center of the simply connected group.
- CartanMatrixPermanentSt000960The permanent of the Cartan matrix: the sum over the ways to choose one entry from each row and column.
- CartanParkingFunctionCountSt000859The parking functions of the type, the points of the torus Q / (h + 1) Q: (h + 1)^n.
- CartanRankSt000113The rank of the type: the number of simple roots.
- CommutativePositiveRootCountSt001173The commutative positive roots: those whose upper ideal in the root poset is abelian.
- CoxeterElementCountSt000865The Coxeter elements of the Weyl group, one conjugacy class of size |W| / h.
- CoxeterElementFactorizationCountSt000858The ways to write a Coxeter element as a product of n reflections: n! h^n / |W|.
- CoxeterNumberSt000139The Coxeter number of the type: the largest degree of its Weyl group.
- DualCoxeterNumberSt001154The dual Coxeter number of the type.
- DynkinIndexSt001156The Dynkin index of the Lie algebra: the greatest common divisor of the indices of its representations.
- ExponentSquareSumSt001680The sum of the squares of the exponents of the Weyl group.
- FirstFundamentalRepresentationDimensionSt000107The dimension of the representation with highest weight the first fundamental weight, in Bourbaki's numbering.
- FullSupportReflectionCountSt000855The reflections whose reduced words use every simple reflection: n h / |W| times the product of the codegrees d - 2 (leaving out the degree 2).
- FullyCommutativeElementCountSt001653The fully commutative elements of the Weyl group: the Catalan number C(n + 1) for A_n, (n + 2) C(n) - 1 for B_n and C_n, (n + 3) C(n) / 2 - 1 for D_n (Stembridge).
- GammaNumberSt001676The gamma number of the Weyl group, from the sum of the squares of its exponents e: γ = 6 Σ e² / n - h² + h.
- HighestRootMaximalCoefficientSt001369The largest coefficient of the highest root in the simple roots.
- HighestRootWeylVectorPairingSt001678The symmetric bilinear form of the highest root with the Weyl vector, normalized so that the shortest roots have squared length 2.
- MinimalFaithfulRepresentationDimensionSt001150The smallest dimension of a faithful representation of the Lie algebra.
- MinusculeWeightCountSt001147The minuscule dominant weights: the simple roots that appear once in the highest root.
- PoincarePolynomialMaximalCoefficientSt001443The largest coefficient of the Poincaré polynomial ∏ [d]_q of the Weyl group.
- PositiveWeylCatalanNumberSt000140The positive Catalan number of the type, ∏ (d + h - 2) / d.
- QuasiMinusculeRepresentationDimensionSt001149The dimension of the quasi-minuscule representation, whose highest weight is the highest short root.
- ReflectionCountSt000857The number of reflections in the Weyl group, which is the number of positive roots: the sum of the exponents.
- ReflectionOrbitCountSt000854The conjugacy classes of reflections in the Weyl group: 2 where the simple roots have two lengths (B, C, F, G), else 1.
- RootPosetRowmotionOrderSt001887The order of rowmotion on the root poset: h where -1 is in the Weyl group, else 2 h.
- SecondPositiveWeylFussCatalanNumberSt001053The positive Fuss-Catalan number of order 2 of the type, ∏ (d + 2h - 2) / d.
- SecondWeylFussCatalanNumberSt000852The Fuss-Catalan number of order 2 of the type, ∏ (d + 2h) / d.
- ThirdPositiveWeylFussCatalanNumberSt001054The positive Fuss-Catalan number of order 3 of the type, ∏ (d + 3h - 2) / d.
- ThirdWeylFussCatalanNumberSt000851The Fuss-Catalan number of order 3 of the type, ∏ (d + 3h) / d.
- WeylCatalanNumberSt000138The Catalan number of the type, ∏ (d + h) / d over the degrees d and the Coxeter number h.
- WeylGroupCenterOrderSt000860The size of the center of the Weyl group: 2 when -1 is in the group (every degree is even), else 1.
- WeylGroupExponentsThe exponents of the Weyl group of the type: one less than each degree.
- WeylGroupOrderSt000106The size of the Weyl group of the type: the product of its degrees.
Maps
Into and out of CartanType, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.
1 more in FindStat, not implemented here
- Mp00148CartanType → PosetsInjectionto root poset