FiniteCartanTypes

The irreducible finite Cartan types of a rank, as a lazy indexed family in FindStat's order.

FiniteCartanTypes(rank)the irreducible finite Cartan types of the given rank, ordered by letter.

Typed signatures

Type	Package	Applies
(integer<0..>) -> indexed_collection<cartan_type>	combinatorics	always

Domain: Collections

Details

• The types of rank $n$ are $A_n$ for $n \ge 1$, $B_n$ for $n \ge 2$, $C_n$ for $n \ge 3$, $D_n$ for $n \ge 4$, $E_n$ for $n = 6, 7, 8$, $F_4$ and $G_2$, ordered by letter: FindStat's collection Finite Cartan types. $C_2$ is $B_2$ and $B_1$, $C_1$ are $A_1$, so they are left out.
• An element is written CartanType(Tuple("A", 3)): the letter and the rank, FindStat's ['A', 3].

Definition

Epsil, as the implementation runs it.

count(_n)

Length(Filter(
               Range(1, 7, 1),
               Function(
                         _n >= 1 if c == 1 else _n >= 2 if c == 2 else _n >= 3 if c == 3 else _n >= 4 if c == 4 else _n >= 6 &&
                                                                                                                     _n <= 8 if c == 5 else _n == 4 if c == 6 else _n == 2,
                         c,
                       ),
             ))

unrank(_n, _r)

[Filter(
         Range(1, 7, 1),
         Function(
                   _n >= 1 if c == 1 else _n >= 2 if c == 2 else _n >= 3 if c ==
                                                                            3 else _n >= 4 if c == 4 else _n >= 6 &&
                                                                                                         _n <= 8 if c == 5 else _n == 4 if c == 6 else _n == 2,
                   c,
                 ),
       )[_r + 1], _n]

rank(_n, _x)

Length(Filter(Filter(
                      Range(1, 7, 1),
                      Function(
                                _n >= 1 if c == 1 else _n >= 2 if c == 2 else _n >=
                                                                              3 if c == 3 else _n >= 4 if c == 4 else _n >= 6 &&
                                                                                                                     _n <= 8 if c == 5 else _n == 4 if c == 6 else _n == 2,
                                c,
                              ),
                    ), Function(c < _x[1], c)))

valid(_n, _x)

_x[2] == _n && Length(Filter(Filter(
                                     Range(1, 7, 1),
                                     Function(
                                               _n >= 1 if c == 1 else _n >= 2 if c == 2 else _n >= 3 if c == 3 else _n >= 4 if c == 4 else _n >= 6 &&
                                                                                                                                           _n <= 8 if c == 5 else _n == 4 if c == 6 else _n == 2,
                                               c,
                                             ),
                                   ), Function(c == _x[1], c))) >= 1 if Length(_x) ==
                                                                       2 else False

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

Enumeration

About

See also: CartanType, WeylGroupOrder, CoxeterNumber