Fast Rank Two Crystals¶

class
sage.combinat.crystals.fast_crystals.
FastCrystal
(ct, shape, format)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
An alternative implementation of rank 2 crystals. The root operators are implemented in memory by table lookup. This means that in comparison with the
CrystalsOfTableaux
, these crystals are slow to instantiate but faster for computation. Implemented for types \(A_2\), \(B_2\), and \(C_2\).INPUT:
cartan_type
– the Cartan type and must be either type \(A_2\), \(B_2\), or \(C_2\)shape
– A shape is of the form[l1,l2]
wherel1
andl2
are either integers or (in type \(B_2\)) half integers such thatl1  l2
is integral. It is assumed thatl1 >= l2 >= 0
. Ifl1
andl2` are integers, this will produce a crystal isomorphic to the one obtained by ``crystals.Tableaux(type, shape=[l1,l2])
. Furthermorecrystals.FastRankTwo(['B', 2], l1+1/2, l2+1/2)
produces a crystal isomorphic to the following crystalT
:sage: C = crystals.Tableaux(['B',2], shape=[l1,l2]) # not tested sage: D = crystals.Spins(['B',2]) # not tested sage: T = crystals.TensorProduct(C, D, C.list()[0], D.list()[0]) # not tested
format
– (default:'string'
) the default representation of elements is in term of theBerensteinZelevinskyLittelmann (BZL) strings[a1, a2, ...]
described under metapost incrystals
. Alternative representations may be obtained by the options'dual_string'
or'simple'
. In the'simple'
format, the element is represented by and integer, and in the'dual_string'
format, it is represented by the BZL string, but the underlying decomposition of the long Weyl group element into simple reflections is changed.

class
Element
(parent, value, format)¶ Bases:
sage.structure.element.Element
EXAMPLES:
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1]) sage: c = C(0); c [0, 0, 0] sage: C[0].parent() The fast crystal for A2 with shape [2,1] sage: TestSuite(c).run()

e
(i)¶ Returns the action of \(e_i\) on self.
EXAMPLES:
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1]) sage: C(1).e(1) [0, 0, 0] sage: C(0).e(1) is None True

f
(i)¶ Returns the action of \(f_i\) on self.
EXAMPLES:
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1]) sage: C(6).f(1) [1, 2, 1] sage: C(7).f(1) is None True

weight
()¶ Returns the weight of self.
EXAMPLES:
sage: [v.weight() for v in crystals.FastRankTwo(['A',2], shape=[2,1])] [(2, 1, 0), (1, 2, 0), (1, 1, 1), (1, 0, 2), (0, 1, 2), (2, 0, 1), (1, 1, 1), (0, 2, 1)] sage: [v.weight() for v in crystals.FastRankTwo(['B',2], shape=[1,0])] [(1, 0), (0, 1), (0, 0), (0, 1), (1, 0)] sage: [v.weight() for v in crystals.FastRankTwo(['B',2], shape=[1/2,1/2])] [(1/2, 1/2), (1/2, 1/2), (1/2, 1/2), (1/2, 1/2)] sage: [v.weight() for v in crystals.FastRankTwo(['C',2], shape=[1,0])] [(1, 0), (0, 1), (0, 1), (1, 0)] sage: [v.weight() for v in crystals.FastRankTwo(['C',2], shape=[1,1])] [(1, 1), (1, 1), (0, 0), (1, 1), (1, 1)]


cmp_elements
(x, y)¶ Returns True if and only if there is a path from x to y in the crystal graph.
Because the crystal graph is classical, it is a directed acyclic graph which can be interpreted as a poset. This function implements the comparison function of this poset.
EXAMPLES:
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1]) sage: x = C(0) sage: y = C(1) sage: C.cmp_elements(x,y) 1 sage: C.cmp_elements(y,x) 1 sage: C.cmp_elements(x,x) 0

digraph
()¶ Returns the digraph associated to self.
EXAMPLES:
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1]) sage: C.digraph() Digraph on 8 vertices

list
()¶ Returns a list of the elements of self.
EXAMPLES:
sage: C = crystals.FastRankTwo(['A',2],shape=[2,1]) sage: C.list() [[0, 0, 0], [1, 0, 0], [0, 1, 1], [0, 2, 1], [1, 2, 1], [0, 1, 0], [1, 1, 0], [2, 1, 0]]