# Regular Crystals¶

class sage.categories.regular_crystals.RegularCrystals(s=None)

The category of regular crystals.

A crystal is called regular if every vertex $$b$$ satisfies

$\varepsilon_i(b) = \max\{ k \mid e_i^k(b) \neq 0 \} \quad \text{and} \quad \varphi_i(b) = \max\{ k \mid f_i^k(b) \neq 0 \}.$

Note

Regular crystals are sometimes referred to as normal. When only one of the conditions (on either $$\varphi_i$$ or $$\varepsilon_i$$) holds, these crystals are sometimes called seminormal or semiregular.

EXAMPLES:

sage: C = RegularCrystals()
sage: C
Category of regular crystals
sage: C.super_categories()
[Category of crystals]
sage: C.example()
Highest weight crystal of type A_3 of highest weight omega_1

class ElementMethods

Bases: object

demazure_operator_simple(i, ring=None)

Return the Demazure operator $$D_i$$ applied to self.

INPUT:

• i – an element of the index set of the underlying crystal
• ring – (default: QQ) a ring

OUTPUT:

An element of the ring-free module indexed by the underlying crystal.

Let $$r = \langle \mathrm{wt}(b), \alpha^{\vee}_i \rangle$$, then $$D_i(b)$$ is defined as follows:

• If $$r \geq 0$$, this returns the sum of the elements obtained from self by application of $$f_i^k$$ for $$0 \leq k \leq r$$.
• If $$r < 0$$, this returns the opposite of the sum of the elements obtained by application of $$e_i^k$$ for $$0 < k < -r$$.

REFERENCES:

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: t = T(rows=[[1,2],])
sage: t.demazure_operator_simple(2)
B[[[1, 2], ]] + B[[[1, 3], ]] + B[[[1, 3], ]]
sage: t.demazure_operator_simple(2).parent()
Algebra of The crystal of tableaux of type ['A', 2] and shape(s) [[2, 1]]
over Integer Ring

sage: t.demazure_operator_simple(1)
0

sage: K = crystals.KirillovReshetikhin(['A',2,1],2,1)
sage: t = K(rows=[,])
sage: t.demazure_operator_simple(0)
B[[[1, 2]]] + B[[[2, 3]]]

dual_equivalence_class(index_set=None)

Return the dual equivalence class indexed by index_set of self.

The dual equivalence class of an element $$b \in B$$ is the set of all elements of $$B$$ reachable from $$b$$ via sequences of $$i$$-elementary dual equivalence relations (i.e., $$i$$-elementary dual equivalence transformations and their inverses) for $$i$$ in the index set of $$B$$.

For this to be well-defined, the element $$b$$ has to be of weight $$0$$ with respect to $$I$$; that is, we need to have $$\varepsilon_j(b) = \varphi_j(b)$$ for all $$j \in I$$.

See [As2008]. See also dual_equivalence_graph() for a definition of $$i$$-elementary dual equivalence transformations.

INPUT:

• index_set – (optional) the index set $$I$$ (default: the whole index set of the crystal); this has to be a subset of the index set of the crystal (as a list or tuple)

OUTPUT:

The dual equivalence class of self indexed by the subset index_set. This class is returned as an undirected edge-colored multigraph. The color of an edge is the index $$i$$ of the dual equivalence relation it encodes.

EXAMPLES:

sage: T = crystals.Tableaux(['A',3], shape=[2,2])
sage: G = T(2,1,4,3).dual_equivalence_class()
sage: sorted(G.edges())
[([[1, 3], [2, 4]], [[1, 2], [3, 4]], 2),
([[1, 3], [2, 4]], [[1, 2], [3, 4]], 3)]
sage: T = crystals.Tableaux(['A',4], shape=[3,2])
sage: G = T(2,1,4,3,5).dual_equivalence_class()
sage: sorted(G.edges())
[([[1, 3, 5], [2, 4]], [[1, 3, 4], [2, 5]], 4),
([[1, 3, 5], [2, 4]], [[1, 2, 5], [3, 4]], 2),
([[1, 3, 5], [2, 4]], [[1, 2, 5], [3, 4]], 3),
([[1, 3, 4], [2, 5]], [[1, 2, 4], [3, 5]], 2),
([[1, 2, 4], [3, 5]], [[1, 2, 3], [4, 5]], 3),
([[1, 2, 4], [3, 5]], [[1, 2, 3], [4, 5]], 4)]

epsilon(i)

Return $$\varepsilon_i$$ of self.

EXAMPLES:

sage: C = crystals.Letters(['A',5])
sage: C(1).epsilon(1)
0
sage: C(2).epsilon(1)
1

phi(i)

Return $$\varphi_i$$ of self.

EXAMPLES:

sage: C = crystals.Letters(['A',5])
sage: C(1).phi(1)
1
sage: C(2).phi(1)
0

stembridgeDel_depth(i, j)

Return the difference in the $$j$$-depth of self and $$f_i$$ of self, where $$i$$ and $$j$$ are in the index set of the underlying crystal. This function is useful for checking the Stembridge local axioms for crystal bases.

The $$i$$-depth of a crystal node $$x$$ is $$\varepsilon_i(x)$$.

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: t=T(rows=[[1,1],])
sage: t.stembridgeDel_depth(1,2)
0
sage: s=T(rows=[[1,3],])
sage: s.stembridgeDel_depth(1,2)
-1

stembridgeDel_rise(i, j)

Return the difference in the $$j$$-rise of self and $$f_i$$ of self, where $$i$$ and $$j$$ are in the index set of the underlying crystal. This function is useful for checking the Stembridge local axioms for crystal bases.

The $$i$$-rise of a crystal node $$x$$ is $$\varphi_i(x)$$.

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: t=T(rows=[[1,1],])
sage: t.stembridgeDel_rise(1,2)
-1
sage: s=T(rows=[[1,3],])
sage: s.stembridgeDel_rise(1,2)
0

stembridgeDelta_depth(i, j)

Return the difference in the $$j$$-depth of self and $$e_i$$ of self, where $$i$$ and $$j$$ are in the index set of the underlying crystal. This function is useful for checking the Stembridge local axioms for crystal bases.

The $$i$$-depth of a crystal node $$x$$ is $$-\varepsilon_i(x)$$.

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: t=T(rows=[[1,2],])
sage: t.stembridgeDelta_depth(1,2)
0
sage: s=T(rows=[[2,3],])
sage: s.stembridgeDelta_depth(1,2)
-1

stembridgeDelta_rise(i, j)

Return the difference in the $$j$$-rise of self and $$e_i$$ of self, where $$i$$ and $$j$$ are in the index set of the underlying crystal. This function is useful for checking the Stembridge local axioms for crystal bases.

The $$i$$-rise of a crystal node $$x$$ is $$\varphi_i(x)$$.

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: t=T(rows=[[1,2],])
sage: t.stembridgeDelta_rise(1,2)
-1
sage: s=T(rows=[[2,3],])
sage: s.stembridgeDelta_rise(1,2)
0

stembridgeTriple(i, j)

Let $$A$$ be the Cartan matrix of the crystal, $$x$$ a crystal element, and let $$i$$ and $$j$$ be in the index set of the crystal. Further, set b=stembridgeDelta_depth(x,i,j), and c=stembridgeDelta_rise(x,i,j)). If x.e(i) is non-empty, this function returns the triple $$( A_{ij}, b, c )$$; otherwise it returns None. By the Stembridge local characterization of crystal bases, one should have $$A_{ij}=b+c$$.

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: t=T(rows=[[1,1],])
sage: t.stembridgeTriple(1,2)
sage: s=T(rows=[[1,2],])
sage: s.stembridgeTriple(1,2)
(-1, 0, -1)

sage: T = crystals.Tableaux(['B',2], shape=[2,1])
sage: t=T(rows=[[1,2],])
sage: t.stembridgeTriple(1,2)
(-2, 0, -2)
sage: s=T(rows=[[-1,-1],])
sage: s.stembridgeTriple(1,2)
(-2, -2, 0)
sage: u=T(rows=[[0,2],])
sage: u.stembridgeTriple(1,2)
(-2, -1, -1)

weight()

Return the weight of this crystal element.

EXAMPLES:

sage: C = crystals.Letters(['A',5])
sage: C(1).weight()
(1, 0, 0, 0, 0, 0)

class MorphismMethods

Bases: object

is_isomorphism()

Check if self is a crystal isomorphism, which is true if and only if this is a strict embedding with the same number of connected components.

EXAMPLES:

sage: La = RootSystem(['A',2,1]).weight_space(extended=True).fundamental_weights()
sage: B = crystals.LSPaths(La)
sage: La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()
sage: C = crystals.GeneralizedYoungWalls(2, La)
sage: H = Hom(B, C)
sage: from sage.categories.highest_weight_crystals import HighestWeightCrystalMorphism
sage: class Psi(HighestWeightCrystalMorphism):
....:     def is_strict(self):
....:         return True
sage: psi = Psi(H, C.module_generators)
sage: psi
['A', 2, 1] Crystal morphism:
From: The crystal of LS paths of type ['A', 2, 1] and weight Lambda
To:   Highest weight crystal of generalized Young walls of Cartan type ['A', 2, 1]
and highest weight Lambda
Defn: (Lambda,) |--> []
sage: psi.is_isomorphism()
True

class ParentMethods

Bases: object

demazure_operator(element, reduced_word)

Returns the application of Demazure operators $$D_i$$ for $$i$$ from reduced_word on element.

INPUT:

• element – an element of a free module indexed by the underlying crystal
• reduced_word – a reduced word of the Weyl group of the same type as the underlying crystal

OUTPUT:

• an element of the free module indexed by the underlying crystal

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: C = CombinatorialFreeModule(QQ,T)
sage: t = T.highest_weight_vector()
sage: b = 2*C(t)
sage: T.demazure_operator(b,[1,2,1])
2*B[[[1, 1], ]] + 2*B[[[1, 2], ]] + 2*B[[[1, 3], ]] + 2*B[[[1, 1], ]]
+ 2*B[[[1, 2], ]] + 2*B[[[1, 3], ]] + 2*B[[[2, 2], ]] + 2*B[[[2, 3], ]]


The Demazure operator is idempotent:

sage: T = crystals.Tableaux("A1",shape=)
sage: C = CombinatorialFreeModule(QQ,T)
sage: b = C(T.module_generators); b
B[[[1, 1, 1, 1]]]
sage: e = T.demazure_operator(b,); e
B[[[1, 1, 1, 1]]] + B[[[1, 1, 1, 2]]] + B[[[1, 1, 2, 2]]] + B[[[1, 2, 2, 2]]] + B[[[2, 2, 2, 2]]]
sage: e == T.demazure_operator(e,)
True

sage: all(T.demazure_operator(T.demazure_operator(C(t),),) == T.demazure_operator(C(t),) for t in T)
True

demazure_subcrystal(element, reduced_word, only_support=True)

Return the subcrystal corresponding to the application of Demazure operators $$D_i$$ for $$i$$ from reduced_word on element.

INPUT:

• element – an element of a free module indexed by the underlying crystal
• reduced_word – a reduced word of the Weyl group of the same type as the underlying crystal
• only_support – (default: True) only include arrows corresponding to the support of reduced_word

OUTPUT:

• the Demazure subcrystal

EXAMPLES:

sage: T = crystals.Tableaux(['A',2], shape=[2,1])
sage: t = T.highest_weight_vector()
sage: S = T.demazure_subcrystal(t, [1,2])
sage: list(S)
[[[1, 1], ], [[1, 2], ], [[1, 1], ],
[[1, 2], ], [[2, 2], ]]
sage: S = T.demazure_subcrystal(t, [2,1])
sage: list(S)
[[[1, 1], ], [[1, 2], ], [[1, 1], ],
[[1, 3], ], [[1, 3], ]]


We construct an example where we don’t only want the arrows indicated by the support of the reduced word:

sage: K = crystals.KirillovReshetikhin(['A',1,1], 1, 2)
sage: mg = K.module_generator()
sage: S = K.demazure_subcrystal(mg, )
sage: S.digraph().edges()
[([[1, 1]], [[1, 2]], 1), ([[1, 2]], [[2, 2]], 1)]
sage: S = K.demazure_subcrystal(mg, , only_support=False)
sage: S.digraph().edges()
[([[1, 1]], [[1, 2]], 1),
([[1, 2]], [[1, 1]], 0),
([[1, 2]], [[2, 2]], 1),
([[2, 2]], [[1, 2]], 0)]

dual_equivalence_graph(X=None, index_set=None, directed=True)

Return the dual equivalence graph indexed by index_set on the subset X of self.

Let $$b \in B$$ be an element of weight $$0$$, so $$\varepsilon_j(b) = \varphi_j(b)$$ for all $$j \in I$$, where $$I$$ is the indexing set. We say $$b'$$ is an $$i$$-elementary dual equivalence transformation of $$b$$ (where $$i \in I$$) if

• $$\varepsilon_i(b) = 1$$ and $$\varepsilon_{i-1}(b) = 0$$, and
• $$b' = f_{i-1} f_i e_{i-1} e_i b$$.

We can do the inverse procedure by interchanging $$i$$ and $$i-1$$ above.

Note

If the index set is not an ordered interval, we let $$i - 1$$ mean the index appearing before $$i$$ in $$I$$.

This definition comes from [As2008] Section 4 (where our $$\varphi_j(b)$$ and $$\varepsilon_j(b)$$ are denoted by $$\epsilon(b, j)$$ and $$-\delta(b, j)$$, respectively).

The dual equivalence graph of $$B$$ is defined to be the colored graph whose vertices are the elements of $$B$$ of weight $$0$$, and whose edges of color $$i$$ (for $$i \in I$$) connect pairs $$\{ b, b' \}$$ such that $$b'$$ is an $$i$$-elementary dual equivalence transformation of $$b$$.

Note

This dual equivalence graph is a generalization of $$\mathcal{G}\left(\mathcal{X}\right)$$ in [As2008] Section 4 except we do not require $$\varepsilon_i(b) = 0, 1$$ for all $$i$$.

This definition can be generalized by choosing a subset $$X$$ of the set of all vertices of $$B$$ of weight $$0$$, and restricting the dual equivalence graph to the vertex set $$X$$.

INPUT:

• X – (optional) the vertex set $$X$$ (default: the whole set of vertices of self of weight $$0$$)
• index_set – (optional) the index set $$I$$ (default: the whole index set of self); this has to be a subset of the index set of self (as a list or tuple)
• directed – (default: True) whether to have the dual equivalence graph be directed, where the head of an edge $$b - b'$$ is $$b$$ and the tail is $$b' = f_{i-1} f_i e_{i-1} e_i b$$)

EXAMPLES:

sage: T = crystals.Tableaux(['A',3], shape=[2,2])
sage: G = T.dual_equivalence_graph()
sage: sorted(G.edges())
[([[1, 3], [2, 4]], [[1, 2], [3, 4]], 2),
([[1, 2], [3, 4]], [[1, 3], [2, 4]], 3)]
sage: T = crystals.Tableaux(['A',4], shape=[3,2])
sage: G = T.dual_equivalence_graph()
sage: sorted(G.edges())
[([[1, 3, 5], [2, 4]], [[1, 3, 4], [2, 5]], 4),
([[1, 3, 5], [2, 4]], [[1, 2, 5], [3, 4]], 2),
([[1, 3, 4], [2, 5]], [[1, 2, 4], [3, 5]], 2),
([[1, 2, 5], [3, 4]], [[1, 3, 5], [2, 4]], 3),
([[1, 2, 4], [3, 5]], [[1, 2, 3], [4, 5]], 3),
([[1, 2, 3], [4, 5]], [[1, 2, 4], [3, 5]], 4)]

sage: T = crystals.Tableaux(['A',4], shape=[3,1])
sage: G = T.dual_equivalence_graph(index_set=[1,2,3])
sage: G.vertices()
[[[1, 3, 4], ], [[1, 2, 4], ], [[1, 2, 3], ]]
sage: G.edges()
[([[1, 3, 4], ], [[1, 2, 4], ], 2),
([[1, 2, 4], ], [[1, 2, 3], ], 3)]

class TensorProducts(category, *args)

The category of regular crystals constructed by tensor product of regular crystals.

extra_super_categories()

EXAMPLES:

sage: RegularCrystals().TensorProducts().extra_super_categories()
[Category of regular crystals]

additional_structure()

Return None.

Indeed, the category of regular crystals defines no new structure: it only relates $$\varepsilon_a$$ and $$\varphi_a$$ to $$e_a$$ and $$f_a$$ respectively.

Todo

Should this category be a CategoryWithAxiom?

EXAMPLES:

sage: RegularCrystals().additional_structure()

example(n=3)

Returns an example of highest weight crystals, as per Category.example().

EXAMPLES:

sage: B = RegularCrystals().example(); B
Highest weight crystal of type A_3 of highest weight omega_1

super_categories()

EXAMPLES:

sage: RegularCrystals().super_categories()
[Category of crystals]