Elementary Crystals¶
Let \(\lambda\) be a weight. The crystals \(T_{\lambda}\), \(R_{\lambda}\), \(B_i\), and \(C\) are important objects in the tensor category of crystals. For example, the crystal \(T_0\) is the neutral object in this category; i.e., \(T_0 \otimes B \cong B \otimes T_0 \cong B\) for any crystal \(B\). We list some other properties of these crystals:
- The crystal \(T_{\lambda} \otimes B(\infty)\) is the crystal of the Verma module with highest weight \(\lambda\), where \(\lambda\) is a dominant integral weight.
- Let \(u_{\infty}\) be the highest weight vector of \(B(\infty)\) and \(\lambda\) be a dominant integral weight. There is an embedding of crystals \(B(\lambda) \longrightarrow T_{\lambda} \otimes B(\infty)\) sending \(u_{\lambda} \mapsto t_{\lambda} \otimes u_{\infty}\) which is not strict, but the embedding \(B(\lambda) \longrightarrow C \otimes T_{\lambda} \otimes B(\infty)\) by \(u_{\lambda} \mapsto c \otimes t_{\lambda} \otimes u_{\infty}\) is a strict embedding.
- For any dominant integral weight \(\lambda\), there is a surjective crystal morphism \(\Psi_{\lambda} \colon R_{\lambda} \otimes B(\infty) \longrightarrow B(\lambda)\). More precisely, if \(B = \{r_{\lambda} \otimes b \in R_{\lambda} \otimes B(\infty) : \Psi_{\lambda}(r_{\lambda} \otimes b) \neq 0 \}\), then \(B \cong B(\lambda)\) as crystals.
- For all Cartan types and all weights \(\lambda\), we have \(R_{\lambda} \cong C \otimes T_{\lambda}\) as crystals.
- For each \(i\), there is a strict crystal morphism \(\Psi_i \colon B(\infty) \longrightarrow B_i \otimes B(\infty)\) defined by \(u_{\infty} \mapsto b_i(0) \otimes u_{\infty}\), where \(u_\infty\) is the highest weight vector of \(B(\infty)\).
For more information on \(B(\infty)\), see
InfinityCrystalOfTableaux
.
Note
As with
TensorProductOfCrystals
,
we are using the opposite of Kashiwara’s convention.
AUTHORS:
- Ben Salisbury: Initial version
REFERENCES:
-
class
sage.combinat.crystals.elementary_crystals.
AbstractSingleCrystalElement
¶ Bases:
sage.structure.element.Element
Abstract base class for elements in crystals with a single element.
-
e
(i)¶ Return \(e_i\) of
self
, which isNone
for all \(i\).INPUT:
i
– An element of the index set
EXAMPLES:
sage: ct = CartanType(['A',2]) sage: la = RootSystem(ct).weight_lattice().fundamental_weights() sage: T = crystals.elementary.T(ct,la[1]) sage: t = T.highest_weight_vector() sage: t.e(1) sage: t.e(2)
-
f
(i)¶ Return \(f_i\) of
self
, which isNone
for all \(i\).INPUT:
i
– An element of the index set
EXAMPLES:
sage: ct = CartanType(['A',2]) sage: la = RootSystem(ct).weight_lattice().fundamental_weights() sage: T = crystals.elementary.T(ct,la[1]) sage: t = T.highest_weight_vector() sage: t.f(1) sage: t.f(2)
-
-
sage.combinat.crystals.elementary_crystals.
ComponentCrystal
¶ The component crystal.
Defined in [Ka1993], the component crystal \(C = \{c\}\) is the single element crystal whose crystal structure is defined by
\[\mathrm{wt}(c) = 0, \quad e_i c = f_i c = 0, \quad \varepsilon_i(c) = \varphi_i(c) = 0.\]Note \(C \cong B(0)\), where \(B(0)\) is the highest weight crystal of highest weight \(0\).
INPUT:
cartan_type
– a Cartan type
-
sage.combinat.crystals.elementary_crystals.
ElementaryCrystal
¶ The elementary crystal \(B_i\).
For \(i\) an element of the index set of type \(X\), the crystal \(B_i\) of type \(X\) is the set
\[B_i = \{ b_i(m) : m \in \ZZ \},\]where the crystal stucture is given by
\[\begin{split}\begin{aligned} \mathrm{wt}\bigl(b_i(m)\bigr) &= m\alpha_i \\ \varphi_j\bigl(b_i(m)\bigr) &= \begin{cases} m & \text{ if } j=i, \\ -\infty & \text{ if } j\neq i, \end{cases} \\ \varepsilon_j\bigl(b_i(m)\bigr) &= \begin{cases} -m & \text{ if } j=i, \\ -\infty & \text{ if } j\neq i, \end{cases} \\ e_j b_i(m) &= \begin{cases} b_i(m+1) & \text{ if } j=i, \\ 0 & \text{ if } j\neq i, \end{cases} \\ f_j b_i(m) &= \begin{cases} b_i(m-1) & \text{ if } j=i, \\ 0 & \text{ if } j\neq i. \end{cases} \end{aligned}\end{split}\]The Kashiwara embedding theorem asserts there is a unique strict crystal embedding of crystals
\[B(\infty) \hookrightarrow B_i \otimes B(\infty),\]satisfying certain properties (see [Ka1993]). The above embedding may be iterated to obtain a new embedding
\[B(\infty) \hookrightarrow B_{i_N} \otimes B_{i_{N-1}} \otimes \cdots \otimes B_{i_2} \otimes B_{i_1} \otimes B(\infty),\]which is a foundational object in the study of polyhedral realizations of crystals (see, for example, [NZ1997]).
-
sage.combinat.crystals.elementary_crystals.
RCrystal
¶ The crystal \(R_{\lambda}\).
For a fixed weight \(\lambda\), the crystal \(R_{\lambda} = \{ r_{\lambda} \}\) is a single element crystal with the crystal structure defined by
\[\mathrm{wt}(r_{\lambda}) = \lambda, \quad e_i r_{\lambda} = f_i r_{\lambda} = 0, \quad \varepsilon_i(r_{\lambda}) = -\langle h_i, \lambda\rangle, \quad \varphi_i(r_{\lambda}) = 0,\]where \(\{h_i\}\) are the simple coroots.
Tensoring \(R_{\lambda}\) with a crystal \(B\) results in shifting the weights of the vertices in \(B\) by \(\lambda\) and may also cut a subset out of the original graph of \(B\). That is, \(\mathrm{wt}(r_{\lambda} \otimes b) = \mathrm{wt}(b) + \lambda\), where \(b \in B\), provided \(r_{\lambda} \otimes b \neq 0\). For example, the crystal graph of \(B(\lambda)\) is the same as the crystal graph of \(R_{\lambda} \otimes B(\infty)\) generated from the component \(r_{\lambda} \otimes u_{\infty}\).
INPUT:
cartan_type
– A Cartan typeweight
– An element of the weight lattice of typecartan_type
EXAMPLES:
We check by tensoring \(R_{\lambda}\) with \(B(\infty)\) results in a component of \(B(\lambda)\):
sage: B = crystals.infinity.Tableaux("A2") sage: R = crystals.elementary.R("A2", B.Lambda()[1]+B.Lambda()[2]) sage: T = crystals.TensorProduct(R, B) sage: mg = T(R.highest_weight_vector(), B.highest_weight_vector()) sage: S = T.subcrystal(generators=[mg]) sage: sorted([x.weight() for x in S], key=str) [(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 1, 1), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0)] sage: C = crystals.Tableaux("A2", shape=[2,1]) sage: sorted([x.weight() for x in C], key=str) [(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 1, 1), (1, 1, 1), (1, 2, 0), (2, 0, 1), (2, 1, 0)] sage: GT = T.digraph(subset=S) sage: GC = C.digraph() sage: GT.is_isomorphic(GC, edge_labels=True) True
-
sage.combinat.crystals.elementary_crystals.
TCrystal
¶ The crystal \(T_{\lambda}\).
Let \(\lambda\) be a weight. As defined in [Ka1993] the crystal \(T_{\lambda} = \{ t_{\lambda} \}\) is a single element crystal with the crystal structure defined by
\[\mathrm{wt}(t_\lambda) = \lambda, \quad e_i t_{\lambda} = f_i t_{\lambda} = 0, \quad \varepsilon_i(t_{\lambda}) = \varphi_i(t_{\lambda}) = -\infty.\]The crystal \(T_{\lambda}\) shifts the weights of the vertices in a crystal \(B\) by \(\lambda\) when tensored with \(B\), but leaves the graph structure of \(B\) unchanged. That is to say, for all \(b \in B\), we have \(\mathrm{wt}(b \otimes t_{\lambda}) = \mathrm{wt}(b) + \lambda\).
INPUT:
cartan_type
– A Cartan typeweight
– An element of the weight lattice of typecartan_type
EXAMPLES:
sage: ct = CartanType(['A',2]) sage: C = crystals.Tableaux(ct, shape=[1]) sage: for x in C: x.weight() (1, 0, 0) (0, 1, 0) (0, 0, 1) sage: La = RootSystem(ct).ambient_space().fundamental_weights() sage: TLa = crystals.elementary.T(ct, 3*(La[1] + La[2])) sage: TP = crystals.TensorProduct(TLa, C) sage: for x in TP: x.weight() (7, 3, 0) (6, 4, 0) (6, 3, 1) sage: G = C.digraph() sage: H = TP.digraph() sage: G.is_isomorphic(H,edge_labels=True) True