Polyhedral Realization of B(\infty)

class sage.combinat.crystals.polyhedral_realization.InfinityCrystalAsPolyhedralRealization(cartan_type, seq)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystals

The polyhedral realization of B(\infty).

Note

Here we are using anti-Kashiwara notation and might differ from some of the literature.

Consider a Kac-Moody algebra \mathfrak{g} of Cartan type X with index set I, and consider a finite sequence J = (j_1, j_2, \ldots, j_m) whose support equals I. We extend this to an infinite sequence by taking \bar{J} = J \cdot J \cdot J \cdots, where \cdot denotes concatenation of sequences. Let

B_J = B_{j_m} \otimes \cdots \otimes B_{j_2} \otimes B_{j_1},

where B_i is an ElementaryCrystal.

As given in Theorem 2.1.1 of [K93], there exists a strict crystal embedding \Psi_i \colon B(\infty) \to B_i \otimes B(\infty) defined by u_{\infty}
\mapsto b_i(0) \otimes u_{\infty}, where b_i(0) \in B_i and u_{\infty} is the (unique) highest weight element in B(\infty). This is sometimes known as the Kashiwara embedding [NZ97] (though, in [NZ97], the target of this map is denoted by \ZZ_J^\infty). By iterating this embedding by taking \Psi_J = \Psi_{j_n} \circ \Psi_{j_{n-1}} \circ \cdots \circ
\Psi_{j_1}, we obtain the following strict crystal embedding:

\Psi_J^n \colon B(\infty) \to B_J^{\otimes n} \otimes B(\infty).

We note there is a natural analog of Lemma 10.6.2 in [HK02] that for any b \in B(\infty), there exists a positive integer N such that

\Psi^N_J(b) = \left( \bigotimes_{k=1}^N b^{(k)} \right)
\otimes u_{\infty}.

Therefore we can model elements b \in B(\infty) by considering an infinite list of elements b^{(k)} \in B_J and defining the crystal structure by:

\begin{aligned}
\mathrm{wt}(b) & = \sum_{k=1}^N \mathrm{wt}(b^{(k)})
\\ e_i(b) & = e_i\left( \left( \bigotimes_{k=1}^N b^{(k)} \right)
\right) \otimes u_{\infty},
\\ f_i(b) & = f_i\left( \left( \bigotimes_{k=1}^N b^{(k)} \right)
\right) \otimes u_{\infty},
\\ \varepsilon_i(b) & = \max_{ e_i^k(b) \neq 0 } k,
\\ \varphi_i(b) & = \varepsilon_i(b) - \langle \mathrm{wt}(b),
h_i^{\vee} \rangle.
\end{aligned}

To translate this into a finite list, we consider a finite sequence b_1 \otimes \cdots \otimes b_N and if

f_i\left( b^{(1)} \otimes \cdots b^{(N-1)} \otimes b^{(N)} \right)
= b^{(1)} \otimes \cdots \otimes b^{(N-1)} \otimes
f_i\left( b^{(N)} \right),

then we take the image as b^{(1)} \otimes \cdots \otimes f_i\left(
b^{(N)} \right) \otimes b^{(N+1)}. Similarly we remove b^{(N)} if we have b^{(N)} = \bigotimes_{k=1}^m b_{j_k}(0). Additionally if

e_i\left( b^{(1)} \otimes \cdots \otimes b^{(N-1)} \otimes
b^{(N)} \right) = b^{(1)} \otimes \cdots \otimes b^{(N-1)}
\otimes e_i\left( b^{(N)} \right),

then we consider this to be 0.

REFERENCES:

[K93]M. Kashiwara. The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71. 1993.

INPUT:

  • cartan_type – a Cartan type
  • seq – (default: None) a finite sequence whose support equals the index set of the Cartan type; if None, then this is the index set

EXAMPLES:

sage: B = crystals.infinity.PolyhedralRealization(['A',2])
sage: mg = B.module_generators[0]; mg
[0, 0]
sage: mg.f_string([2,1,2,2])
[0, -3, -1, 0, 0, 0]

An example of type B_2:

sage: B = crystals.infinity.PolyhedralRealization(['B',2])
sage: mg = B.module_generators[0]; mg
[0, 0]
sage: mg.f_string([2,1,2,2])
[0, -2, -1, -1, 0, 0]

An example of type G_2:

sage: B = crystals.infinity.PolyhedralRealization(['G',2])
sage: mg = B.module_generators[0]; mg
[0, 0]
sage: mg.f_string([2,1,2,2])
[0, -3, -1, 0, 0, 0]
class Element(parent, *args, **kwds)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystalsElement

An element in the polyhedral realization of B(\infty).

e(i)

Return the action of e_i on self.

EXAMPLES:

sage: B = crystals.infinity.PolyhedralRealization(['A',2])
sage: mg = B.module_generators[0]
sage: all(mg.e(i) is None for i in B.index_set())
True
sage: mg.f(1).e(1) == mg
True
epsilon(i)

Return \varepsilon_i of self.

EXAMPLES:

sage: B = crystals.infinity.PolyhedralRealization(['A',2,1])
sage: mg = B.module_generators[0]
sage: [mg.epsilon(i) for i in B.index_set()]
[0, 0, 0]
sage: elt = mg.f(0)
sage: [elt.epsilon(i) for i in B.index_set()]
[1, 0, 0]
sage: elt = mg.f_string([0,1,2])
sage: [elt.epsilon(i) for i in B.index_set()]
[0, 0, 1]
sage: elt = mg.f_string([0,1,2,2])
sage: [elt.epsilon(i) for i in B.index_set()]
[0, 0, 2]
f(i)

Return the action of f_i on self.

EXAMPLES:

sage: B = crystals.infinity.PolyhedralRealization(['A',2])
sage: mg = B.module_generators[0]
sage: mg.f(1)
[-1, 0, 0, 0]
sage: mg.f_string([1,2,2,1])
[-1, -2, -1, 0, 0, 0]
phi(i)

Return \varphi_i of self.

EXAMPLES:

sage: B = crystals.infinity.PolyhedralRealization(['A',2,1])
sage: mg = B.module_generators[0]
sage: [mg.phi(i) for i in B.index_set()]
[0, 0, 0]
sage: elt = mg.f(0)
sage: [elt.phi(i) for i in B.index_set()]
[-1, 1, 1]
sage: elt = mg.f_string([0,1])
sage: [elt.phi(i) for i in B.index_set()]
[-1, 0, 2]
sage: elt = mg.f_string([0,1,2,2])
sage: [elt.phi(i) for i in B.index_set()]
[1, 1, 0]
truncate(k=None)

Truncate self to have length k and return as an element in a (finite) tensor product of crystals.

INPUT:

  • k – (optional) the length of the truncation; if not specified, then returns one more than the current non-ground-state elements (i.e. the current list in self)

EXAMPLES:

sage: B = crystals.infinity.PolyhedralRealization(['A',2])
sage: mg = B.module_generators[0]
sage: elt = mg.f_string([1,2,2,1]); elt
[-1, -2, -1, 0, 0, 0]
sage: t = elt.truncate(); t
[-1, -2, -1, 0, 0, 0]
sage: t.parent() is B.finite_tensor_product(6)
True
sage: elt.truncate(2)
[-1, -2]
sage: elt.truncate(10)
[-1, -2, -1, 0, 0, 0, 0, 0, 0, 0]
InfinityCrystalAsPolyhedralRealization.finite_tensor_product(k)

Return the finite tensor product of crystals of length k by truncating self.

EXAMPLES:

sage: B = crystals.infinity.PolyhedralRealization(['A',2])
sage: B.finite_tensor_product(5)
Full tensor product of the crystals
 [The 1-elementary crystal of type ['A', 2],
  The 2-elementary crystal of type ['A', 2],
  The 1-elementary crystal of type ['A', 2],
  The 2-elementary crystal of type ['A', 2],
  The 1-elementary crystal of type ['A', 2]]