# Polyhedral Realization of $$B(\infty)$$¶

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

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{split}\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}\end{split}

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

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]

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]]