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 [Ka1993], 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 [NZ1997] (though, in [NZ1997], 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 [HK2002] 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 \(0\).
INPUT:
cartan_type
– a Cartan typeseq
– (default:None
) a finite sequence whose support equals the index set of the Cartan type; ifNone
, 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#
Bases:
sage.combinat.crystals.tensor_product_element.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 lengthk
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 inself
)
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 truncatingself
.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]]