Crystal of Bernstein-Zelevinsky Multisegments

class sage.combinat.crystals.multisegments.InfinityCrystalOfMultisegments(n)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

The type \(A_n^{(1)}\) crystal \(B(\infty)\) realized using Bernstein-Zelevinsky (BZ) multisegments.

Using (a modified version of the) notation from [JL2009], for \(\ell \in \ZZ_{>0}\) and \(i \in \ZZ / (n+1)\ZZ\), a segment of length \(\ell\) and head \(i\) is the sequence of consecutive residues \([i,i+1,\dots,i+\ell-1]\). The notation for a segment of length \(\ell\) and head \(i\) is simplified to \([i; \ell)\). Similarly, a segment of length \(\ell\) and tail \(i\) is the sequence of consecutive residues \([i-\ell+1, \ldots, i-1, i]\). The latter is denoted simply by \((\ell;i]\). Finally, a multisegment is a formal linear combination of segments, usually written in the form

\[\begin{split}\psi = \sum_{\substack{i \in \ZZ/(n+1)\ZZ \\ \ell \in \ZZ_{>0}}} m_{(\ell;i]} (\ell; i].\end{split}\]

Such a multisegment is called aperiodic if, for every \(\ell > 0\), there exists some \(i \in \ZZ / (n+1)\ZZ\) such that \((\ell; i]\) does not appear in \(\psi\). Denote the set of all periodic multisegments, together with the empty multisegment \(\varnothing\), by \(\Psi\). We define a crystal structure on multisegments as follows. Set \(S_{\ell,i} = \sum_{k \ge \ell} (m_{(k;i-1]} - m_{(k;i]})\) and let \(\ell_f\) be the minimal \(\ell\) that attains the value \(\min_{\ell > 0} S_{\ell,i}\). Then we have

\[\begin{split}f_i \psi = \begin{cases} \psi + (1;i] & \text{ if } \ell_f = 1,\\ \psi + (\ell_f;i] - (\ell_f-1;i-1] & \text{ if } \ell_f > 1. \end{cases}\end{split}\]

Similarly, let \(\ell_e\) be the maximal \(\ell\) that attains the value \(\min_{\ell > 0} S_{\ell,i}\). Then we have

\[\begin{split}e_i \psi = \begin{cases} 0 & \text{ if } \min_{\ell > 0} S_{\ell,i} = 0, \\ \psi + (1; i] & \text{ if } \ell_e = 1,\\ \psi - (\ell_e; i] + (\ell_e-1; i-1] & \text{ if } \ell_e > 1. \end{cases}\end{split}\]

Alternatively, the crystal operators may be defined using a signature rule, as detailed in Section 4 of [JL2009] (following [AJL2011]). For \(\psi \in \Psi\) and \(i \in \ZZ/(n+1)\ZZ\), encode all segments in \(\psi\) with tail \(i\) by the symbol \(R\) and all segments in \(\psi\) with tail \(i-1\) by \(A\). For \(\ell > 0\), set \(w_{i,\ell} = R^{m_{(\ell;i]}} A^{m_{(\ell;i-1]}}\) and \(w_i = \prod_{\ell\ge 1} w_{i,\ell}\). By successively canceling out as many \(RA\) factors as possible, set \(\widetilde{w}_i = A^{a_i(\psi)} R^{r_i(\psi)}\). If \(a_i(\psi) > 0\), denote by \(\ell_f > 0\) the length of the rightmost segment \(A\) in \(\widetilde{w}_i\). If \(a_i(\psi) = 0\), set \(\ell_f = 0\). Then

\[\begin{split}f_i \psi = \begin{cases} \psi + (1; i] & \text{ if } a_i(\psi) = 0,\\ \psi + (\ell_f; i] - (\ell_f-1; i-1] & \text{ if } a_i(\psi) > 0. \end{cases}\end{split}\]

The rule for computing \(e_i \psi\) is similar.

INPUT:

  • n – for type \(A_n^{(1)}\)

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: x = B([(8,1),(6,0),(5,1),(5,0),(4,0),(4,1),(4,1),(3,0),(3,0),(3,1),(3,1),(1,0),(1,2),(1,2)]); x
(8; 1] + (6; 0] + (5; 0] + (5; 1] + (4; 0] + 2 * (4; 1]
 + 2 * (3; 0] + 2 * (3; 1] + (1; 0] + 2 * (1; 2]
sage: x.f(1)
(8; 1] + (6; 0] + (5; 0] + (5; 1] + (4; 0] + 2 * (4; 1]
 + 2 * (3; 0] + 2 * (3; 1] + (2; 1] + 2 * (1; 2]
sage: x.f(1).f(1)
(8; 1] + (6; 0] + (5; 0] + (5; 1] + (4; 0] + 2 * (4; 1]
 + 2 * (3; 0] + 2 * (3; 1] + (2; 1] + (1; 1] + 2 * (1; 2]
sage: x.e(1)
(7; 0] + (6; 0] + (5; 0] + (5; 1] + (4; 0] + 2 * (4; 1]
 + 2 * (3; 0] + 2 * (3; 1] + (1; 0] + 2 * (1; 2]
sage: x.e(1).e(1)
sage: x.f(0)
(8; 1] + (6; 0] + (5; 0] + (5; 1] + (4; 0] + 2 * (4; 1]
 + 2 * (3; 0] + 2 * (3; 1] + (2; 0] + (1; 0] + (1; 2]

We check an \(\widehat{\mathfrak{sl}}_2\) example against the generalized Young walls:

sage: B = crystals.infinity.Multisegments(1)
sage: G = B.subcrystal(max_depth=4).digraph()
sage: C = crystals.infinity.GeneralizedYoungWalls(1)
sage: GC = C.subcrystal(max_depth=4).digraph()
sage: G.is_isomorphic(GC, edge_labels=True)
True

REFERENCES:

class Element(parent, value)

Bases: sage.structure.element_wrapper.ElementWrapper

An element in a BZ multisegments crystal.

e(i)

Return the action of \(e_i\) on self.

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: b = B([(4,2), (3,0), (3,1), (1,1), (1,0)])
sage: b.e(0)
(4; 2] + (3; 0] + (3; 1] + (1; 1]
sage: b.e(1)
sage: b.e(2)
(3; 0] + 2 * (3; 1] + (1; 0] + (1; 1]
epsilon(i)

Return \(\varepsilon_i\) of self.

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: b = B([(4,2), (3,0), (3,1), (1,1), (1,0)])
sage: b.epsilon(0)
1
sage: b.epsilon(1)
0
sage: b.epsilon(2)
1
f(i)

Return the action of \(f_i\) on self.

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: b = B([(4,2), (3,0), (3,1), (1,1), (1,0)])
sage: b.f(0)
(4; 2] + (3; 0] + (3; 1] + 2 * (1; 0] + (1; 1]
sage: b.f(1)
(4; 2] + (3; 0] + (3; 1] + (1; 0] + 2 * (1; 1]
sage: b.f(2)
2 * (4; 2] + (3; 0] + (1; 0] + (1; 1]
phi(i)

Return \(\varphi_i\) of self.

Let \(\psi \in \Psi\). Define \(\varphi_i(\psi) := \varepsilon_i(\psi) + \langle h_i, \mathrm{wt}(\psi) \rangle\), where \(h_i\) is the \(i\)-th simple coroot and \(\mathrm{wt}(\psi)\) is the weight() of \(\psi\).

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: b = B([(4,2), (3,0), (3,1), (1,1), (1,0)])
sage: b.phi(0)
1
sage: b.phi(1)
0
sage: mg = B.highest_weight_vector()
sage: mg.f(1).phi(0)
1
weight()

Return the weight of self.

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: b = B([(4,2), (3,0), (3,1), (1,1), (1,0)])
sage: b.weight()
-4*delta
InfinityCrystalOfMultisegments.highest_weight_vector()

Return the highest weight vector of self.

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: B.highest_weight_vector()
0
InfinityCrystalOfMultisegments.weight_lattice_realization()

Return a realization of the weight lattice of self.

EXAMPLES:

sage: B = crystals.infinity.Multisegments(2)
sage: B.weight_lattice_realization()
Extended weight lattice of the Root system of type ['A', 2, 1]