Crystal of Bernstein-Zelevinsky Multisegments#
- class sage.combinat.crystals.multisegments.InfinityCrystalOfMultisegments(n)#
Bases:
Parent,UniqueRepresentationThe 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:
ElementWrapperAn 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
- highest_weight_vector()#
Return the highest weight vector of
self.EXAMPLES:
sage: B = crystals.infinity.Multisegments(2) sage: B.highest_weight_vector() 0
- 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]