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# -*- coding: utf-8 -*- r""" `\mathcal{B}(\infty)` Crystal Of PBW Monomials.
AUTHORS:
- Dinakar Muthiah (2015-05-11): initial version
.. SEEALSO::
For information on PBW datum, see :ref:`sage.combinat.crystals.pbw_datum`. """
#***************************************************************************** # Copyright (C) 2015 Dinakar Muthiah <muthiah at ualberta.ca> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from sage.misc.cachefunc import cached_method from sage.structure.element import Element from sage.structure.parent import Parent from sage.structure.unique_representation import UniqueRepresentation from sage.structure.richcmp import richcmp from sage.categories.highest_weight_crystals import HighestWeightCrystals from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets from sage.combinat.root_system.cartan_type import CartanType from sage.combinat.crystals.pbw_datum import PBWData, PBWDatum
class PBWCrystalElement(Element): """ A crystal element in the PBW model. """ def __init__(self, parent, lusztig_datum, long_word=None): """ Initialize ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['F', 4]) sage: u = B.highest_weight_vector() sage: b = u.f_string([1,2,3,4,2,3,2,3,4,1,2]) sage: TestSuite(b).run() """
def _repr_(self): """ Return a string representation of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['B', 4]) sage: u = B.highest_weight_vector() sage: u.f_string([1,2,3,4,2,3,2,3,4,1,2]) PBW monomial with Lusztig datum (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 2) """
def _latex_(self): r""" Return a latex representation of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['F', 4]) sage: u = B.highest_weight_vector() sage: b = u.f_string([1,2,3,4,2,3,2,3,4,1,2]) sage: latex(b) f_{\alpha_{4}}^{2} f_{\alpha_{3}} f_{\alpha_{1} + \alpha_{2} + 2\alpha_{3}} f_{\alpha_{1} + \alpha_{2}} f_{\alpha_{2}}^{2} """ for i, exp in enumerate(lusztig_datum) if exp) return '1'
def lusztig_datum(self, word=None): """ Return the Lusztig datum of ``self`` with respect to the reduced expression of the long word ``word``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 2]) sage: u = B.highest_weight_vector() sage: b = u.f_string([2,1,2,2,2,2,1,1,2,1,2,1,2,1,2,2]) sage: b.lusztig_datum() (6, 0, 10) sage: b.lusztig_datum(word=[2,1,2]) (4, 6, 0) """ else:
def __eq__(self, other): """ Check equality of ``self`` with ``other``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 2]) sage: u = B.highest_weight_vector() sage: b = u.f_string([2,1,2,2,2,2,1,1,2,1,2,1,2,1,2,2]) sage: bp = u.f_string([2,1,2,2,1,1,2,2,2,1,2,1,2,2,1,2]) sage: b == bp True """
def __ne__(self, other): """ Check inequality of ``self`` with ``other``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 2]) sage: u = B.highest_weight_vector() sage: b = u.f_string([2,1,2,2,2,2,1,1,2,1,2,1,2,1,2,2]) sage: bp = u.f_string([2,1,2,2,1,1,2,2,2,1,2,1,2,2,1,2]) sage: b != bp False """
# Necessary for displaying subcrystals def _richcmp_(self, other, op): """ Return comparison of ``self`` and ``other``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 2]) sage: u = B.highest_weight_vector() sage: b = u.f_string([2,1,2,2,2,2,1,1,2,1,2,1,2,1,2,2]) sage: bp = u.f_string([2,1,2,2,1,1,2,2,2,1,2,1,2]) sage: w = [1, 2, 1] sage: (b < bp) == (b.lusztig_datum(w) < bp.lusztig_datum(w)) True sage: (b > bp) == (b.lusztig_datum(w) > bp.lusztig_datum(w)) True """
@cached_method def __hash__(self): """ Return the hash of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 2]) sage: u = B.highest_weight_vector() sage: b = u.f_string([2,1,2,2,2,2,1,1,2,1,2,1,2,1,2,2]) sage: bp = u.f_string([2,1,2,2,1,1,2,2,2,1,2,1,2,2,1,2]) sage: hash(b) == hash(bp) True """
def e(self, i): """ Return the action of `e_i` on ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['B', 3]) sage: b = B.highest_weight_vector() sage: c = b.f_string([2,1,3,2,1,3,2,2]); c PBW monomial with Lusztig datum (0, 1, 0, 1, 0, 0, 0, 1, 2) sage: c.e(2) PBW monomial with Lusztig datum (0, 1, 0, 1, 0, 0, 0, 1, 1) sage: c.e_string([2,2,1,3,2,1,3,2]) == b True """
def f(self, i): """ Return the action of `f_i` on ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW("D4") sage: b = B.highest_weight_vector() sage: c = b.f_string([1,2,3,1,2,3,4]); c PBW monomial with Lusztig datum (0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0) sage: c == b.f_string([1,2,4,1,2,3,3]) True """
def epsilon(self, i): r""" Return `\varepsilon_i` of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(["A2"]) sage: s = B((3,0,0), (1,2,1)) sage: s.epsilon(1) 3 sage: s.epsilon(2) 0 """
def phi(self, i): r""" Return `\varphi_i` of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 2]) sage: s = B((3,0,0), (1,2,1)) sage: s.phi(1) -3 sage: s.phi(2) 3 """
def weight(self): """ Return weight of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 2]) sage: s = B((2,2,2), (1,2,1)) sage: s.weight() (-4, 0, 4) """
def star(self): r""" Return the starred crystal element corresponding to ``self``.
Let `b` be an element of ``self`` with Lusztig datum `(b_1, \ldots, b_N)` with respect to `w_0 = s_{i_1} \cdots s_{i_N}`. Then `b^*` is the element with Lusztig datum `(b_N, \ldots, b_1)` with respect to `w_0 = s_{i_N^*} \cdots s_{i_1^*}`, where `i_j^* = \omega(i_j)` with `\omega` being the :meth:`automorphism <sage.combinat.root_system.cartan_type.CartanType_standard_finite.opposition_automorphism>` given by the action of `w_0` on the simple roots.
EXAMPLES::
sage: P = crystals.infinity.PBW(['A', 2]) sage: P((1,2,3), (1,2,1)).star() == P((3,2,1), (2,1,2)) True
sage: B = crystals.infinity.PBW(['E', 6]) sage: b = B.highest_weight_vector() sage: c = b.f_string([1,2,6,3,4,2,5,2,3,4,1,6]) sage: c == c.star().star() True
TESTS::
sage: from itertools import product sage: def test_star(PBW, depth): ....: S = crystals.infinity.Star(PBW) ....: for f_str in product(*([PBW.index_set()]*depth)): ....: x = PBW.highest_weight_vector().f_string(f_str).star() ....: y = S.highest_weight_vector().f_string(f_str) ....: assert x.lusztig_datum() == y.value.lusztig_datum() sage: P = crystals.infinity.PBW(['A', 2]) sage: test_star(P, 5) sage: P = crystals.infinity.PBW(['A', 3]) sage: test_star(P, 5) sage: P = crystals.infinity.PBW(['B', 3]) sage: test_star(P, 5) sage: P = crystals.infinity.PBW(['C', 3]) sage: test_star(P, 5) sage: P = crystals.infinity.PBW(['D', 4]) sage: test_star(P, 5) # long time sage: P = crystals.infinity.PBW(['D', 5]) sage: test_star(P, 4) # long time sage: P = crystals.infinity.PBW(['E', 6]) sage: test_star(P, 4) # long time sage: P = crystals.infinity.PBW(['F', 4]) sage: test_star(P, 4) # long time sage: P = crystals.infinity.PBW(['G', 2]) sage: test_star(P, 5) """ starred_pbw_datum.long_word)
class PBWCrystal(Parent, UniqueRepresentation): r""" Crystal of `\mathcal{B}(\infty)` given by PBW monomials.
A model of the crystal `\mathcal{B}(\infty)` whose elements are PBW datum up to equivalence by the tropical Plücker relations. The crystal structure on Lusztig data `x = (x_1, \ldots, x_m)` for the reduced word `s_{i_1} \cdots s_{i_m} = w_0` is given as follows. Suppose `i_1 = j`, then `f_j x = (x_1 + 1, x_2, \ldots, x_m)`. If `i_1 \neq j`, then we use the tropical Plücker relations to change the reduced expression such that `i_1' = j` and then we change back to the original word.
EXAMPLES::
sage: PBW = crystals.infinity.PBW(['B', 3]) sage: hw = PBW.highest_weight_vector() sage: x = hw.f_string([1,2,2,3,3,1,3,3,2,3,2,1,3,1,2,3,1,2,1,3,2]); x PBW monomial with Lusztig datum (1, 1, 1, 3, 1, 0, 0, 1, 1)
Elements are expressed in terms of Lusztig datum for a fixed reduced expression of `w_0`::
sage: PBW.default_long_word() [1, 3, 2, 3, 1, 2, 3, 1, 2] sage: PBW.set_default_long_word([2,1,3,2,1,3,2,3,1]) sage: x PBW monomial with Lusztig datum (3, 1, 1, 0, 1, 0, 1, 3, 4) sage: PBW.set_default_long_word([1, 3, 2, 3, 1, 2, 3, 1, 2])
We can construct elements by giving it Lusztig data (with respect to the default long word)::
sage: PBW([1,1,1,3,1,0,0,1,1]) PBW monomial with Lusztig datum (1, 1, 1, 3, 1, 0, 0, 1, 1)
We can also construct elements by passing in a reduced expression for a long word::
sage: x = PBW([1,1,1,3,1,0,0,1,1], [3,2,1,3,2,3,2,1,2]); x PBW monomial with Lusztig datum (1, 1, 1, 0, 1, 0, 5, 1, 1) sage: x.to_highest_weight()[1] [1, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 1, 3] """ @staticmethod def __classcall__(cls, cartan_type): """ Normalize input to ensure a unique representation.
EXAMPLES::
sage: B1 = crystals.infinity.PBW(['A', 2]) sage: B2 = crystals.infinity.PBW("A2") sage: B3 = crystals.infinity.PBW(CartanType("A2")) sage: B1 is B2 and B2 is B3 True """ raise NotImplementedError("only implemented for finite types")
def __init__(self, cartan_type): """ Initialize ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['B', 2]) sage: TestSuite(B).run() """
# There must be a better way to do the following zero_lusztig_datum, self._default_word),)
def _repr_(self): """ Return a string representation of ``self``.
EXAMPLES::
sage: crystals.infinity.PBW(['C', 3]) Crystal of PBW data of type ['C', 3] """
def default_long_word(self): """ Return the default long word used to express elements of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['E', 6]) sage: B.default_long_word() [1, 3, 4, 5, 6, 2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6, 2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 3, 4, 1, 3, 2, 4, 1, 3, 2, 1] """
def _check_is_long_word(self, word): """ Check if ``word`` is a reduced expression of the long of the Coxeter group of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['A', 3]) sage: B._check_is_long_word([1,2,1,3,2,1]) sage: B._check_is_long_word([1,3,2,3,2,1]) Traceback (most recent call last): ... ValueError: not a reduced word of the long element sage: B._check_is_long_word([1,2,1,3,2]) Traceback (most recent call last): ... ValueError: not a reduced word of the long element sage: B._check_is_long_word([1,2,1,3,2,1,2]) Traceback (most recent call last): ... ValueError: not a reduced word of the long element """ or W.from_reduced_word(word) != W.long_element()):
def set_default_long_word(self, word): """ Set the default long word used to express elements of ``self``.
EXAMPLES::
sage: B = crystals.infinity.PBW(['C', 3]) sage: B.default_long_word() [1, 3, 2, 3, 1, 2, 3, 1, 2] sage: x = B.highest_weight_vector().f_string([2,1,3,2,3,1,2,3,3,1]) sage: x PBW monomial with Lusztig datum (1, 2, 2, 0, 0, 0, 0, 0, 1) sage: B.set_default_long_word([2,1,3,2,1,3,2,3,1]) sage: B.default_long_word() [2, 1, 3, 2, 1, 3, 2, 3, 1] sage: x PBW monomial with Lusztig datum (2, 0, 0, 0, 0, 0, 1, 3, 2)
TESTS::
sage: B = crystals.infinity.PBW(['A', 3]) sage: B._check_is_long_word([1,2,1,3,2,1,2]) Traceback (most recent call last): ... ValueError: not a reduced word of the long element """
Element = PBWCrystalElement
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