def heap(self, **kargs):
r"""
Create the heap poset of ``self``.
The heap of an FC element `w` is a labeled poset that can be defined
from any reduced word of `w`. Different reduced words yield isomorphic
labeled posets, so the heap is well defined.
Heaps are very useful for visualizing and studying FC elements; see, for
example, [Ste1996]_ and [GX2020]_.
INPUT:
- ``self`` -- list, a reduced word `w=s_0... s_{k-1}` of an FC element
- ``one_index`` -- boolean (default: False). Setting the value to True
will change the underlying set of the poset to `\{1, 2, \dots, n\}`
- ``display_labeling`` -- boolean (default: False). Setting the value to
True will display the label `s_i` for each element `i` of the poset
OUTPUT: A labeled poset where the underlying set is `\{0,1,...,k-1\}`
and where each element `i` carries `s_i` as its label. The partial order
`\prec` on the poset is defined by declaring `i\prec j` if `i<j` and
`m(s_i,s_j)\neq 2`.
EXAMPLES::
sage: FC = CoxeterGroup(['A', 5]).fully_commutative_elements()
sage: FC([1, 4, 3, 5, 2, 4]).heap().cover_relations()
[[1, 2], [1, 3], [2, 5], [2, 4], [3, 5], [0, 4]]
sage: FC([1, 4, 3, 5, 4, 2]).heap(one_index=True).cover_relations()
[[2, 3], [2, 4], [3, 6], [3, 5], [4, 6], [1, 5]]
"""
m = self.parent().coxeter_group().coxeter_matrix()
one_index = kargs.get('one_index', False)
display_labeling = kargs.get('display_labeling', False)
# elements of the poset:
elements = list(range(1,
len(self) +
1)) if one_index else list(range(len(self)))
# get the label of each poset element:
def letter(index):
return self[index - 1] if one_index else self[index]
# specify the partial order:
relations = [(i, j) for [i, j] in itertools.product(elements, repeat=2)
if i < j and m[letter(i), letter(j)] != 2]
p = Poset((elements, relations))
if not display_labeling:
return p
else:
return p.relabel(lambda i: (i, letter(i)))
