def __init__(self, n):
r"""
Initializes the class of all standard super tableaux of size ``n``.
TESTS::
sage: TestSuite( StandardSuperTableaux(4) ).run()
"""
StandardSuperTableaux.__init__(self)
from sage.combinat.partition import Partitions_n
DisjointUnionEnumeratedSets.__init__(self,
Family(Partitions_n(n),
StandardSuperTableaux_shape),
category=FiniteEnumeratedSets(),
facade=True, keepkey=False)
self.size = Integer(n)
def __init__(self, domain, part):
"""
Initialize ``self``.
EXAMPLES::
sage: G = SymmetricGroup(5)
sage: g = G([(1,2), (3,4,5)])
sage: C = G.conjugacy_class(Partition([3,2]))
sage: type(C._part)
<class 'sage.combinat.partition.Partitions_n_with_category.element_class'>
"""
P = Partitions_n(len(domain))
self._part = P(part)
self._domain = domain
self._set = None
def wsum(m): # expansion of h_m in w-basis, for m > 0
return self._from_dict({lam: 1 for lam in Partitions_n(m)})
def _precompute_cache(self, n, to_self_cache, from_self_cache, transition_matrices, inverse_transition_matrices, to_self_gen_function):
"""
Compute the transition matrices between ``self`` and another
multiplicative homogeneous basis in the homogeneous components of
degree `n`.
The results are not returned, but rather stored in the caches.
This assumes that the transition matrices in all degrees smaller
than `n` have already been computed and cached!
INPUT:
- ``n`` -- nonnegative integer
- ``to_self_cache`` -- a cache which stores the coordinates of
the elements of the other basis with respect to the
basis ``self``
- ``from_self_cache`` -- a cache which stores the coordinates
of the elements of ``self`` with respect to the other
basis
- ``transition_matrices`` -- a cache for transition matrices
which contain the coordinates of the elements of the other
basis with respect to ``self``
- ``inverse_transition_matrices`` -- a cache for transition
matrices which contain the coordinates of the elements of
``self`` with respect to the other basis
- ``to_self_gen_function`` -- a function which takes a
positive integer `n` and returns the element of the other
basis corresponding to the partition `[n]` expanded with
respect to the Witt basis ``self`` (as an element of
``self``, not as a dictionary)
Examples for usage of this function are the ``_precompute_h``,
``_precompute_e`` and ``_precompute_p`` methods of this class.
EXAMPLES::
The examples below demonstrate how the caches are built
step by step using the ``_precompute_cache`` method. In order
not to influence the outcome of other doctests, we make sure
not to use the caches internally used by this class, but
rather to create new caches::
sage: Sym = SymmetricFunctions(QQ)
sage: w = Sym.w()
sage: toy_to_self_cache = {}
sage: toy_from_self_cache = {}
sage: toy_transition_matrices = {}
sage: toy_inverse_transition_matrices = {}
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l(toy_to_self_cache)
[]
sage: def toy_gen_function(n):
....: if n > 1:
....: return w(Partition([n])) + n * w(Partition([n-1,1]))
....: return w(Partition([n]))
sage: w._precompute_cache(0, toy_to_self_cache,
....: toy_from_self_cache,
....: toy_transition_matrices,
....: toy_inverse_transition_matrices,
....: toy_gen_function)
sage: l(toy_to_self_cache)
[([], [([], 1)])]
sage: w._precompute_cache(1, toy_to_self_cache,
....: toy_from_self_cache,
....: toy_transition_matrices,
....: toy_inverse_transition_matrices,
....: toy_gen_function)
sage: l(toy_to_self_cache)
[([], [([], 1)]), ([1], [([1], 1)])]
sage: w._precompute_cache(2, toy_to_self_cache,
....: toy_from_self_cache,
....: toy_transition_matrices,
....: toy_inverse_transition_matrices,
....: toy_gen_function)
sage: l(toy_to_self_cache)
[([], [([], 1)]),
([1], [([1], 1)]),
([1, 1], [([1, 1], 1)]),
([2], [([1, 1], 2), ([2], 1)])]
sage: toy_transition_matrices[2]
[1 2]
[0 1]
sage: toy_inverse_transition_matrices[2]
[ 1 -2]
[ 0 1]
sage: toy_transition_matrices.keys()
[0, 1, 2]
"""
# Much of this code is adapted from dual.py
base_ring = self.base_ring()
zero = base_ring(0)
from sage.combinat.partition import Partition, Partitions_n
# Handle the n == 0 case separately
if n == 0:
part = Partition([])
to_self_cache[ part ] = { part: base_ring(1) }
from_self_cache[ part ] = { part: base_ring(1) }
transition_matrices[n] = matrix(base_ring, [[1]])
inverse_transition_matrices[n] = matrix(base_ring, [[1]])
return
partitions_n = Partitions_n(n).list()
# The other basis will be called B from now on.
# This contains the data for the transition matrix from the
# basis B to the Witt basis self.
transition_matrix_n = matrix(base_ring, len(partitions_n), len(partitions_n))
# This first section calculates how the basis elements of the
# basis B are expressed in terms of the Witt basis ``self``.
# For every partition p of size n, expand B[p] in terms of
# the Witt basis self using multiplicativity and
# to_self_gen_function.
i = 0
for s_part in partitions_n:
# s_mcs will be self(B[s_part])._monomial_coefficients
s_mcs = {}
# We need to compute the coordinates of B[s_part] in the Witt basis.
hsp_in_w_basis = self.one()
for p in s_part:
hsp_in_w_basis *= to_self_gen_function(p)
# Now, hsp_in_w_basis is B[s_part] expanded in the Witt
# basis self (this is the same as the coercion self(B[s_part]).
j = 0
for p_part in partitions_n:
if p_part in hsp_in_w_basis._monomial_coefficients:
sp = hsp_in_w_basis._monomial_coefficients[p_part]
s_mcs[p_part] = sp
transition_matrix_n[i,j] = sp
j += 1
to_self_cache[ s_part ] = s_mcs
i += 1
# Save the transition matrix
transition_matrices[n] = transition_matrix_n
# This second section calculates how the basis elements of
# self expand in terms of the basis B. We do this by
# computing the inverse of the matrix transition_matrix_n
# obtained above.
# TODO: Possibly this can be sped up by using properties
# of this matrix (e. g., it being triangular in most standard cases).
# Are there significantly faster ways to invert a triangular
# matrix (compared to the usual matrix inversion algorithms)?
inverse_transition = ~transition_matrix_n
for i in range(len(partitions_n)):
d_mcs = {}
for j in range(len(partitions_n)):
if inverse_transition[i,j] != zero:
d_mcs[ partitions_n[j] ] = inverse_transition[i,j]
from_self_cache[ partitions_n[i] ] = d_mcs
inverse_transition_matrices[n] = inverse_transition