def _creation_by_determinant_helper(self, k, part):
"""
EXAMPLES::
sage: S = MacdonaldPolynomialsS(QQ)
sage: a = S([2,1])
sage: a._creation_by_determinant_helper(2,[1])
(q^3*t-q^2*t-q+1)*McdS[2, 1] + (q^3-q^2*t-q+t)*McdS[3]
"""
S = self.parent()
q,t = S.q, S.t
part += [0]*(k-len(part))
if len(part) > k:
raise ValueError, "the column to add is too small"
#Create the matrix over the homogeneous symmetric
#functions and take its determinant
MS = MatrixSpace(sfa.SFAHomogeneous(S.base_ring()), k, k)
h = MS.base_ring()
m = []
for i in range(k):
row = [0]*max(0, (i+1)-2-part[i])
for j in range(max(0, (i+1)-2-part[i]),k):
value = part[i]+j-i+1
p = [value] if value > 0 else []
row.append( (1-q**(part[i]+j-i+1)*t**(k-(j+1)))*h(p) )
m.append(row)
M = MS(m)
res = M.det()
#Convert to the Schurs
res = S._s( res )
return S._from_element(res)
def matrix_clean_up(mat):
r"""
Apply get_cleaned_up_version to each object within
each entry and return a cleaned up matrix
"""
dim = mat.nrows()
L = list()
mat_space = MatrixSpace(CombinatorialScalarRing(),dim)
prnt = mat_space.base_ring()
for i in range(dim):
L.append(list())
for j in range(dim):
L[i].append(CombinatorialScalarWrapper(mat[i,j].get_cleaned_up_version()))
return mat_space(L)
def identity_matrix(dim):
r"""
Returns standard combinatorial identity matrix
"""
mat_space = MatrixSpace(CombinatorialScalarRing(),dim)
prnt = mat_space.base_ring()
L = list()
for i in range(dim):
L.append(list())
for j in range(dim):
if i==j:
L[i].append(prnt._one())
else:
L[i].append(prnt._zero())
return mat_space(L)
def identity_matrix(dim):
r"""
Returns standard combinatorial identity matrix
"""
mat_space = MatrixSpace(CombinatorialScalarRing(), dim)
prnt = mat_space.base_ring()
L = list()
for i in range(dim):
L.append(list())
for j in range(dim):
if i == j:
L[i].append(prnt._one())
else:
L[i].append(prnt._zero())
return mat_space(L)
def matrix_clean_up(mat):
r"""
Apply get_cleaned_up_version to each object within
each entry and return a cleaned up matrix
"""
dim = mat.nrows()
L = list()
mat_space = MatrixSpace(CombinatorialScalarRing(), dim)
prnt = mat_space.base_ring()
for i in range(dim):
L.append(list())
for j in range(dim):
L[i].append(
CombinatorialScalarWrapper(mat[i, j].get_cleaned_up_version()))
return mat_space(L)
def matrix_combinatorial_adjoint(mat):
r"""
Return Combinatorial Adjoint.
"""
dim = mat.nrows()
M = list()
mat_space = MatrixSpace(CombinatorialScalarRing(), dim)
prnt = mat_space.base_ring()
#create list L of lists of sets, dimension is increased by one to mitigate index confusion
L = list()
for i in range(dim + 1):
L.append(list())
for j in range(dim + 1):
L[i].append(set())
P = Permutations(dim)
for p in P:
p_comb = CombinatorialScalarWrapper(
[CombinatorialObject(p, p.signature())])
l = list()
for i in range(1, dim + 1):
l.append(mat[p(i) - 1, i - 1])
#This list will have empty sets, which will yield to an empty cartesian product
#especially when the matrix input is triangular (except for the identity permutation).
#We will now iterate through the selected entries in each column
#and create a set of a singleton of an empty string that corresponds
#to the "missing" element of the tuple described in definition 39.
for i in range(1, dim + 1):
copy_l = copy(l)
copy_l[i - 1] = CombinatorialScalarWrapper(
[CombinatorialObject('_', 1)])
cp = CartesianProduct(p_comb, *copy_l)
for tupel in cp:
tupel = tuple(tupel)
tupel_weight = 1
tupel_sign = 1
for elm in tupel:
tupel_sign = tupel_sign * elm.get_sign()
tupel_weight = tupel_weight * elm.get_weight()
L[i][p(i)].add(
CombinatorialObject(tupel, tupel_sign, tupel_weight))
#turn these sets into CombinatorialScalars
for i in range(1, dim + 1):
l = list()
for j in range(1, dim + 1):
l.append(CombinatorialScalarWrapper(L[i][j]))
M.append(l)
return mat_space(M)
def matrix_combinatorial_adjoint(mat):
r"""
Return Combinatorial Adjoint.
"""
dim = mat.nrows()
M = list()
mat_space = MatrixSpace(CombinatorialScalarRing(),dim)
prnt = mat_space.base_ring()
#create list L of lists of sets, dimension is increased by one to mitigate index confusion
L = list()
for i in range(dim+1):
L.append(list())
for j in range(dim+1):
L[i].append(set())
P = Permutations(dim)
for p in P:
p_comb = CombinatorialScalarWrapper([CombinatorialObject(p,p.signature())])
l = list()
for i in range(1,dim+1):
l.append(mat[p(i)-1,i-1])
#This list will have empty sets, which will yield to an empty cartesian product
#especially when the matrix input is triangular (except for the identity permutation).
#We will now iterate through the selected entries in each column
#and create a set of a singleton of an empty string that corresponds
#to the "missing" element of the tuple described in definition 39.
for i in range(1,dim+1):
copy_l = copy(l)
copy_l[i-1]=CombinatorialScalarWrapper([CombinatorialObject('_',1)])
cp = CartesianProduct(p_comb,*copy_l)
for tupel in cp:
tupel = tuple(tupel)
tupel_weight = 1
tupel_sign = 1
for elm in tupel:
tupel_sign = tupel_sign*elm.get_sign()
tupel_weight = tupel_weight*elm.get_weight()
L[i][p(i)].add(CombinatorialObject(tupel,tupel_sign,tupel_weight))
#turn these sets into CombinatorialScalars
for i in range(1,dim+1):
l = list()
for j in range(1,dim+1):
l.append(CombinatorialScalarWrapper(L[i][j]))
M.append(l)
return mat_space(M)
def _creation_by_determinant_helper(self, k, part):
"""
EXAMPLES::
sage: S = MacdonaldPolynomialsS(QQ)
sage: a = S([2,1])
sage: a._creation_by_determinant_helper(2,[1])
(q^3*t-q^2*t-q+1)*McdS[2, 1] + (q^3-q^2*t-q+t)*McdS[3]
"""
S = self.parent()
q, t = S.q, S.t
part += [0] * (k - len(part))
if len(part) > k:
raise ValueError, "the column to add is too small"
#Create the matrix over the homogeneous symmetric
#functions and take its determinant
MS = MatrixSpace(sfa.SFAHomogeneous(S.base_ring()), k, k)
h = MS.base_ring()
m = []
for i in range(k):
row = [0] * max(0, (i + 1) - 2 - part[i])
for j in range(max(0, (i + 1) - 2 - part[i]), k):
value = part[i] + j - i + 1
p = [value] if value > 0 else []
row.append(
(1 - q**(part[i] + j - i + 1) * t**(k -
(j + 1))) * h(p))
m.append(row)
M = MS(m)
res = M.det()
#Convert to the Schurs
res = S._s(res)
return S._from_element(res)
