def _to_m(self, part):
r"""
Returns a function which gives the coefficient of part2 in the
monomial expansion of self(part).
INPUT:
- ``self`` -- an instance of the LLT hcospin basis
- ``part`` -- a partition
OUTPUT:
- returns a function which accepts a partition and returns the coefficient
in the expansion of the monomial basis
EXAMPLES::
sage: HCosp3 = SymmetricFunctions(FractionField(QQ['t'])).llt(3).hcospin()
sage: f21 = HCosp3._to_m(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[1, t + 1, 2*t + 1]
sage: HCosp3.symmetric_function_ring().m()( HCosp3[2,1] )
(2*t+1)*m[1, 1, 1] + (t+1)*m[2, 1] + m[3]
"""
level = self.level()
f = lambda part2: QQt(ribbon_tableau.cospin_polynomial([level*i for i in part], part2, level))
return f
def _to_m(self, part):
"""
Returns a function which gives the coefficient of part2 in the
monomial expansion of self(part).
EXAMPLES::
sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: f21 = HCosp3._to_m(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[1, t + 1, 2*t + 1]
"""
BR = self.base_ring()
level = self.level()
f = lambda part2: BR(ribbon_tableau.cospin_polynomial([level*i for i in part], part2, level))
return f
def _to_m(self, part):
"""
Returns a function which gives the coefficient of part2 in the
monomial expansion of self(part).
EXAMPLES::
sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: f21 = HCosp3._to_m(Partition([2,1]))
sage: [f21(p) for p in Partitions(3)]
[1, t + 1, 2*t + 1]
"""
BR = self.base_ring()
level = self.level()
f = lambda part2: BR(
ribbon_tableau.cospin_polynomial([level * i
for i in part], part2, level))
return f
