sage: Qp._s_cache(2)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l(Qp._s_to_self_cache[2])
[([1, 1], [([1, 1], 1), ([2], -t)]), ([2], [([2], 1)])]
sage: l(Qp._self_to_s_cache[2])
[([1, 1], [([1, 1], 1), ([2], t)]), ([2], [([2], 1)])]
"""
self._invert_morphism(n, ZZ['t'], self._self_to_s_cache, \
self._s_to_self_cache, to_other_function = self._to_s, \
lower_triangular=True, ones_on_diagonal=True)
#############
# Cache #
#############
from sage.misc.cache import Cache
cache_p = Cache(HallLittlewood_p)
cache_q = Cache(HallLittlewood_q)
cache_qp = Cache(HallLittlewood_qp)
# Unpickling backward compatibility
sage.structure.sage_object.register_unpickle_override(
'sage.combinat.sf.hall_littlewood', 'HallLittlewoodElement_p',
HallLittlewood_p.Element)
sage.structure.sage_object.register_unpickle_override(
'sage.combinat.sf.hall_littlewood', 'HallLittlewoodElement_q',
HallLittlewood_q.Element)
sage.structure.sage_object.register_unpickle_override(
'sage.combinat.sf.hall_littlewood', 'HallLittlewoodElement_qp',
HallLittlewood_qp.Element)
((v1, e1), (v2, e2)) = list(list(to_commute)[0][1])
assert e1 == 1
assert e2 == 1
assert v1 > v2
c_coef = None
d_poly = None
for (c, m) in commuted:
if list(m) == [(v2, 1), (v1, 1)]:
c_coef = c
#buggy coercion workaround
d_poly = commuted - self(c) * self(m)
break
assert not c_coef is None, list(m)
v2_ind = self.gens().index(v2)
v1_ind = self.gens().index(v1)
cmat[v2_ind, v1_ind] = c_coef
if d_poly:
dmat[v2_ind, v1_ind] = d_poly
from sage.rings.polynomial.plural import g_Algebra
return g_Algebra(base_ring,
cmat,
dmat,
names=names or self.variable_names(),
order=order,
check=check)
from sage.misc.cache import Cache
cache = Cache(FreeAlgebra_generic)
p = sage.combinat.partition.Partition([])
self._self_to_s_cache[0] = {p: {p: R(1)}}
return
else:
self._self_to_s_cache[n] = {}
#Fill in the cache from the k-Schurs to the Schurs
for p in sage.combinat.partition.Partitions_n(n):
if max(p) > self.k:
self._self_to_s_cache[n][p] = {}
continue
katom = p.k_atom(self.k)
res = sum([t**tab.charge() * s(tab.shape()) for tab in katom],
zero)
self._self_to_s_cache[n][p] = res.monomial_coefficients()
class Element(kSchurFunctions_generic.Element):
pass
#############
# Cache #
#############
from sage.misc.cache import Cache
cache_t = Cache(kSchurFunctions_t)
# Backward compatibility for unpickling
from sage.structure.sage_object import register_unpickle_override
register_unpickle_override('sage.combinat.sf.kschur', 'kSchurFunction_t',
kSchurFunctions_t.Element)
sage: S2 is S #indirect doctest
True
"""
br = self.base_ring()
q, t = self.q, self.t
c[c.key(br, q=q, t=t)] = self
c[c.key(br, q, t)] = self
c[c.key(br, q, t)] = self
c[c.key(br, q, t=t)] = self
#############
# Cache #
#############
from sage.misc.cache import Cache
cache_p = Cache(MacdonaldPolynomials_p)
cache_j = Cache(MacdonaldPolynomials_j)
cache_q = Cache(MacdonaldPolynomials_q)
cache_h = Cache(MacdonaldPolynomials_h)
cache_ht = Cache(MacdonaldPolynomials_ht)
cache_s = Cache(MacdonaldPolynomials_s)
# Backward compatibility for unpickling
from sage.structure.sage_object import register_unpickle_override
register_unpickle_override('sage.combinat.sf.macdonald',
'MacdonaldPolynomial_h',
MacdonaldPolynomials_h.Element)
register_unpickle_override('sage.combinat.sf.macdonald',
'MacdonaldPolynomial_ht',
MacdonaldPolynomials_ht.Element)
register_unpickle_override('sage.combinat.sf.macdonald',
sage: Q([1])^2 # indirect doctest
JackQ[1, 1] + (2/(t+1))*JackQ[2]
sage: Q([2])^2
JackQ[2, 2] + (2/(t+1))*JackQ[3, 1] + ((6*t+6)/(6*t^2+5*t+1))*JackQ[4]
"""
return self(self._P(left) * self._P(right))
class Element(JackPolynomials_generic.Element):
pass
#############
# Cache #
#############
from sage.misc.cache import Cache
cache_p = Cache(JackPolynomials_p)
cache_j = Cache(JackPolynomials_j)
cache_q = Cache(JackPolynomials_q)
cache_z = Cache(ZonalPolynomials)
# Backward compatibility for unpickling
from sage.structure.sage_object import register_unpickle_override
register_unpickle_override('sage.combinat.sf.jack', 'JackPolynomial_j',
JackPolynomials_j.Element)
register_unpickle_override('sage.combinat.sf.jack', 'JackPolynomial_p',
JackPolynomials_p.Element)
register_unpickle_override('sage.combinat.sf.jack', 'JackPolynomial_q',
JackPolynomials_q.Element)
EXAMPLES::
sage: from sage.combinat.sf.llt import *
sage: HCosp3 = LLT_cospin(QQ, 3)
sage: HCosp3._m_cache(2)
sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())]
sage: l( HCosp3._self_to_m_cache[2] )
[([1, 1], [([1, 1], t + 1), ([2], 1)]), ([2], [([1, 1], 1), ([2], 1)])]
sage: l( HCosp3._m_to_self_cache[2] )
[([1, 1], [([1, 1], 1/t), ([2], -1/t)]),
([2], [([1, 1], -1/t), ([2], (t + 1)/t)])]
"""
self._invert_morphism(n, self.base_ring(), self._self_to_m_cache, \
self._m_to_self_cache, to_other_function = self._to_m)
class Element(LLT_generic.Element):
pass
from sage.misc.cache import Cache
cache_llt = Cache(LLT_class)
cache_llt_cospin = Cache(LLT_cospin)
cache_llt_spin = Cache(LLT_spin)
# Backward compatibility for unpickling
from sage.structure.sage_object import register_unpickle_override
register_unpickle_override('sage.combinat.sf.llt', 'LLTElement_spin',
LLT_spin.Element)
register_unpickle_override('sage.combinat.sf.llt', 'LLTElement_cospin',
LLT_cospin.Element)