def _product_coroot_root(self, i, j):
r"""
Return the product `\alpha^{\vee}_i \alpha_j`.
EXAMPLES::
sage: k = QQ['c,t']
sage: R = algebras.RationalCherednik(['A',3], k.gen(0), k.gen(1))
sage: R._product_coroot_root(1, 1)
((1, 2*t), (s1*s2*s3*s2*s1, 1/2*c), (s2*s3*s2, 1/2*c),
(s1*s2*s1, 1/2*c), (s1, 2*c), (s3, 0), (s2, 1/2*c))
sage: R._product_coroot_root(1, 2)
((1, -t), (s1*s2*s3*s2*s1, 0), (s2*s3*s2, -1/2*c),
(s1*s2*s1, 1/2*c), (s1, -c), (s3, 0), (s2, -c))
sage: R._product_coroot_root(1, 3)
((1, 0), (s1*s2*s3*s2*s1, 1/2*c), (s2*s3*s2, -1/2*c),
(s1*s2*s1, -1/2*c), (s1, 0), (s3, 0), (s2, 1/2*c))
"""
Q = RootSystem(self._cartan_type).root_lattice()
ac = Q.simple_coroot(i)
al = Q.simple_root(j)
R = self.base_ring()
terms = [(self._weyl.one(), self._t * R(ac.scalar(al)))]
for s in self._reflections:
# p[0] is the root, p[1] is the coroot, p[2] the value c_s
pr, pc, c = self._reflections[s]
terms.append(
(s, c * R(ac.scalar(pr) * pc.scalar(al) / pc.scalar(pr))))
return tuple(terms)
def _product_coroot_root(self, i, j):
r"""
Return the product `\alpha^{\vee}_i \alpha_j`.
EXAMPLES::
sage: k = QQ['c,t']
sage: R = algebras.RationalCherednik(['A',3], k.gen(0), k.gen(1))
sage: R._product_coroot_root(1, 1)
((1, 2*t), (s1*s2*s3*s2*s1, 1/2*c), (s2*s3*s2, 1/2*c),
(s1*s2*s1, 1/2*c), (s1, 2*c), (s3, 0), (s2, 1/2*c))
sage: R._product_coroot_root(1, 2)
((1, -t), (s1*s2*s3*s2*s1, 0), (s2*s3*s2, -1/2*c),
(s1*s2*s1, 1/2*c), (s1, -c), (s3, 0), (s2, -c))
sage: R._product_coroot_root(1, 3)
((1, 0), (s1*s2*s3*s2*s1, 1/2*c), (s2*s3*s2, -1/2*c),
(s1*s2*s1, -1/2*c), (s1, 0), (s3, 0), (s2, 1/2*c))
"""
Q = RootSystem(self._cartan_type).root_lattice()
ac = Q.simple_coroot(i)
al = Q.simple_root(j)
R = self.base_ring()
terms = [( self._weyl.one(), self._t * R(ac.scalar(al)) )]
for s in self._reflections:
# p[0] is the root, p[1] is the coroot, p[2] the value c_s
pr, pc, c = self._reflections[s]
terms.append(( s, c * R(ac.scalar(pr) * pc.scalar(al)
/ pc.scalar(pr)) ))
return tuple(terms)
