class RationalCherednikAlgebra(CombinatorialFreeModule):
r"""
A rational Cherednik algebra.
Let `k` be a field. Let `W` be a complex reflection group acting on
a vector space `\mathfrak{h}` (over `k`). Let `\mathfrak{h}^*` denote
the corresponding dual vector space. Let `\cdot` denote the
natural action of `w` on `\mathfrak{h}` and `\mathfrak{h}^*`. Let
`\mathcal{S}` denote the set of reflections of `W` and `\alpha_s`
and `\alpha_s^{\vee}` are the associated root and coroot of `s`. Let
`c = (c_s)_{s \in W}` such that `c_s = c_{tst^{-1}}` for all `t \in W`.
The *rational Cherednik algebra* is the `k`-algebra
`H_{c,t}(W) = T(\mathfrak{h} \oplus \mathfrak{h}^*) \otimes kW` with
parameters `c, t \in k` that is subject to the relations:
.. MATH::
\begin{aligned}
w \alpha & = (w \cdot \alpha) w,
\\ \alpha^{\vee} w & = w (w^{-1} \cdot \alpha^{\vee}),
\\ \alpha \alpha^{\vee} & = \alpha^{\vee} \alpha
+ t \langle \alpha^{\vee}, \alpha \rangle
+ \sum_{s \in \mathcal{S}} c_s \frac{\langle \alpha^{\vee},
\alpha_s \rangle \langle \alpha^{\vee}_s, \alpha \rangle}{
\langle \alpha^{\vee}, \alpha \rangle} s,
\end{aligned}
where `w \in W` and `\alpha \in \mathfrak{h}` and
`\alpha^{\vee} \in \mathfrak{h}^*`.
INPUT:
- ``ct`` -- a finite Cartan type
- ``c`` -- the parameters `c_s` given as an element or a tuple, where
the first entry is the one for the long roots and (for
non-simply-laced types) the second is for the short roots
- ``t`` -- the parameter `t`
- ``base_ring`` -- (optional) the base ring
- ``prefix`` -- (default: ``('a', 's', 'ac')``) the prefixes
.. TODO::
Implement a version for complex reflection groups.
REFERENCES:
- [GGOR2003]_
- [EM2001]_
"""
@staticmethod
def __classcall_private__(cls,
ct,
c=1,
t=None,
base_ring=None,
prefix=('a', 's', 'ac')):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: R1 = algebras.RationalCherednik(['B',2], 1, 1, QQ)
sage: R2 = algebras.RationalCherednik(CartanType(['B',2]), [1,1], 1, QQ, ('a', 's', 'ac'))
sage: R1 is R2
True
"""
ct = CartanType(ct)
if not ct.is_finite():
raise ValueError("the Cartan type must be finite")
if base_ring is None:
if t is None:
base_ring = QQ
else:
base_ring = t.parent()
if t is None:
t = base_ring.one()
else:
t = base_ring(t)
# Normalize the parameter c
if isinstance(c, (tuple, list)):
if ct.is_simply_laced():
if len(c) != 1:
raise ValueError(
"1 parameter c_s must be given for simply-laced types")
c = (base_ring(c[0]), )
else:
if len(c) != 2:
raise ValueError(
"2 parameters c_s must be given for non-simply-laced types"
)
c = (base_ring(c[0]), base_ring(c[1]))
else:
c = base_ring(c)
if ct.is_simply_laced():
c = (c, )
else:
c = (c, c)
return super(RationalCherednikAlgebra,
cls).__classcall__(cls, ct, c, t, base_ring,
tuple(prefix))
def __init__(self, ct, c, t, base_ring, prefix):
r"""
Initialize ``self``.
EXAMPLES::
sage: k = QQ['c,t']
sage: R = algebras.RationalCherednik(['A',2], k.gen(0), k.gen(1))
sage: TestSuite(R).run() # long time
"""
self._c = c
self._t = t
self._cartan_type = ct
self._weyl = RootSystem(ct).root_lattice().weyl_group(prefix=prefix[1])
self._hd = IndexedFreeAbelianMonoid(ct.index_set(),
prefix=prefix[0],
bracket=False)
self._h = IndexedFreeAbelianMonoid(ct.index_set(),
prefix=prefix[2],
bracket=False)
indices = DisjointUnionEnumeratedSets([self._hd, self._weyl, self._h])
CombinatorialFreeModule.__init__(
self,
base_ring,
indices,
category=Algebras(base_ring).WithBasis().Graded(),
sorting_key=self._genkey)
def _genkey(self, t):
"""
Construct a key for comparison for a term indexed by ``t``.
The key we create is the tuple in the following order:
- overall degree
- length of the weyl group element
- the weyl group element
- the element of `\mathfrak{h}`
- the element of `\mathfrak{h}^*`
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.an_element()**2 # indirect doctest
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2
"""
return (self.degree_on_basis(t), t[1].length(), t[1], str(t[0]),
str(t[2]))
@lazy_attribute
def _reflections(self):
"""
A dictionary of reflections to a pair of the associated root
and coroot.
EXAMPLES::
sage: R = algebras.RationalCherednik(['B',2], [1,2], 1, QQ)
sage: [R._reflections[k] for k in sorted(R._reflections, key=str)]
[(alpha[1], alphacheck[1], 1),
(alpha[1] + alpha[2], 2*alphacheck[1] + alphacheck[2], 2),
(alpha[2], alphacheck[2], 2),
(alpha[1] + 2*alpha[2], alphacheck[1] + alphacheck[2], 1)]
"""
d = {}
for r in RootSystem(self._cartan_type).root_lattice().positive_roots():
s = self._weyl.from_reduced_word(r.associated_reflection())
if r.is_short_root():
c = self._c[1]
else:
c = self._c[0]
d[s] = (r, r.associated_coroot(), c)
return d
def _repr_(self):
r"""
Return a string representation of ``self``.
EXAMPLES ::
sage: RationalCherednikAlgebra(['A',4], 2, 1, QQ)
Rational Cherednik Algebra of type ['A', 4] with c=2 and t=1
over Rational Field
sage: algebras.RationalCherednik(['B',2], [1,2], 1, QQ)
Rational Cherednik Algebra of type ['B', 2] with c_L=1 and c_S=2
and t=1 over Rational Field
"""
ret = "Rational Cherednik Algebra of type {} with ".format(
self._cartan_type)
if self._cartan_type.is_simply_laced():
ret += "c={}".format(self._c[0])
else:
ret += "c_L={} and c_S={}".format(*self._c)
return ret + " and t={} over {}".format(self._t, self.base_ring())
def _repr_term(self, t):
"""
Return a string representation of the term indexed by ``t``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.an_element() # indirect doctest
3*ac1 + 2*s1 + a1
sage: R.one() # indirect doctest
I
"""
r = []
if t[0] != self._hd.one():
r.append(t[0])
if t[1] != self._weyl.one():
r.append(t[1])
if t[2] != self._h.one():
r.append(t[2])
if not r:
return 'I'
return '*'.join(repr(x) for x in r)
def algebra_generators(self):
"""
Return the algebra generators of ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: list(R.algebra_generators())
[a1, a2, s1, s2, ac1, ac2]
"""
keys = ['a' + str(i) for i in self._cartan_type.index_set()]
keys += ['s' + str(i) for i in self._cartan_type.index_set()]
keys += ['ac' + str(i) for i in self._cartan_type.index_set()]
def gen_map(k):
if k[0] == 's':
i = int(k[1:])
return self.monomial(
(self._hd.one(), self._weyl.group_generators()[i],
self._h.one()))
if k[1] == 'c':
i = int(k[2:])
return self.monomial((self._hd.one(), self._weyl.one(),
self._h.monoid_generators()[i]))
i = int(k[1:])
return self.monomial((self._hd.monoid_generators()[i],
self._weyl.one(), self._h.one()))
return Family(keys, gen_map)
@cached_method
def one_basis(self):
"""
Return the index of the element `1`.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.one_basis()
(1, 1, 1)
"""
return (self._hd.one(), self._weyl.one(), self._h.one())
def product_on_basis(self, left, right):
r"""
Return ``left`` multiplied by ``right`` in ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: a2 = R.algebra_generators()['a2']
sage: ac1 = R.algebra_generators()['ac1']
sage: a2 * ac1 # indirect doctest
a2*ac1
sage: ac1 * a2
-I + a2*ac1 - s1 - s2 + 1/2*s1*s2*s1
sage: x = R.an_element()
sage: [y * x for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
3*a1*ac1 + 2*a1*s1 + a1^2,
3*a2*ac1 + 2*a2*s1 + a1*a2,
3*s1*ac1 + 2*I - a1*s1,
3*s2*ac1 + 2*s2*s1 + a1*s2 + a2*s2,
3*ac1^2 - 2*s1*ac1 + 2*I + a1*ac1 + 2*s1 + 1/2*s2 + 1/2*s1*s2*s1,
3*ac1*ac2 + 2*s1*ac1 + 2*s1*ac2 - I + a1*ac2 - s1 - s2 + 1/2*s1*s2*s1]
sage: [x * y for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
6*I + 3*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 - 2*a1*s1 + a1^2,
-3*I + 3*a2*ac1 - 3*s1 - 3*s2 + 3/2*s1*s2*s1 + 2*a1*s1 + 2*a2*s1 + a1*a2,
-3*s1*ac1 + 2*I + a1*s1,
3*s2*ac1 + 3*s2*ac2 + 2*s1*s2 + a1*s2,
3*ac1^2 + 2*s1*ac1 + a1*ac1,
3*ac1*ac2 + 2*s1*ac2 + a1*ac2]
"""
# Make copies of the internal dictionaries
dl = dict(left[2]._monomial)
dr = dict(right[0]._monomial)
# If there is nothing to commute
if not dl and not dr:
return self.monomial((left[0], left[1] * right[1], right[2]))
R = self.base_ring()
I = self._cartan_type.index_set()
P = PolynomialRing(R, 'x', len(I))
G = P.gens()
gens_dict = {a: G[i] for i, a in enumerate(I)}
Q = RootSystem(self._cartan_type).root_lattice()
alpha = Q.simple_roots()
alphacheck = Q.simple_coroots()
def commute_w_hd(w, al): # al is given as a dictionary
ret = P.one()
for k in al:
x = sum(c * gens_dict[i] for i, c in alpha[k].weyl_action(w))
ret *= x**al[k]
ret = ret.dict()
for k in ret:
yield (self._hd({I[i]: e
for i, e in enumerate(k) if e != 0}), ret[k])
# Do Lac Ra if they are both non-trivial
if dl and dr:
il = dl.keys()[0]
ir = dr.keys()[0]
# Compute the commutator
terms = self._product_coroot_root(il, ir)
# remove the generator from the elements
dl[il] -= 1
if dl[il] == 0:
del dl[il]
dr[ir] -= 1
if dr[ir] == 0:
del dr[ir]
# We now commute right roots past the left reflections: s Ra = Ra' s
cur = self._from_dict({(hd, s * right[1], right[2]): c * cc
for s, c in terms
for hd, cc in commute_w_hd(s, dr)})
cur = self.monomial((left[0], left[1], self._h(dl))) * cur
# Add back in the commuted h and hd elements
rem = self.monomial((left[0], left[1], self._h(dl)))
rem = rem * self.monomial(
(self._hd({ir: 1}), self._weyl.one(), self._h({il: 1})))
rem = rem * self.monomial((self._hd(dr), right[1], right[2]))
return cur + rem
if dl:
# We have La Ls Lac Rs Rac,
# so we must commute Lac Rs = Rs Lac'
# and obtain La (Ls Rs) (Lac' Rac)
ret = P.one()
for k in dl:
x = sum(c * gens_dict[i] for i, c in alphacheck[k].weyl_action(
right[1].reduced_word(), inverse=True))
ret *= x**dl[k]
ret = ret.dict()
w = left[1] * right[1]
return self._from_dict({
(left[0], w,
self._h({I[i]: e
for i, e in enumerate(k) if e != 0}) * right[2]):
ret[k]
for k in ret
})
# Otherwise dr is non-trivial and we have La Ls Ra Rs Rac,
# so we must commute Ls Ra = Ra' Ls
w = left[1] * right[1]
return self._from_dict({(left[0] * hd, w, right[2]): c
for hd, c in commute_w_hd(left[1], dr)})
@cached_method
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 degree_on_basis(self, m):
"""
Return the degree on the monomial indexed by ``m``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: [R.degree_on_basis(g.leading_support())
....: for g in R.algebra_generators()]
[1, 1, 0, 0, -1, -1]
"""
return m[0].length() - m[2].length()
@cached_method
def trivial_idempotent(self):
"""
Return the trivial idempotent of ``self``.
Let `e = |W|^{-1} \sum_{w \in W} w` is the trivial idempotent.
Thus `e^2 = e` and `eW = We`. The trivial idempotent is used
in the construction of the spherical Cherednik algebra from
the rational Cherednik algebra by `U_{c,t}(W) = e H_{c,t}(W) e`.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.trivial_idempotent()
1/6*I + 1/6*s1 + 1/6*s2 + 1/6*s2*s1 + 1/6*s1*s2 + 1/6*s1*s2*s1
"""
coeff = self.base_ring()(~self._weyl.cardinality())
hd_one = self._hd.one() # root - a
h_one = self._h.one() # coroot - ac
return self._from_dict({(hd_one, w, h_one): coeff
for w in self._weyl},
remove_zeros=False)
@cached_method
def deformed_euler(self):
"""
Return the element `eu_k`.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.deformed_euler()
2*I + 2/3*a1*ac1 + 1/3*a1*ac2 + 1/3*a2*ac1 + 2/3*a2*ac2
+ s1 + s2 + s1*s2*s1
"""
I = self._cartan_type.index_set()
G = self.algebra_generators()
cm = ~CartanMatrix(self._cartan_type)
n = len(I)
ac = [G['ac' + str(i)] for i in I]
la = [
sum(cm[i, j] * G['a' + str(I[i])] for i in range(n))
for j in range(n)
]
return self.sum(ac[i] * la[i] for i in range(n))
@cached_method
def an_element(self):
"""
Return an element of ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.an_element()
3*ac1 + 2*s1 + a1
"""
G = self.algebra_generators()
i = str(self._cartan_type.index_set()[0])
return G['a' + i] + 2 * G['s' + i] + 3 * G['ac' + i]
def some_elements(self):
"""
Return some elements of ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.some_elements()
[0, I, 3*ac1 + 2*s1 + a1, a1, a2, s1, s2, ac1, ac2]
"""
ret = [self.zero(), self.one(), self.an_element()]
ret += list(self.algebra_generators())
return ret
class RationalCherednikAlgebra(CombinatorialFreeModule):
r"""
A rational Cherednik algebra.
Let `k` be a field. Let `W` be a complex reflection group acting on
a vector space `\mathfrak{h}` (over `k`). Let `\mathfrak{h}^*` denote
the corresponding dual vector space. Let `\cdot` denote the
natural action of `w` on `\mathfrak{h}` and `\mathfrak{h}^*`. Let
`\mathcal{S}` denote the set of reflections of `W` and `\alpha_s`
and `\alpha_s^{\vee}` are the associated root and coroot of `s`. Let
`c = (c_s)_{s \in W}` such that `c_s = c_{tst^{-1}}` for all `t \in W`.
The *rational Cherednik algebra* is the `k`-algebra
`H_{c,t}(W) = T(\mathfrak{h} \oplus \mathfrak{h}^*) \otimes kW` with
parameters `c, t \in k` that is subject to the relations:
.. MATH::
\begin{aligned}
w \alpha & = (w \cdot \alpha) w,
\\ \alpha^{\vee} w & = w (w^{-1} \cdot \alpha^{\vee}),
\\ \alpha \alpha^{\vee} & = \alpha^{\vee} \alpha
+ t \langle \alpha^{\vee}, \alpha \rangle
+ \sum_{s \in \mathcal{S}} c_s \frac{\langle \alpha^{\vee},
\alpha_s \rangle \langle \alpha^{\vee}_s, \alpha \rangle}{
\langle \alpha^{\vee}, \alpha \rangle} s,
\end{aligned}
where `w \in W` and `\alpha \in \mathfrak{h}` and
`\alpha^{\vee} \in \mathfrak{h}^*`.
INPUT:
- ``ct`` -- a finite Cartan type
- ``c`` -- the parameters `c_s` given as an element or a tuple, where
the first entry is the one for the long roots and (for
non-simply-laced types) the second is for the short roots
- ``t`` -- the parameter `t`
- ``base_ring`` -- (optional) the base ring
- ``prefix`` -- (default: ``('a', 's', 'ac')``) the prefixes
.. TODO::
Implement a version for complex reflection groups.
REFERENCES:
- [GGOR2003]_
- [EM2001]_
"""
@staticmethod
def __classcall_private__(cls, ct, c=1, t=None, base_ring=None, prefix=('a', 's', 'ac')):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: R1 = algebras.RationalCherednik(['B',2], 1, 1, QQ)
sage: R2 = algebras.RationalCherednik(CartanType(['B',2]), [1,1], 1, QQ, ('a', 's', 'ac'))
sage: R1 is R2
True
"""
ct = CartanType(ct)
if not ct.is_finite():
raise ValueError("the Cartan type must be finite")
if base_ring is None:
if t is None:
base_ring = QQ
else:
base_ring = t.parent()
if t is None:
t = base_ring.one()
else:
t = base_ring(t)
# Normalize the parameter c
if isinstance(c, (tuple, list)):
if ct.is_simply_laced():
if len(c) != 1:
raise ValueError("1 parameter c_s must be given for simply-laced types")
c = (base_ring(c[0]),)
else:
if len(c) != 2:
raise ValueError("2 parameters c_s must be given for non-simply-laced types")
c = (base_ring(c[0]), base_ring(c[1]))
else:
c = base_ring(c)
if ct.is_simply_laced():
c = (c,)
else:
c = (c, c)
return super(RationalCherednikAlgebra, cls).__classcall__(cls, ct, c, t, base_ring, tuple(prefix))
def __init__(self, ct, c, t, base_ring, prefix):
r"""
Initialize ``self``.
EXAMPLES::
sage: k = QQ['c,t']
sage: R = algebras.RationalCherednik(['A',2], k.gen(0), k.gen(1))
sage: TestSuite(R).run() # long time
"""
self._c = c
self._t = t
self._cartan_type = ct
self._weyl = RootSystem(ct).root_lattice().weyl_group(prefix=prefix[1])
self._hd = IndexedFreeAbelianMonoid(ct.index_set(), prefix=prefix[0],
bracket=False)
self._h = IndexedFreeAbelianMonoid(ct.index_set(), prefix=prefix[2],
bracket=False)
indices = DisjointUnionEnumeratedSets([self._hd, self._weyl, self._h])
CombinatorialFreeModule.__init__(self, base_ring, indices,
category=Algebras(base_ring).WithBasis().Graded(),
sorting_key=self._genkey)
def _genkey(self, t):
"""
Construct a key for comparison for a term indexed by ``t``.
The key we create is the tuple in the following order:
- overall degree
- length of the Weyl group element
- the Weyl group element
- the element of `\mathfrak{h}`
- the element of `\mathfrak{h}^*`
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.an_element()**2 # indirect doctest
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2
"""
return (self.degree_on_basis(t), t[1].length(), t[1], str(t[0]), str(t[2]))
@lazy_attribute
def _reflections(self):
"""
A dictionary of reflections to a pair of the associated root
and coroot.
EXAMPLES::
sage: R = algebras.RationalCherednik(['B',2], [1,2], 1, QQ)
sage: [R._reflections[k] for k in sorted(R._reflections, key=str)]
[(alpha[1], alphacheck[1], 1),
(alpha[1] + alpha[2], 2*alphacheck[1] + alphacheck[2], 2),
(alpha[2], alphacheck[2], 2),
(alpha[1] + 2*alpha[2], alphacheck[1] + alphacheck[2], 1)]
"""
d = {}
for r in RootSystem(self._cartan_type).root_lattice().positive_roots():
s = self._weyl.from_reduced_word(r.associated_reflection())
if r.is_short_root():
c = self._c[1]
else:
c = self._c[0]
d[s] = (r, r.associated_coroot(), c)
return d
def _repr_(self):
r"""
Return a string representation of ``self``.
EXAMPLES ::
sage: RationalCherednikAlgebra(['A',4], 2, 1, QQ)
Rational Cherednik Algebra of type ['A', 4] with c=2 and t=1
over Rational Field
sage: algebras.RationalCherednik(['B',2], [1,2], 1, QQ)
Rational Cherednik Algebra of type ['B', 2] with c_L=1 and c_S=2
and t=1 over Rational Field
"""
ret = "Rational Cherednik Algebra of type {} with ".format(self._cartan_type)
if self._cartan_type.is_simply_laced():
ret += "c={}".format(self._c[0])
else:
ret += "c_L={} and c_S={}".format(*self._c)
return ret + " and t={} over {}".format(self._t, self.base_ring())
def _repr_term(self, t):
"""
Return a string representation of the term indexed by ``t``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.an_element() # indirect doctest
3*ac1 + 2*s1 + a1
sage: R.one() # indirect doctest
I
"""
r = []
if t[0] != self._hd.one():
r.append(t[0])
if t[1] != self._weyl.one():
r.append(t[1])
if t[2] != self._h.one():
r.append(t[2])
if not r:
return 'I'
return '*'.join(repr(x) for x in r)
def algebra_generators(self):
"""
Return the algebra generators of ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: list(R.algebra_generators())
[a1, a2, s1, s2, ac1, ac2]
"""
keys = ['a'+str(i) for i in self._cartan_type.index_set()]
keys += ['s'+str(i) for i in self._cartan_type.index_set()]
keys += ['ac'+str(i) for i in self._cartan_type.index_set()]
def gen_map(k):
if k[0] == 's':
i = int(k[1:])
return self.monomial( (self._hd.one(),
self._weyl.group_generators()[i],
self._h.one()) )
if k[1] == 'c':
i = int(k[2:])
return self.monomial( (self._hd.one(),
self._weyl.one(),
self._h.monoid_generators()[i]) )
i = int(k[1:])
return self.monomial( (self._hd.monoid_generators()[i],
self._weyl.one(),
self._h.one()) )
return Family(keys, gen_map)
@cached_method
def one_basis(self):
"""
Return the index of the element `1`.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.one_basis()
(1, 1, 1)
"""
return (self._hd.one(), self._weyl.one(), self._h.one())
def product_on_basis(self, left, right):
r"""
Return ``left`` multiplied by ``right`` in ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: a2 = R.algebra_generators()['a2']
sage: ac1 = R.algebra_generators()['ac1']
sage: a2 * ac1 # indirect doctest
a2*ac1
sage: ac1 * a2
-I + a2*ac1 - s1 - s2 + 1/2*s1*s2*s1
sage: x = R.an_element()
sage: [y * x for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
3*a1*ac1 + 2*a1*s1 + a1^2,
3*a2*ac1 + 2*a2*s1 + a1*a2,
3*s1*ac1 + 2*I - a1*s1,
3*s2*ac1 + 2*s2*s1 + a1*s2 + a2*s2,
3*ac1^2 - 2*s1*ac1 + 2*I + a1*ac1 + 2*s1 + 1/2*s2 + 1/2*s1*s2*s1,
3*ac1*ac2 + 2*s1*ac1 + 2*s1*ac2 - I + a1*ac2 - s1 - s2 + 1/2*s1*s2*s1]
sage: [x * y for y in R.some_elements()]
[0,
3*ac1 + 2*s1 + a1,
9*ac1^2 + 10*I + 6*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 + a1^2,
6*I + 3*a1*ac1 + 6*s1 + 3/2*s2 + 3/2*s1*s2*s1 - 2*a1*s1 + a1^2,
-3*I + 3*a2*ac1 - 3*s1 - 3*s2 + 3/2*s1*s2*s1 + 2*a1*s1 + 2*a2*s1 + a1*a2,
-3*s1*ac1 + 2*I + a1*s1,
3*s2*ac1 + 3*s2*ac2 + 2*s1*s2 + a1*s2,
3*ac1^2 + 2*s1*ac1 + a1*ac1,
3*ac1*ac2 + 2*s1*ac2 + a1*ac2]
"""
# Make copies of the internal dictionaries
dl = dict(left[2]._monomial)
dr = dict(right[0]._monomial)
# If there is nothing to commute
if not dl and not dr:
return self.monomial((left[0], left[1] * right[1], right[2]))
R = self.base_ring()
I = self._cartan_type.index_set()
P = PolynomialRing(R, 'x', len(I))
G = P.gens()
gens_dict = {a:G[i] for i,a in enumerate(I)}
Q = RootSystem(self._cartan_type).root_lattice()
alpha = Q.simple_roots()
alphacheck = Q.simple_coroots()
def commute_w_hd(w, al): # al is given as a dictionary
ret = P.one()
for k in al:
x = sum(c * gens_dict[i] for i,c in alpha[k].weyl_action(w))
ret *= x**al[k]
ret = ret.dict()
for k in ret:
yield (self._hd({I[i]: e for i,e in enumerate(k) if e != 0}), ret[k])
# Do Lac Ra if they are both non-trivial
if dl and dr:
il = dl.keys()[0]
ir = dr.keys()[0]
# Compute the commutator
terms = self._product_coroot_root(il, ir)
# remove the generator from the elements
dl[il] -= 1
if dl[il] == 0:
del dl[il]
dr[ir] -= 1
if dr[ir] == 0:
del dr[ir]
# We now commute right roots past the left reflections: s Ra = Ra' s
cur = self._from_dict({ (hd, s*right[1], right[2]): c * cc
for s,c in terms
for hd, cc in commute_w_hd(s, dr) })
cur = self.monomial( (left[0], left[1], self._h(dl)) ) * cur
# Add back in the commuted h and hd elements
rem = self.monomial( (left[0], left[1], self._h(dl)) )
rem = rem * self.monomial( (self._hd({ir:1}), self._weyl.one(),
self._h({il:1})) )
rem = rem * self.monomial( (self._hd(dr), right[1], right[2]) )
return cur + rem
if dl:
# We have La Ls Lac Rs Rac,
# so we must commute Lac Rs = Rs Lac'
# and obtain La (Ls Rs) (Lac' Rac)
ret = P.one()
for k in dl:
x = sum(c * gens_dict[i]
for i,c in alphacheck[k].weyl_action(right[1].reduced_word(),
inverse=True))
ret *= x**dl[k]
ret = ret.dict()
w = left[1]*right[1]
return self._from_dict({ (left[0], w,
self._h({I[i]: e for i,e in enumerate(k)
if e != 0}) * right[2]
): ret[k]
for k in ret })
# Otherwise dr is non-trivial and we have La Ls Ra Rs Rac,
# so we must commute Ls Ra = Ra' Ls
w = left[1]*right[1]
return self._from_dict({ (left[0] * hd, w, right[2]): c
for hd, c in commute_w_hd(left[1], dr) })
@cached_method
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 degree_on_basis(self, m):
"""
Return the degree on the monomial indexed by ``m``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: [R.degree_on_basis(g.leading_support())
....: for g in R.algebra_generators()]
[1, 1, 0, 0, -1, -1]
"""
return m[0].length() - m[2].length()
@cached_method
def trivial_idempotent(self):
"""
Return the trivial idempotent of ``self``.
Let `e = |W|^{-1} \sum_{w \in W} w` is the trivial idempotent.
Thus `e^2 = e` and `eW = We`. The trivial idempotent is used
in the construction of the spherical Cherednik algebra from
the rational Cherednik algebra by `U_{c,t}(W) = e H_{c,t}(W) e`.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.trivial_idempotent()
1/6*I + 1/6*s1 + 1/6*s2 + 1/6*s2*s1 + 1/6*s1*s2 + 1/6*s1*s2*s1
"""
coeff = self.base_ring()(~self._weyl.cardinality())
hd_one = self._hd.one() # root - a
h_one = self._h.one() # coroot - ac
return self._from_dict({(hd_one, w, h_one): coeff for w in self._weyl},
remove_zeros=False)
@cached_method
def deformed_euler(self):
"""
Return the element `eu_k`.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.deformed_euler()
2*I + 2/3*a1*ac1 + 1/3*a1*ac2 + 1/3*a2*ac1 + 2/3*a2*ac2
+ s1 + s2 + s1*s2*s1
"""
I = self._cartan_type.index_set()
G = self.algebra_generators()
cm = ~CartanMatrix(self._cartan_type)
n = len(I)
ac = [G['ac'+str(i)] for i in I]
la = [sum(cm[i,j]*G['a'+str(I[i])] for i in range(n)) for j in range(n)]
return self.sum(ac[i]*la[i] for i in range(n))
@cached_method
def an_element(self):
"""
Return an element of ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.an_element()
3*ac1 + 2*s1 + a1
"""
G = self.algebra_generators()
i = str(self._cartan_type.index_set()[0])
return G['a'+i] + 2*G['s'+i] + 3*G['ac'+i]
def some_elements(self):
"""
Return some elements of ``self``.
EXAMPLES::
sage: R = algebras.RationalCherednik(['A',2], 1, 1, QQ)
sage: R.some_elements()
[0, I, 3*ac1 + 2*s1 + a1, a1, a2, s1, s2, ac1, ac2]
"""
ret = [self.zero(), self.one(), self.an_element()]
ret += list(self.algebra_generators())
return ret
