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""" 

Rational Cherednik Algebras 

""" 

#***************************************************************************** 

# Copyright (C) 2015 Travis Scrimshaw <tscrim at ucdavis.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

 

from sage.misc.cachefunc import cached_method 

from sage.misc.lazy_attribute import lazy_attribute 

from sage.categories.algebras import Algebras 

from sage.combinat.free_module import CombinatorialFreeModule 

from sage.combinat.root_system.cartan_type import CartanType 

from sage.combinat.root_system.cartan_matrix import CartanMatrix 

from sage.combinat.root_system.root_system import RootSystem 

from sage.sets.disjoint_union_enumerated_sets import DisjointUnionEnumeratedSets 

from sage.sets.family import Family 

from sage.monoids.indexed_free_monoid import IndexedFreeAbelianMonoid 

from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing 

from sage.rings.all import QQ 

 

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 = next(iter(dl.keys())) 

ir = next(iter(dr.keys())) 

 

# 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), 

(s2, 1/2*c), 

(s3, 0), 

(s1*s2*s3*s2*s1, 1/2*c), 

(s2*s3*s2, 1/2*c), 

(s1, 2*c), 

(s1*s2*s1, 1/2*c)) 

sage: R._product_coroot_root(1, 2) 

((1, -t), 

(s2, -c), 

(s3, 0), 

(s1*s2*s3*s2*s1, 0), 

(s2*s3*s2, -1/2*c), 

(s1, -c), 

(s1*s2*s1, 1/2*c)) 

sage: R._product_coroot_root(1, 3) 

((1, 0), 

(s2, 1/2*c), 

(s3, 0), 

(s1*s2*s3*s2*s1, 1/2*c), 

(s2*s3*s2, -1/2*c), 

(s1, 0), 

(s1*s2*s1, -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