Coverage for local/lib/python2.7/site-packages/sage/algebras/nil_coxeter_algebra.py : 97%

""" 

Nil-Coxeter Algebra 

""" 

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

# Copyright (C) 2011 Chris Berg <cberg at fields.utoronto.ca> 

# Anne Schilling <anne at math.ucdavis.edu> 

# 

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

# http://www.gnu.org/licenses/ 

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

from sage.algebras.iwahori_hecke_algebra import IwahoriHeckeAlgebra 

from sage.combinat.sf.sf import SymmetricFunctions 

from sage.misc.misc_c import prod 

from sage.rings.rational_field import QQ 

from sage.combinat.partition import Partitions 

class NilCoxeterAlgebra(IwahoriHeckeAlgebra.T): 

r""" 

Construct the Nil-Coxeter algebra of given type. 

This is the algebra 

with generators `u_i` for every node `i` of the corresponding Dynkin 

diagram. It has the usual braid relations (from the Weyl group) as well 

as the quadratic relation `u_i^2 = 0`. 

INPUT: 

- ``W`` -- a Weyl group 

OPTIONAL ARGUMENTS: 

- ``base_ring`` -- a ring (default is the rational numbers) 

- ``prefix`` -- a label for the generators (default "u") 

EXAMPLES:: 

sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1])) 

sage: u0, u1, u2, u3 = U.algebra_generators() 

sage: u1*u1 

0 

sage: u2*u1*u2 == u1*u2*u1 

True 

sage: U.an_element() 

u[0,1,2,3] + 2*u[0] + 3*u[1] + 1 

""" 

def __init__(self, W, base_ring = QQ, prefix='u'): 

r""" 

Initiate the affine nil-Coxeter algebra corresponding to the Weyl 

group `W` over the base ring. 

EXAMPLES:: 

sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1])); U 

The Nil-Coxeter Algebra of Type A3~ over Rational Field 

sage: TestSuite(U).run() 

sage: U = NilCoxeterAlgebra(WeylGroup(['C',3]), ZZ); U 

The Nil-Coxeter Algebra of Type C3 over Integer Ring 

sage: TestSuite(U).run() 

""" 

self._W = W 

self._n = W.n 

self._base_ring = base_ring 

self._cartan_type = W.cartan_type() 

H = IwahoriHeckeAlgebra(W, 0, 0, base_ring=base_ring) 

super(IwahoriHeckeAlgebra.T,self).__init__(H, prefix=prefix) 

def _repr_(self): 

r""" 

EXAMPLES :: 

sage: NilCoxeterAlgebra(WeylGroup(['A',3,1])) # indirect doctest 

The Nil-Coxeter Algebra of Type A3~ over Rational Field 

""" 

return "The Nil-Coxeter Algebra of Type %s over %s"%(self._cartan_type._repr_(compact=True), self.base_ring()) 

def homogeneous_generator_noncommutative_variables(self, r): 

r""" 

Give the `r^{th}` homogeneous function inside the Nil-Coxeter algebra. 

In finite type `A` this is the sum of all decreasing elements of length `r`. 

In affine type `A` this is the sum of all cyclically decreasing elements of length `r`. 

This is only defined in finite type `A`, `B` and affine types `A^{(1)}`, `B^{(1)}`, `C^{(1)}`, `D^{(1)}`. 

INPUT: 

- ``r`` -- a positive integer at most the rank of the Weyl group 

EXAMPLES:: 

sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1])) 

sage: U.homogeneous_generator_noncommutative_variables(2) 

u[1,0] + u[2,0] + u[0,3] + u[3,2] + u[3,1] + u[2,1] 

sage: U = NilCoxeterAlgebra(WeylGroup(['B',4])) 

sage: U.homogeneous_generator_noncommutative_variables(2) 

u[1,2] + u[2,1] + u[3,1] + u[4,1] + u[2,3] + u[3,2] + u[4,2] + u[3,4] + u[4,3] 

sage: U = NilCoxeterAlgebra(WeylGroup(['C',3])) 

sage: U.homogeneous_generator_noncommutative_variables(2) 

Traceback (most recent call last): 

... 

AssertionError: Analogue of symmetric functions in noncommutative variables is not defined in type ['C', 3] 

TESTS:: 

sage: U = NilCoxeterAlgebra(WeylGroup(['B',3,1])) 

sage: U.homogeneous_generator_noncommutative_variables(-1) 

0 

sage: U.homogeneous_generator_noncommutative_variables(0) 

1 

""" 

assert (len(self._cartan_type) == 2 and self._cartan_type[0] in ['A','B']) or (len(self._cartan_type) == 3 and self._cartan_type[2] == 1), "Analogue of symmetric functions in noncommutative variables is not defined in type %s"%(self._cartan_type) 

if r >= self._n: 

return self.zero() 

return self.sum_of_monomials(w for w in self._W.pieri_factors() if w.length() == r) 

def homogeneous_noncommutative_variables(self,la): 

r""" 

Give the homogeneous function indexed by `la`, viewed inside the Nil-Coxeter algebra. 

This is only defined in finite type `A`, `B` and affine types `A^{(1)}`, `B^{(1)}`, `C^{(1)}`, `D^{(1)}`. 

INPUT: 

- ``la`` -- a partition with first part bounded by the rank of the Weyl group 

EXAMPLES:: 

sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1])) 

sage: U.homogeneous_noncommutative_variables([2,1]) 

u[1,2,0] + 2*u[2,1,0] + u[0,2,0] + u[0,2,1] + u[1,2,1] + u[2,1,2] + u[2,0,2] + u[1,0,2] 

TESTS:: 

sage: U = NilCoxeterAlgebra(WeylGroup(['B',2,1])) 

sage: U.homogeneous_noncommutative_variables([]) 

1 

""" 

return prod(self.homogeneous_generator_noncommutative_variables(p) for p in la) 

def k_schur_noncommutative_variables(self, la): 

r""" 

In type `A^{(1)}` this is the `k`-Schur function in noncommutative variables 

defined by Thomas Lam [Lam2005]_. 

This function is currently only defined in type `A^{(1)}`. 

INPUT: 

- ``la`` -- a partition with first part bounded by the rank of the Weyl group 

EXAMPLES:: 

sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1])) 

sage: A.k_schur_noncommutative_variables([2,2]) 

u[0,3,1,0] + u[3,1,2,0] + u[1,2,0,1] + u[3,2,0,3] + u[2,0,3,1] + u[2,3,1,2] 

TESTS:: 

sage: A = NilCoxeterAlgebra(WeylGroup(['A',3,1])) 

sage: A.k_schur_noncommutative_variables([]) 

1 

sage: A.k_schur_noncommutative_variables([1,2]) 

Traceback (most recent call last): 

... 

AssertionError: [1, 2] is not a partition. 

sage: A.k_schur_noncommutative_variables([4,2]) 

Traceback (most recent call last): 

... 

AssertionError: [4, 2] is not a 3-bounded partition. 

sage: C = NilCoxeterAlgebra(WeylGroup(['C',3,1])) 

sage: C.k_schur_noncommutative_variables([2,2]) 

Traceback (most recent call last): 

... 

AssertionError: Weyl Group of type ['C', 3, 1] (as a matrix group acting on the root space) is not affine type A. 

""" 

assert self._cartan_type[0] == 'A' and len(self._cartan_type) == 3 and self._cartan_type[2] == 1, "%s is not affine type A."%(self._W) 

assert la in Partitions(), "%s is not a partition."%(la) 

assert (len(la) == 0 or la[0] < self._W.n), "%s is not a %s-bounded partition."%(la, self._W.n-1) 

Sym = SymmetricFunctions(self._base_ring) 

h = Sym.homogeneous() 

ks = Sym.kschur(self._n-1,1) 

f = h(ks[la]) 

return sum(f.coefficient(x)*self.homogeneous_noncommutative_variables(x) for x in f.support())