# This file was *autogenerated* from the file local_braid.sage
from sage.all_cmdline import *   # import sage library
_sage_const_2 = Integer(2); _sage_const_1 = Integer(1); _sage_const_0 = Integer(0)
"""
Local Braid Groups

This module contains several extensions to the sage braid group class.
It fills in additional methods and extend the existing method burau_matrix

##############################################################################
#       Copyright (C) 2016 Sebastian Oehms <[email protected]>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
##############################################################################

This module contains extensions of the following classes from sage.groups.braid

sage class                                     |  extended class
-------------------------------------------------------------------------------
Braid                                          |  local_Braid
BraidGroup_class                               |  local_BraidGroup_class

For further information see the documentation of the corresponding classes and methods

sage: print local_Braid.__doc__
sage: print local_Braid.burau_matrix_unitary.__doc__
sage: print local_Braid.__burau_matrix_wikipedia__.__doc__
sage: print local_Braid.__burau_matrix_unitary__.__doc__

sage: print local_BraidGroup_class.__doc__
sage: print local_BraidGroup_class.__doc__.__unitary_form__

EXAMPLES:

see the documentation of the corresponding classes and methods

TESTS:

see the documentation of the corresponding classes and methods

AUTHOR

- Sebastian Oehms, Oct. 2016

"""

####################################################################################################
# Extension to Braid Element Class defining the _burau_matrix_unitary_
####################################################################################################
from sage.groups.braid import *

class local_Braid(Braid):
"""
This Class contains extensions to the Braid Element class.

If you don't see this well formatted type

sage: print local_Braid.__doc__

This class has two new methods and one overwriting the corresponding methods of the class Braid.

New methods:
- __burau_matrix_wikipedia__
- __burau_matrix_unitary__

Modified:
. burau_matrix (using __burau_matrix_wikipedia__, __burau_matrix_unitary__ )

For more information type

sage: print local_Braid.__burau_matrix_wikipedia__.__doc__
sage: print local_Braid.__burau_matrix_unitary__.__doc__
sage: print local_Braid.burau_matrix.__doc__

AUTHOR

- Sebastian Oehms, Oct. 2016

"""

def __burau_matrix_wikipedia__( self, var='t'):
"""
The explicit reduced Burau representation given on Wikipedia since 11.03.2014 is different from the version
implemented in the braid-basis class. The version according to the recent Wikipedia-Page is implemented here.

If you don't see this well formatted type

sage: print local_Braid.__burau_matrix_wikipedia__.__doc__

Moreover it is the version used by Squier and Coxeter and will be used here to implement the unitary Burau
reoresentation

INPUT:

- "var":  string (default: 't'); the name of the variable in the entries of the matrix. See also:
print sage.groups.braid.Braid.burau_matrix.__doc__

OUTPUT:

The Burau matrix of the braid. It is a matrix whose entries are Laurent polynomials in the variable "var".

EXAMPLES:

sage: sage.groups.braid.BraidGroup_class = local_BraidGroup_class
sage: B4 = BraidGroup(4)
sage: b1, b2, b3 = B4.gens()
sage: b = b1*b2/b3/b2
sage: type(b)
<class 'lib.local_braid.local_BraidGroup_class_with_category.element_class'>
sage: b.__burau_matrix_wikipedia__()
[    1 - t -t^-1 + 1        -1]
[        1 -t^-1 + 1        -1]
[        1     -t^-1         0]
sage: b.__burau_matrix_wikipedia__(var='x')
[    1 - x -x^-1 + 1        -1]
[        1 -x^-1 + 1        -1]
[        1     -x^-1         0]

compare with the original method

sage: b.burau_matrix()
[       1 - t            0      t - t^2          t^2]
[           1            0            0            0]
[           0            0            1            0]
[           0         t^-2 -t^-2 + t^-1    -t^-1 + 1]
sage: b.burau_matrix(reduced=True)
[                     0               -t + t^2                   -t^2]
[                     0            1 - t + t^2                   -t^2]
[                  t^-2 -t^-2 + t^-1 - t + t^2        -t^-1 + 1 - t^2]
sage:

REFERENCES:

- wikipedia:'Burau_representation'

AUTHOR

- Sebastian Oehms, Oct. 2016

"""
R = LaurentPolynomialRing(IntegerRing(), var)
t = R.gen()
n = self.strands()
M = identity_matrix(R, n-_sage_const_1 )
for i in self.Tietze():
A = identity_matrix(R, n-_sage_const_1 )
if i > _sage_const_0 :
A[i-_sage_const_1 , i-_sage_const_1 ] = -t
if i > _sage_const_1 :
A[i-_sage_const_1 , i-_sage_const_2 ] = t
if i < n-_sage_const_1 :
A[i-_sage_const_1 , i] = _sage_const_1
if i < _sage_const_0 :
A[-i-_sage_const_1 , -i-_sage_const_1 ] = -t**(-_sage_const_1 )
if -i > _sage_const_1 :
A[-i-_sage_const_1 , -i-_sage_const_2 ] = _sage_const_1
if -i < n-_sage_const_1 :
A[-i-_sage_const_1 , -i] = t**(-_sage_const_1 )
M = M * A
return M

def __burau_matrix_unitary__(self, var='s'):
"""
Return the unitary form of the Burau matrix of the braid according to

CRAIG C. SQUIE: THE BURAU REPRESENTATION IS UNITARY, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY,
Volume 90. Number 2, February 1984

If you don't see this well formatted type

sage: print local_Braid.__burau_matrix_unitary__.__doc__

INPUT:

- "var":  string (default: 's'); the name of the variable in the entries of the matrix. The connection with the
variable t of the original burau_matrix is t=s**2. See also:
print sage.groups.braid.Braid.burau_matrix.__doc__

OUTPUT:

The Burau matrix of the braid in the unitary form. It is obtained from the original burau_matrix by a base change
in order to preserve a hermitian form. It is a matrix whose entries are Laurent polynomials in the variable "var".
The original Burau matrix can be obtained by the method local_Braid.__burau_matrix_wikipedia__

EXAMPLES:

sage: sage.groups.braid.BraidGroup_class = local_BraidGroup_class
sage: B4 = BraidGroup(4)
sage: b1, b2, b3 = B4.gens()
sage: b = b1*b2/b3/b2
sage: type(b)
<class 'lib.local_braid.local_BraidGroup_class_with_category.element_class'>
sage: b.__burau_matrix_unitary__()
[  1 - s^2 -s^-1 + s      -s^2]
[     s^-1 -s^-2 + 1        -s]
[     s^-2     -s^-3         0]
sage: b.__burau_matrix_unitary__(var='x')
[  1 - x^2 -x^-1 + x      -x^2]
[     x^-1 -x^-2 + 1        -x]
[     x^-2     -x^-3         0]
sage:

compare with the version given on wikipedia:

sage: b.__burau_matrix_wikipedia__()
[    1 - t -t^-1 + 1        -1]
[        1 -t^-1 + 1        -1]
[        1     -t^-1         0]

compare with the original method:

sage: b.burau_matrix()
[       1 - t            0      t - t^2          t^2]
[           1            0            0            0]
[           0            0            1            0]
[           0         t^-2 -t^-2 + t^-1    -t^-1 + 1]
sage: b.burau_matrix(reduced=True)
[                     0               -t + t^2                   -t^2]
[                     0            1 - t + t^2                   -t^2]
[                  t^-2 -t^-2 + t^-1 - t + t^2        -t^-1 + 1 - t^2]
sage:

REFERENCES:

- Coxeter, H.S.M: "Factor groups of the braid groups, Proceedings of the Fourth Candian Mathematical Congress
(Vancover      1957), pp. 95-122".

- C. C. Squier:THE BURAU REPRESENTATION IS UNITARY, PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 90. Number 2, February 1984
- Tyakay Venkataramana: Image of the Burau Representation at $d$-th Roots of unity. ANNALS OF MATHEMATICS MAY 2014

AUTHOR

- Sebastian Oehms, Oct. 2016

"""

BurauOri = self.__burau_matrix_wikipedia__( )
d = BurauOri.dimensions()[_sage_const_0 ]
oriDomain = LaurentPolynomialRing(IntegerRing(), names=('t',)); (t,) = oriDomain._first_ngens(1)
newDomain = LaurentPolynomialRing(IntegerRing(), var)
s = newDomain.gen()
subsVar = oriDomain.hom( [s**_sage_const_2 ], codomain = newDomain )

BurauMat = matrix(d,d, lambda i,j:  subsVar(BurauOri[i,j] ) )
transformP = matrix( newDomain, d,d , lambda i, j: _sage_const_0 )
transformPI = matrix( newDomain, d,d , lambda i, j: _sage_const_0  )
for i in range(d):
transformP[i,i]=s**(i+_sage_const_1 )
transformPI[i,i]=s**(-i-_sage_const_1 )
res = transformPI * BurauMat * transformP
return res

def burau_matrix(self, var='t', reduced=False, version='default'):
"""
This method is a modification of the original burau_matrix-method. It contains an additional keyword-parameter
"version". If this keyword is not set or is set to the value 'default' it behaves like the original one.

If you don't see this well formatted type:

sage: print local_Braid.burau_matrix.__doc__

To read the original docstring type:

sage: print Braid.burau_matrix.__doc__

INPUT:

- "var":  string (default: 't'); the name of the variable in the entries of the matrix. See also:
print sage.groups.braid.Braid.burau_matrix.__doc__
- "reduced": boolean (default: 'False'); whether to return the reduced or unreduced Burau representation.
Note: if version is set to a value different from 'default' this keyword is ignored and treated
as set to 'True' (this means: no unreduced form for other versions)
- "version": string (default = 'default' ). The following values are possible

- "default" the method behaves like the original one. For more information on this see
sage: print Braid.burau_matrix.__doc__

- "unitary" gives the unitary form according to Squier. For more information on this see
sage: print local_Braid.__burau_matrix_unitary__.__doc__

- any value else gives the reduced form given on wikipedia. For more information on this see
sage: print local_Braid.__burau_matrix_wikipedia__.__doc__

OUTPUT:

The Burau matrix of the braid. It is a matrix whose entries are Laurent polynomials in the variable "var".

EXAMPLES:

sage: sage.groups.braid.BraidGroup_class = local_BraidGroup_class
sage: B4 = BraidGroup(4)
sage: b1, b2, b3 = B4.gens()
sage: b = b1*b2/b3/b2
sage: type(b)
<class 'lib.local_braid.local_BraidGroup_class_with_category.element_class'>
sage: b.burau_matrix()
[       1 - t            0      t - t^2          t^2]
[           1            0            0            0]
[           0            0            1            0]
[           0         t^-2 -t^-2 + t^-1    -t^-1 + 1]
sage: b.burau_matrix(version='unitary')
[  1 - t^2 -t^-1 + t      -t^2]
[     t^-1 -t^-2 + 1        -t]
[     t^-2     -t^-3         0]
sage: b.burau_matrix(version='wiki')
[    1 - t -t^-1 + 1        -1]
[        1 -t^-1 + 1        -1]
[        1     -t^-1         0]
sage: b.burau_matrix(version='wiki', reduced=True)
[    1 - t -t^-1 + 1        -1]
[        1 -t^-1 + 1        -1]
[        1     -t^-1         0]
sage: b.burau_matrix(version='wiki', reduced=False)
[    1 - t -t^-1 + 1        -1]
[        1 -t^-1 + 1        -1]
[        1     -t^-1         0]
sage: b.burau_matrix(reduced=True)
[                     0               -t + t^2                   -t^2]
[                     0            1 - t + t^2                   -t^2]
[                  t^-2 -t^-2 + t^-1 - t + t^2        -t^-1 + 1 - t^2]
sage: b.burau_matrix(var='s', version='unitary')
[  1 - s^2 -s^-1 + s      -s^2]
[     s^-1 -s^-2 + 1        -s]
[     s^-2     -s^-3         0]
sage:

AUTHOR

- Sebastian Oehms, Oct. 2016

"""

if version == 'default':
return sage.groups.braid.Braid.burau_matrix(self, var=var, reduced=reduced )
elif version == 'unitary':
return self.__burau_matrix_unitary__(var=var)
else:
return self.__burau_matrix_wikipedia__(var=var)

class local_BraidGroup_class(BraidGroup_class):
"""
This Class contains extensions to the sage BraidGroup class.

If you don't see this well formatted type:

sage: print local_BraidGroup_class.__doc__

New methods:

- __unitary_form__

Modified:

. Element Class is local_Braid instead of Braid. This contains extension of the burau_matrix -method
For more information type:

sage: print local_Braid.__doc__

AUTHOR

- Sebastian Oehms, Oct. 2016

"""
Element = local_Braid

def __unitary_form__( self, var='s' ):
"""
Returns the hermitian form with respect to the __unitary_burau_matrix__ of the Element Class

If you don't see this well formatted type:

sage: print local_BraidGroup_class.__unitary_form__.__doc__

The hermitian form returned by this method is kept invariant by the unitary Burau matrices returned by the
local_Braid -method burau_matrix() setting the version keyword to 'unitary'

For more information on the unitary Burau matrices type

sage: print local_Braid.__burau_matrix_unitary__.__doc__
sage: print local_Braid.burau_matrix.__doc__

INPUT:

- "var":  string (default: 's'); the name of the variable in the entries of the matrix of the unitary form.

OUTPUT:

The hermitian form as a quadratic matrix whose entries are Laurent polynomials in the variable "var".

EXAMPLES:

sage: sage.groups.braid.BraidGroup_class = local_BraidGroup_class
sage: B4 = BraidGroup(4)
sage: B4.__unitary_form__()
[s^-1 + s       -1        0]
[      -1 s^-1 + s       -1]
[       0       -1 s^-1 + s]
sage:
sage:
sage: B4.__unitary_form__(var='x')
[x^-1 + x       -1        0]
[      -1 x^-1 + x       -1]
[       0       -1 x^-1 + x]
sage:

REFERENCES:

- Coxeter, H.S.M: "Factor groups of the braid groups, Proceedings of the Fourth Candian Mathematical Congress
(Vancover      1957), pp. 95-122".

- C. C. Squier:THE BURAU REPRESENTATION IS UNITARY, PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 90. Number 2, February 1984

AUTHOR

- Sebastian Oehms, Oct. 2016

"""
R = LaurentPolynomialRing(IntegerRing(), var)
t = R.gen()
n = self.strands()
M = (t+t**(-_sage_const_1 ))*identity_matrix(R, n-_sage_const_1 )
for i in range(n-_sage_const_2 ):
M[i,i+_sage_const_1 ] =-_sage_const_1
M[i+_sage_const_1 ,i] =-_sage_const_1
return M