# Today we'll talk about what we can do in Sage, especially things related to Coxeter groups.
# We can ...
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# do simple calculations:
1+3
4 4 |
# create variables and carry out symbolic computations:
f=(x+1)^2
x, f.canonicalize_radical(), ((x+1)^2).canonicalize_radical(), f.simplify_exp()
(x, x^2 + 2*x + 1, x^2 + 2*x + 1, x^2 + 2*x + 1) (x, x^2 + 2*x + 1, x^2 + 2*x + 1, x^2 + 2*x + 1) |
# create some special, pre-programmed Coxeter groups by so-called 'Cartan type':
A5=CoxeterGroup(["A",5],implementation='matrix')
A5
Weyl Group of type ['A', 5] (as a matrix group acting on the ambient space) Weyl Group of type ['A', 5] (as a matrix group acting on the ambient space) |
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# More generally we can create an arbitrary Coxeter group using a Coxeter matrice:
h3matrix=matrix(3,[1,5,2,5,1,3,2,3,1])
H3=CoxeterGroup(h3matrix) # this is using the default 'reflection' implementation
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H3
Finite Coxeter group over Number Field in a with defining polynomial x^2 - 5 with Coxeter matrix: [1 5 2] [5 1 3] [2 3 1] Finite Coxeter group over Number Field in a with defining polynomial x^2 - 5 with Coxeter matrix: [1 5 2] [5 1 3] [2 3 1] |
# Later we may want to use the 'coxeter3' implementation for faster computations.
h3=CoxeterGroup(h3matrix,implementation='coxeter3')
h3
Finite Coxeter group over Number Field in a with defining polynomial x^2 - 5 with Coxeter matrix: [1 5 2] [5 1 3] [2 3 1] Finite Coxeter group over Number Field in a with defining polynomial x^2 - 5 with Coxeter matrix: [1 5 2] [5 1 3] [2 3 1] |
# We can access methods on the groups:
H3.is_finite(), h3.is_commutative()
(True, False) (True, False) |
# get the generators as a list:
gens= H3.gens()
(gens[0]*gens[1])^5
[1 0 0] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [0 0 1] |
# change the symbols for the generators:
a,b,c=H3.gens() # may not do much; still returns things like [2,1,3] as reduced words.
H3.gens()
( [ -1 1/2*a + 1/2 0] [ 0 1 0] [ 0 0 1], [ 1 0 0] [ 1 0 0] [1/2*a + 1/2 -1 1] [ 0 1 0] [ 0 0 1], [ 0 1 -1] ) ( [ -1 1/2*a + 1/2 0] [ 0 1 0] [ 0 0 1], [ 1 0 0] [ 1 0 0] [1/2*a + 1/2 -1 1] [ 0 1 0] [ 0 0 1], [ 0 1 -1] ) |
# create an element using a reduced word:
w=H3.from_reduced_word([1,2])
# by default the generators "are" 1,2,3,...; the exact relations can be ambiguous though.
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w=H3.from_reduced_word([1,2,1,2,1,3,2,1])
w
[-1/2*a - 3/2 1/2*a + 3/2 -1] [ -a - 1 1/2*a + 3/2 0] [-1/2*a - 1/2 1/2*a + 1/2 0] [-1/2*a - 3/2 1/2*a + 3/2 -1] [ -a - 1 1/2*a + 3/2 0] [-1/2*a - 1/2 1/2*a + 1/2 0] |
# create the reduced word graph and show it:
w_rwg=w.reduced_word_graph()
w_rwg, w_rwg.show()
(Graph on 7 vertices, None) (Graph on 7 vertices, None) 
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# Actually the input for from_reduced_word doesn't have to be reduced (!):
x=H3.from_reduced_word([2,3,2,3])
x
[ 1 0 0] [1/2*a + 1/2 -1 1] [1/2*a + 1/2 -1 0] [ 1 0 0] [1/2*a + 1/2 -1 1] [1/2*a + 1/2 -1 0] |
x.reduced_words() # returns the list of all reduced words of the elements
[[3, 2]] [[3, 2]] |
x.is_reduced() # there is no direct command to check if a word is reduced
Traceback (click to the left of this block for traceback) ... AttributeError: 'CoxeterMatrixGroup_with_category.element_class' object has no attribute 'is_reduced' Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_33.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("eC5pc19yZWR1Y2VkKCkgIyB0aGVyZSBpcyBubyBkaXJlY3QgY29tbWFuZCB0byBjaGVjayBpZiBhIHdvcmQgaXMgcmVkdWNlZA=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpy4PGP3/___code___.py", line 2, in <module>
exec compile(u'x.is_reduced() # there is no direct command to check if a word is reduced
File "", line 1, in <module>
File "sage/structure/element.pyx", line 493, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4550)
File "sage/structure/element.pyx", line 506, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4659)
File "sage/cpython/getattr.pyx", line 389, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2469)
AttributeError: 'CoxeterMatrixGroup_with_category.element_class' object has no attribute 'is_reduced'
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# but since we have the list of reduced words, we can easily check if a word is reduced
l=[2,3,2,3]
y=H3.from_reduced_word(l)
l in y.reduced_words() # not in, so l is not reduced
False False |
# of the reduced words, Sage has a way of picking a canonical reduced word;
# we don't know how Sage picks the canonical word, same as in Bjorner--Brenti?
x.reduced_word(), w.reduced_word()
([3, 2], [1, 2, 1, 2, 3, 1, 2, 1]) ([3, 2], [1, 2, 1, 2, 3, 1, 2, 1]) |
# find the left or right descent set of an element, better to specify a side:
x.reduced_words(), x.descents(side='left')
([[3, 2]], [3]) ([[3, 2]], [3]) |
# without specifying a side Sage picks a side randomly:
x.descents()
[2] [2] |
w.reduced_words(), w.descents(side='right')
([[1, 2, 1, 2, 3, 1, 2, 1], [1, 2, 1, 2, 1, 3, 2, 1], [2, 1, 2, 3, 1, 2, 1, 3], [2, 1, 2, 1, 3, 2, 1, 3], [2, 1, 2, 1, 2, 3, 2, 1], [2, 1, 2, 3, 1, 2, 3, 1], [2, 1, 2, 1, 3, 2, 3, 1]], [1, 3]) ([[1, 2, 1, 2, 3, 1, 2, 1], [1, 2, 1, 2, 1, 3, 2, 1], [2, 1, 2, 3, 1, 2, 1, 3], [2, 1, 2, 1, 3, 2, 1, 3], [2, 1, 2, 1, 2, 3, 2, 1], [2, 1, 2, 3, 1, 2, 3, 1], [2, 1, 2, 1, 3, 2, 3, 1]], [1, 3]) |
# Sage can't check if an element is fully commutative (FC) in general groups
w.is_fully_commutative()
Traceback (click to the left of this block for traceback) ... AttributeError: 'CoxeterMatrixGroup_with_category.element_class' object has no attribute 'is_fully_commutative' Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_39.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("IyBTYWdlIGNhbid0IGNoZWNrIGlmIGFuIGVsZW1lbnQgaXMgZnVsbHkgY29tbXV0YXRpdmUgKEZDKSBpbiBnZW5lcmFsIGdyb3Vwcwp3LmlzX2Z1bGx5X2NvbW11dGF0aXZlKCk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmp4li5V_/___code___.py", line 3, in <module>
exec compile(u'w.is_fully_commutative()
File "", line 1, in <module>
File "sage/structure/element.pyx", line 493, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4550)
File "sage/structure/element.pyx", line 506, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4659)
File "sage/cpython/getattr.pyx", line 389, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2469)
AttributeError: 'CoxeterMatrixGroup_with_category.element_class' object has no attribute 'is_fully_commutative'
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# But it can check FC-ness in affine Coxeter groups
W=CoxeterGroup(['A',5,1]) # the 1 indicates we have an "affine" group
W, W is A5
(Coxeter group over Integer Ring with Coxeter matrix: [1 3 2 2 2 3] [3 1 3 2 2 2] [2 3 1 3 2 2] [2 2 3 1 3 2] [2 2 2 3 1 3] [3 2 2 2 3 1], False) (Coxeter group over Integer Ring with Coxeter matrix: [1 3 2 2 2 3] [3 1 3 2 2 2] [2 3 1 3 2 2] [2 2 3 1 3 2] [2 2 2 3 1 3] [3 2 2 2 3 1], False) |
z=W.from_reduced_word([1,2,1,3])
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z.is_fully_commutative() # probably can't access the check if W is created this way
Traceback (click to the left of this block for traceback) ... AttributeError: 'CoxeterMatrixGroup_with_category.element_class' object has no attribute 'is_fully_commutative' Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_42.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ei5pc19mdWxseV9jb21tdXRhdGl2ZSgpICMgcHJvYmFibHkgY2FuJ3QgYWNjZXNzIHRoZSBjaGVjayBpZiBXIGlzIGNyZWF0ZWQgdGhpcyB3YXk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpdGmCnX/___code___.py", line 2, in <module>
exec compile(u"z.is_fully_commutative() # probably can't access the check if W is created this way" + '\n', '', 'single')
File "", line 1, in <module>
File "sage/structure/element.pyx", line 493, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4550)
File "sage/structure/element.pyx", line 506, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4659)
File "sage/cpython/getattr.pyx", line 389, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2469)
AttributeError: 'CoxeterMatrixGroup_with_category.element_class' object has no attribute 'is_fully_commutative'
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# But the check works if we create the group specifically as an affine group
WW=AffinePermutationGroup(['A',5,1])
W, WW # Sage thinks of W and WW as different things because they were created using different methods; consequently their elements have access different methods.
(Coxeter group over Integer Ring with Coxeter matrix: [1 3 2 2 2 3] [3 1 3 2 2 2] [2 3 1 3 2 2] [2 2 3 1 3 2] [2 2 2 3 1 3] [3 2 2 2 3 1], The group of affine permutations of type ['A', 5, 1]) (Coxeter group over Integer Ring with Coxeter matrix: [1 3 2 2 2 3] [3 1 3 2 2 2] [2 3 1 3 2 2] [2 2 3 1 3 2] [2 2 2 3 1 3] [3 2 2 2 3 1], The group of affine permutations of type ['A', 5, 1]) |
v=WW.from_reduced_word([1,2,1,3])
v, v.is_fully_commutative() # we don't know how this is done or whether it easily generalizes
(Type A affine permutation with window [3, 2, 4, 1, 5, 6], False) (Type A affine permutation with window [3, 2, 4, 1, 5, 6], False) |
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