Affine Permutations

class sage.combinat.affine_permutation.AffinePermutation(parent, lst, check=True)

Bases: sage.structure.list_clone.ClonableArray

An affine permutation, representated in the window notation, and considered as a bijection from \(\ZZ\) to \(\ZZ\).

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
apply_simple_reflection(i, side='right')

Applies a simple reflection.

INPUT:

  • i – an integer.
  • side – Determines whether to apply the reflection on the ‘right’ or ‘left’. Default ‘right’.

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.apply_simple_reflection(3)
Type A affine permutation with window [3, -1, 6, 0, 5, 4, 10, 9]
sage: p.apply_simple_reflection(11)
Type A affine permutation with window [3, -1, 6, 0, 5, 4, 10, 9]
sage: p.apply_simple_reflection(3, 'left')
Type A affine permutation with window [4, -1, 0, 6, 5, 3, 10, 9]
sage: p.apply_simple_reflection(11, 'left')
Type A affine permutation with window [4, -1, 0, 6, 5, 3, 10, 9]
grassmannian_quotient(i=0, side='right')

Return Grassmannian quotient.

Factors self into a unique product of a Grassmannian and a finite-type element. Returns a tuple containing the Grassmannain and finite elements, in order according to side.

INPUT:

  • i – An element of the index set; the descent checked for. Defaults to 0.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: gq=p.grassmannian_quotient()
sage: gq
(Type A affine permutation with window [-1, 0, 3, 4, 5, 6, 9, 10], Type A affine permutation with window [3, 1, 2, 6, 5, 4, 8, 7])
sage: gq[0].is_i_grassmannian()
True
sage: 0 not in gq[1].reduced_word()
True
sage: prod(gq)==p
True

sage: gqLeft=p.grassmannian_quotient(side='left')
sage: 0 not in gqLeft[0].reduced_word()
True
sage: gqLeft[1].is_i_grassmannian(side='left')
True
sage: prod(gqLeft)==p
True
index_set()

Index set of the affine permutation group.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: A.index_set()
(0, 1, 2, 3, 4, 5, 6, 7)
inverse()

Finds the inverse affine permutation.

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.inverse()
Type A affine permutation with window [0, -1, 1, 6, 5, 4, 10, 11]
is_i_grassmannian(i=0, side='right')

Test whether self is \(i\)-grassmannian, ie, either is the identity or has i as the sole descent.

INPUT:

  • i – An element of the index set.
  • side – determines the side on which to check the descents.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.is_i_grassmannian()
False
sage: q=A.from_word([3,2,1,0])
sage: q.is_i_grassmannian()
True
sage: q=A.from_word([2,3,4,5])
sage: q.is_i_grassmannian(5)
True
sage: q.is_i_grassmannian(2, side='left')
True
is_one()

Tests whether the affine permutation is the identity.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.is_one()
False
sage: q=A.one()
sage: q.is_one()
True
lower_covers(side='right')

Return lower covers of self.

The set of affine permutations of one less length related by multiplication by a simple transposition on the indicated side. These are the elements that self covers in weak order.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.lower_covers()
[Type A affine permutation with window [-1, 3, 0, 6, 5, 4, 10, 9], Type A affine permutation with window [3, -1, 0, 5, 6, 4, 10, 9], Type A affine permutation with window [3, -1, 0, 6, 4, 5, 10, 9], Type A affine permutation with window [3, -1, 0, 6, 5, 4, 9, 10]]
reduced_word()

Returns a reduced word for the affine permutation.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.reduced_word()
[0, 7, 4, 1, 0, 7, 5, 4, 2, 1]
signature()

Signature of the affine permutation, \((-1)^l\), where \(l\) is the length of the permutation.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.signature()
1
to_weyl_group_element()

The affine Weyl group element corresponding to the affine permutation.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.to_weyl_group_element()
[ 0 -1  0  1  0  0  1  0]
[ 1 -1  0  1  0  0  1 -1]
[ 1 -1  0  1  0  0  0  0]
[ 0  0  0  1  0  0  0  0]
[ 0  0  0  1  0 -1  1  0]
[ 0  0  0  1 -1  0  1  0]
[ 0  0  0  0  0  0  1  0]
[ 0 -1  1  0  0  0  1  0]
sage.combinat.affine_permutation.AffinePermutationGroup(cartan_type)

Wrapper function for specific affine permutation groups.

These are combinatorial implmentations of the affine Weyl groups of types \(A\), \(B\), \(C\), \(D\), and \(G\) as permutations of the set of all integers. the basic algorithms are derived from [BjBr] and [Erik].

REFERENCES:

[BjBr]Bjorner and Brenti. Combinatorics of Coxeter Groups. Springer, 2005.
[Erik]H. Erikson. Computational and Combinatorial Aspects of Coxeter Groups. Thesis, 1995.

EXAMPLES:

sage: ct=CartanType(['A',7,1])
sage: A=AffinePermutationGroup(ct)
sage: A
The group of affine permutations of type ['A', 7, 1]

We define an element of A:

sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]

We find the value \(p(1)\), considering \(p\) as a bijection on the integers. This is the same as calling the ‘value’ method:

sage: p.value(1)
3
sage: p(1)==p.value(1)
True

We can also find the position of the integer 3 in \(p\) considered as a sequence, equivalent to finding \(p^{-1}(3)\):

sage: p.position(3)
1
sage: (p^-1)(3)
1

Since the affine permutation group is a group, we demonstrate its group properties:

sage: A.one()
Type A affine permutation with window [1, 2, 3, 4, 5, 6, 7, 8]

sage: q=A([0, 2, 3, 4, 5, 6, 7, 9])
sage: p*q
Type A affine permutation with window [1, -1, 0, 6, 5, 4, 10, 11]
sage: q*p
Type A affine permutation with window [3, -1, 1, 6, 5, 4, 10, 8]

sage: p^-1
Type A affine permutation with window [0, -1, 1, 6, 5, 4, 10, 11]
sage: p^-1*p==A.one()
True
sage: p*p^-1==A.one()
True

If we decide we prefer the Weyl Group implementation of the affine Weyl group, we can easily get it:

sage: p.to_weyl_group_element()
[ 0 -1  0  1  0  0  1  0]
[ 1 -1  0  1  0  0  1 -1]
[ 1 -1  0  1  0  0  0  0]
[ 0  0  0  1  0  0  0  0]
[ 0  0  0  1  0 -1  1  0]
[ 0  0  0  1 -1  0  1  0]
[ 0  0  0  0  0  0  1  0]
[ 0 -1  1  0  0  0  1  0]

We can find a reduced word and do all of the other things one expects in a Coxeter group:

sage: p.has_right_descent(1)
True
sage: p.apply_simple_reflection(1)
Type A affine permutation with window [-1, 3, 0, 6, 5, 4, 10, 9]
sage: p.apply_simple_reflection(0)
Type A affine permutation with window [1, -1, 0, 6, 5, 4, 10, 11]
sage: p.reduced_word()
[0, 7, 4, 1, 0, 7, 5, 4, 2, 1]
sage: p.length()
10

The following methods are particular to Type \(A\). We can check if the element is fully commutative:

sage: p.is_fully_commutative()
False
sage: q.is_fully_commutative()
True

And we can also compute the affine Lehmer code of the permutation, a weak composition with \(k+1\) entries:

sage: p.to_lehmer_code()
[0, 3, 3, 0, 1, 2, 0, 1]

Once we have the Lehmer code, we can obtain a \(k\)-bounded partition by sorting the Lehmer code, and then reading the row lengths. There is a unique 0-Grassmanian (dominant) affine permutation associated to this \(k\)-bounded partition, and a \(k\)-core as well.

sage: p.to_bounded_partition()
[5, 3, 2]
sage: p.to_dominant()
Type A affine permutation with window [-2, -1, 1, 3, 4, 8, 10, 13]
sage: p.to_core()
[5, 3, 2]

Finally, we can take a reduced word for \(p\) and insert it to find a standard composition tableau associated uniquely to that word.

sage: p.tableau_of_word(p.reduced_word())
[[], [1, 6, 9], [2, 7, 10], [], [3], [4, 8], [], [5]]

We can also form affine permutation groups in types \(B\), \(C\), \(D\), and \(G\).

sage: B=AffinePermutationGroup(['B',4,1])
sage: B.an_element()
Type B affine permutation with window [-1, 3, 4, 11]

sage: C=AffinePermutationGroup(['C',4,1])
sage: C.an_element()
Type C affine permutation with window [2, 3, 4, 10]

sage: D=AffinePermutationGroup(['D',4,1])
sage: D.an_element()
Type D affine permutation with window [-1, 3, 11, 5]

sage: G=AffinePermutationGroup(['G',2,1])
sage: G.an_element()
Type G affine permutation with window [0, 4, -1, 8, 3, 7]
class sage.combinat.affine_permutation.AffinePermutationGroupGeneric(cartan_type)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

The generic affine permutation group class, in which we define all type-free methods for the specific affine permutation groups.

cartan_matrix()

Returns the Cartan matrix of self.

EXAMPLES:

sage: AffinePermutationGroup(['A',7,1]).cartan_matrix()
[ 2 -1  0  0  0  0  0 -1]
[-1  2 -1  0  0  0  0  0]
[ 0 -1  2 -1  0  0  0  0]
[ 0  0 -1  2 -1  0  0  0]
[ 0  0  0 -1  2 -1  0  0]
[ 0  0  0  0 -1  2 -1  0]
[ 0  0  0  0  0 -1  2 -1]
[-1  0  0  0  0  0 -1  2]
cartan_type()

Returns the Cartan type of self.

EXAMPLES:

sage: AffinePermutationGroup(['A',7,1]).cartan_type()
['A', 7, 1]
classical()

Returns the finite permutation group.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: A.classical()
Symmetric group of order 8! as a permutation group
elements_of_length(n)

Returns all elements of length \(n\).

EXAMPLES:

sage: A=AffinePermutationGroup(['A',2,1])
sage: [len(list(A.elements_of_length(i))) for i in [0..5]]
[1, 3, 6, 9, 12, 15]
from_word(w)

Builds an affine permutation from a given word. Note: Already in category as from_reduced_word, but this is less typing!

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: A.from_word([0, 7, 4, 1, 0, 7, 5, 4, 2, 1])
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
index_set()

EXAMPLES:

sage: AffinePermutationGroup(['A',7,1]).index_set()
(0, 1, 2, 3, 4, 5, 6, 7)
is_crystallographic()

Tells whether the affine permutation group is crystallographic.

EXAMPLES:

sage: AffinePermutationGroup(['A',7,1]).is_crystallographic()
True
random_element(n)

Returns a random affine permutation of length \(n\).

Starts at the identity, then chooses an upper cover at random. Not very uniform: actually constructs a uniformly random reduced word of length \(n\). Thus we most likely get elements with lots of reduced words!

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A.random_element(10)
sage: p.length()==10
True
rank()

Rank of the affine permutation group, equal to \(k+1\).

EXAMPLES:

sage: AffinePermutationGroup(['A',7,1]).rank()
8
reflection_index_set()

EXAMPLES:

sage: AffinePermutationGroup(['A',7,1]).index_set()
(0, 1, 2, 3, 4, 5, 6, 7)
weyl_group()

Returns the Weyl Group of the same type as self.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: A.weyl_group()
Weyl Group of type ['A', 7, 1] (as a matrix group acting on the root space)
class sage.combinat.affine_permutation.AffinePermutationGroupTypeA(cartan_type)

Bases: sage.combinat.affine_permutation.AffinePermutationGroupGeneric

TESTS:

sage: AffinePermutationGroup(['A',7,1])
The group of affine permutations of type ['A', 7, 1]
Element

alias of AffinePermutationTypeA

from_lehmer_code(C, typ='decreasing', side='right')

Returns the affine permutation with the supplied Lehmer code (a weak composition with \(k+1\) parts, at least one of which is 0).

INPUT:

  • typ – ‘increasing’ or ‘decreasing’: type of product.

    (default: ‘decreasing’.)

  • side – ‘right’ or ‘left’: Whether the decomposition is from

    the right or left. (default: ‘right’.)

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.to_lehmer_code()
[0, 3, 3, 0, 1, 2, 0, 1]
sage: A.from_lehmer_code(p.to_lehmer_code())==p
True
sage: CP=CartesianProduct( ('increasing','decreasing'),('left','right') )
sage: for a in CP:
....:   A.from_lehmer_code(p.to_lehmer_code(a[0],a[1]),a[0],a[1])==p
True
True
True
True
one()

Returns the identity element.

EXAMPLES:

sage: AffinePermutationGroup(['A',7,1]).one()
Type A affine permutation with window [1, 2, 3, 4, 5, 6, 7, 8]

TESTS:

sage: A=AffinePermutationGroup(['A',5,1])
sage: A==loads(dumps(A))
True
sage: TestSuite(A).run()
class sage.combinat.affine_permutation.AffinePermutationGroupTypeB(cartan_type)

Bases: sage.combinat.affine_permutation.AffinePermutationGroupTypeC

TESTS:

sage: AffinePermutationGroup(['A',7,1])
The group of affine permutations of type ['A', 7, 1]
Element

alias of AffinePermutationTypeB

class sage.combinat.affine_permutation.AffinePermutationGroupTypeC(cartan_type)

Bases: sage.combinat.affine_permutation.AffinePermutationGroupTypeC

TESTS:

sage: AffinePermutationGroup(['A',7,1])
The group of affine permutations of type ['A', 7, 1]
Element

alias of AffinePermutationTypeC

class sage.combinat.affine_permutation.AffinePermutationGroupTypeD(cartan_type)

Bases: sage.combinat.affine_permutation.AffinePermutationGroupTypeC

TESTS:

sage: AffinePermutationGroup(['A',7,1])
The group of affine permutations of type ['A', 7, 1]
Element

alias of AffinePermutationTypeD

class sage.combinat.affine_permutation.AffinePermutationGroupTypeG(cartan_type)

Bases: sage.combinat.affine_permutation.AffinePermutationGroupGeneric

TESTS:

sage: AffinePermutationGroup(['A',7,1])
The group of affine permutations of type ['A', 7, 1]
Element

alias of AffinePermutationTypeG

one()

Returns the identity element.

EXAMPLES:

sage: AffinePermutationGroup(['G',2,1]).one()
Type G affine permutation with window [1, 2, 3, 4, 5, 6]

TESTS:

sage: G=AffinePermutationGroup(['G',2,1])
sage: G==loads(dumps(G))
True
sage: TestSuite(G).run()
class sage.combinat.affine_permutation.AffinePermutationTypeA(parent, lst, check=True)

Bases: sage.combinat.affine_permutation.AffinePermutation

Initialize self

INPUT:

  • parent – The parent affine permutation group.
  • lst – List giving the base window of the affine permutation.
  • check– Chooses whether to test that the affine permutation is legit.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9]) #indirect doctest
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
apply_simple_reflection_left(i)

Applies simple reflection to the values \(i\), \(i+1\).

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.apply_simple_reflection_left(3)
Type A affine permutation with window [4, -1, 0, 6, 5, 3, 10, 9]
sage: p.apply_simple_reflection_left(11)
Type A affine permutation with window [4, -1, 0, 6, 5, 3, 10, 9]
apply_simple_reflection_right(i)

Applies the simple reflection to positions \(i\), \(i+1\). \(i\) is allowed to be any integer.

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.apply_simple_reflection_right(3)
Type A affine permutation with window [3, -1, 6, 0, 5, 4, 10, 9]
sage: p.apply_simple_reflection_right(11)
Type A affine permutation with window [3, -1, 6, 0, 5, 4, 10, 9]
check()

Check that self is an affine permutation.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
sage: q=A([1,2,3])
Traceback (most recent call last):
...
ValueError: Length of list must be k+1=8.
sage: q=A([1,2,3,4,5,6,7,0])
Traceback (most recent call last):
...
ValueError: Window does not sum to 36.
sage: q=A([1,1,3,4,5,6,7,9])
Traceback (most recent call last):
...
ValueError: Entries must have distinct residues.
flip_automorphism()

The Dynkin diagram automorphism which fixes \(s_0\) and reverses all other indices.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.flip_automorphism()
Type A affine permutation with window [0, -1, 5, 4, 3, 9, 10, 6]
has_left_descent(i)

Determines whether there is a descent at \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.has_left_descent(1)
True
sage: p.has_left_descent(9)
True
sage: p.has_left_descent(0)
True
has_right_descent(i)

Determines whether there is a descent at \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.has_right_descent(1)
True
sage: p.has_right_descent(9)
True
sage: p.has_right_descent(0)
False
is_fully_commutative()

Determines whether self is fully commutative, ie, has no reduced words with a braid.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.is_fully_commutative()
False
sage: q=A([-3, -2, 0, 7, 9, 2, 11, 12])
sage: q.is_fully_commutative()
True
maximal_cyclic_decomposition(typ='decreasing', side='right', verbose=False)

Finds the unique maximal decomposition of self into cyclically decreasing/increasing elements.

INPUT:

  • typ – ‘increasing’ or ‘decreasing’ (default: ‘decreasing’.) Chooses whether to find increasing or deacreasing sets.
  • side – ‘right’ or ‘left’ (default: ‘right’.) Chooses whether to find maximal sets starting from the left or the right.
  • verbose – Print extra information while finding the decomposition.

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.maximal_cyclic_decomposition()
[[0, 7], [4, 1, 0], [7, 5, 4, 2, 1]]
sage: p.maximal_cyclic_decomposition(side='left')
[[1, 0, 7, 5, 4], [1, 0, 5], [2, 1]]
sage: p.maximal_cyclic_decomposition(typ='increasing', side='right')
[[1], [5, 0, 1, 2], [4, 5, 7, 0, 1]]
sage: p.maximal_cyclic_decomposition(typ='increasing', side='left')
[[0, 1, 2, 4, 5], [4, 7, 0, 1], [7]]

TESTS:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: S=p.maximal_cyclic_decomposition()
sage: p==prod(A.from_word(l) for l in S)
True
sage: S=p.maximal_cyclic_decomposition(typ='increasing', side='left')
sage: p==prod(A.from_word(l) for l in S)
True
sage: S=p.maximal_cyclic_decomposition(typ='increasing', side='right')
sage: p==prod(A.from_word(l) for l in S)
True
sage: S=p.maximal_cyclic_decomposition(typ='decreasing', side='right')
sage: p==prod(A.from_word(l) for l in S)
True
maximal_cyclic_factor(typ='decreasing', side='right', verbose=False)

For an affine permutation \(x\), finds the unique maximal subset \(A\) of the index set such that \(x=yd_A\) is a reduced product.

INPUT:

  • typ – ‘increasing’ or ‘decreasing.’ Determines the type of maximal cyclic element found.
  • side – ‘right’ or ‘left’.
  • verbose – True or False. If True, outputs information about how the cyclically increasing element was found.

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.maximal_cyclic_factor()
[7, 5, 4, 2, 1]
sage: p.maximal_cyclic_factor(side='left')
[1, 0, 7, 5, 4]
sage: p.maximal_cyclic_factor('increasing','right')
[4, 5, 7, 0, 1]
sage: p.maximal_cyclic_factor('increasing','left')
[0, 1, 2, 4, 5]
position(i)

Find the position \(j\) such the self.value(j)=i

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.position(3)
1
sage: p.position(11)
9
promotion()

The Dynkin diagram automorphism which sends \(s_i\) to \(s_{i+1}\).

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.promotion()
Type A affine permutation with window [2, 4, 0, 1, 7, 6, 5, 11]
tableau_of_word(w, typ='decreasing', side='right', alpha=None)

Finds a tableau on the Lehmer code of self corresponding to the given reduced word.

For a full description of this algorithm, see [D2012].

INPUT:

  • w – a reduced word for self.
  • typ – ‘increasing’ or ‘decreasing.’ The type of Lehmer code used.
  • side – ‘right’ or ‘left.’
  • alpha – A content vector. w should be of type alpha. Specifying alpha produces semistandard tableaux.

REFERENCES:

[D2012]tom denton. Canonical Decompositions of Affine Permutations, Affine Codes, and Split \(k\)-Schur Functions. Electronic Journal of Combinatorics, 2012.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.tableau_of_word(p.reduced_word())
[[], [1, 6, 9], [2, 7, 10], [], [3], [4, 8], [], [5]]
sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: w=p.reduced_word()
sage: w
[0, 7, 4, 1, 0, 7, 5, 4, 2, 1]
sage: alpha=[5,3,2]
sage: p.tableau_of_word(p.reduced_word(), alpha=alpha)
[[], [1, 2, 3], [1, 2, 3], [], [1], [1, 2], [], [1]]
sage: p.tableau_of_word(p.reduced_word(), side='left')
[[1, 4, 9], [6], [], [], [3, 7], [8], [], [2, 5, 10]]
sage: p.tableau_of_word(p.reduced_word(), typ='increasing', side='right')
[[9, 10], [1, 2], [], [], [3, 4], [8], [], [5, 6, 7]]
sage: p.tableau_of_word(p.reduced_word(), typ='increasing', side='left')
[[1, 2], [4, 5, 6], [9, 10], [], [3], [7, 8], [], []]
to_bounded_partition(typ='decreasing', side='right')

Returns the \(k\)-bounded partition associated to the dominant element obtained by sorting the Lehmer code.

INPUT:

  • typ – ‘increasing’ or ‘decreasing’ (default: ‘decreasing’.) Chooses whether to find increasing or deacreasing sets.
  • side – ‘right’ or ‘left’ (default: ‘right’.) Chooses whether to find maximal sets starting from the left or the right.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',2,1])
sage: p=A.from_lehmer_code([4,1,0])
sage: p.to_bounded_partition()
[2, 1, 1, 1]
to_core(typ='decreasing', side='right')

Returns the core associated to the dominant element obtained by sorting the Lehmer code.

INPUT:

  • typ – ‘increasing’ or ‘decreasing’ (default: ‘decreasing’.)
  • side – ‘right’ or ‘left’ (default: ‘right’.) Chooses whether to find maximal sets starting from the left or the right.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',2,1])
sage: p=A.from_lehmer_code([4,1,0])
sage: p.to_bounded_partition()
[2, 1, 1, 1]
sage: p.to_core()
[4, 2, 1, 1]
to_dominant(typ='decreasing', side='right')

Finds the Lehmer code and then sorts it. Returns the affine permutation with the given sorted Lehmer code; this element is 0-dominant.

INPUT:

  • typ – ‘increasing’ or ‘decreasing’ (default: ‘decreasing’.) Chooses whether to find increasing or deacreasing sets.
  • side – ‘right’ or ‘left’ (default: ‘right’.) Chooses whether to find maximal sets starting from the left or the right.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.to_dominant()
Type A affine permutation with window [-2, -1, 1, 3, 4, 8, 10, 13]
sage: p.to_dominant(typ='increasing', side='left')
Type A affine permutation with window [3, 4, -1, 5, 0, 9, 6, 10]
to_lehmer_code(typ='decreasing', side='right')

Returns the affine Lehmer code.

There are four such codes; the options typ and side determine which code is generated. The codes generated are the shape of the maximal cyclic decompositions of self according to the given typ and side options.

INPUT:

  • typ – ‘increasing’ or ‘decreasing’ (default: ‘decreasing’.) Chooses whether to find increasing or deacreasing sets.
  • side – ‘right’ or ‘left’ (default: ‘right’.) Chooses whether to find maximal sets starting from the left or the right.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: CP=CartesianProduct( ('increasing','decreasing'),('left','right') )
sage: for a in CP:
....:   p.to_lehmer_code(a[0],a[1])
[2, 3, 2, 0, 1, 2, 0, 0]
[2, 2, 0, 0, 2, 1, 0, 3]
[3, 1, 0, 0, 2, 1, 0, 3]
[0, 3, 3, 0, 1, 2, 0, 1]
sage: for a in CP:
....:   A.from_lehmer_code(p.to_lehmer_code(a[0],a[1]), a[0],a[1])==p
True
True
True
True
to_type_a()

Returns an embedding of self into the affine permutation group of type A. (For Type \(A\), just returns self.)

EXAMPLES:

sage: p=AffinePermutationGroup(['A',7,1])([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.to_type_a()==p
True
value(i, base_window=False)

Return the image of the integer i under this permutation.

INPUT:

  • base_window – a Boolean, indicating whether \(i\) is in the base window. If True, will run a bit faster, but the method will screw up if \(i\) is not actually in the index set.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9])
sage: p.value(1)
3
sage: p.value(9)
11
class sage.combinat.affine_permutation.AffinePermutationTypeB(parent, lst, check=True)

Bases: sage.combinat.affine_permutation.AffinePermutationTypeC

Initialize self

INPUT:

  • parent – The parent affine permutation group.
  • lst – List giving the base window of the affine permutation.
  • check– Chooses whether to test that the affine permutation is legit.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9]) #indirect doctest
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
apply_simple_reflection_left(i)

Applies simple reflection indexed by \(i\) on values.

EXAMPLES:

sage: B=AffinePermutationGroup(['B',4,1])
sage: p=B([-5,1,6,-2])
sage: p.apply_simple_reflection_left(0)
Type B affine permutation with window [-5, -2, 6, 1]
sage: p.apply_simple_reflection_left(2)
Type B affine permutation with window [-5, 1, 7, -3]
sage: p.apply_simple_reflection_left(4)
Type B affine permutation with window [-4, 1, 6, -2]
apply_simple_reflection_right(i)

Applies the simple reflection indexed by \(i\) on positions.

EXAMPLES:

sage: B=AffinePermutationGroup(['B',4,1])
sage: p=B([-5,1,6,-2])
sage: p.apply_simple_reflection_right(1)
Type B affine permutation with window [1, -5, 6, -2]
sage: p.apply_simple_reflection_right(0)
Type B affine permutation with window [-1, 5, 6, -2]
sage: p.apply_simple_reflection_right(4)
Type B affine permutation with window [-5, 1, 6, 11]
check()

Check that self is an affine permutation.

EXAMPLES:

sage: B=AffinePermutationGroup(['B',4,1])
sage: x=B([-5,1,6,-2])
sage: x
Type B affine permutation with window [-5, 1, 6, -2]
has_left_descent(i)

Determines whether there is a descent at \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: B=AffinePermutationGroup(['B',4,1])
sage: p=B([-5,1,6,-2])
sage: [p.has_left_descent(i) for i in B.index_set()]
[True, True, False, False, True]
has_right_descent(i)

Determines whether there is a descent at index \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: B=AffinePermutationGroup(['B',4,1])
sage: p=B([-5,1,6,-2])
sage: [p.has_right_descent(i) for i in B.index_set()]
[True, False, False, True, False]
class sage.combinat.affine_permutation.AffinePermutationTypeC(parent, lst, check=True)

Bases: sage.combinat.affine_permutation.AffinePermutation

Initialize self

INPUT:

  • parent – The parent affine permutation group.
  • lst – List giving the base window of the affine permutation.
  • check– Chooses whether to test that the affine permutation is legit.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9]) #indirect doctest
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
apply_simple_reflection_left(i)

Applies simple reflection indexed by \(i\) on values.

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C([-1,5,3,7])
sage: for i in C.index_set(): x.apply_simple_reflection_left(i)
Type C affine permutation with window [1, 5, 3, 7]
Type C affine permutation with window [-2, 5, 3, 8]
Type C affine permutation with window [-1, 5, 2, 6]
Type C affine permutation with window [-1, 6, 4, 7]
Type C affine permutation with window [-1, 4, 3, 7]
apply_simple_reflection_right(i)

Applies the simple reflection indexed by \(i\) on positions.

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C([-1,5,3,7])
sage: for i in C.index_set(): x.apply_simple_reflection_right(i)
Type C affine permutation with window [1, 5, 3, 7]
Type C affine permutation with window [5, -1, 3, 7]
Type C affine permutation with window [-1, 3, 5, 7]
Type C affine permutation with window [-1, 5, 7, 3]
Type C affine permutation with window [-1, 5, 3, 2]
check()

Check that self is an affine permutation.

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C([-1,5,3,7])
sage: x
Type C affine permutation with window [-1, 5, 3, 7]
has_left_descent(i)

Determines whether there is a descent at \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C([-1,5,3,7])
sage: for i in C.index_set(): x.has_left_descent(i)
True
False
True
False
True
has_right_descent(i)

Determines whether there is a descent at index \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C([-1,5,3,7])
sage: for i in C.index_set(): x.has_right_descent(i)
True
False
True
False
True
position(i)

Find the position \(j\) such the self.value(j)=i

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C.one()
sage: [x.position(i) for i in range(-10,10)]==range(-10,10)
True
to_type_a()

Returns an embedding of self into the affine permutation group of type \(A\).

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C([-1,5,3,7])
sage: x.to_type_a()
Type A affine permutation with window [-1, 5, 3, 7, 2, 6, 4, 10, 9]
value(i)

Returns the image of the integer \(i\) under this permutation.

EXAMPLES:

sage: C=AffinePermutationGroup(['C',4,1])
sage: x=C.one()
sage: [x.value(i) for i in range(-10,10)]==range(-10,10)
True
class sage.combinat.affine_permutation.AffinePermutationTypeD(parent, lst, check=True)

Bases: sage.combinat.affine_permutation.AffinePermutationTypeC

Initialize self

INPUT:

  • parent – The parent affine permutation group.
  • lst – List giving the base window of the affine permutation.
  • check– Chooses whether to test that the affine permutation is legit.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9]) #indirect doctest
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
apply_simple_reflection_left(i)

Applies simple reflection indexed by \(i\) on values.

EXAMPLES:

sage: D=AffinePermutationGroup(['D',4,1])
sage: p=D([1,-6,5,-2])
sage: p.apply_simple_reflection_left(0)
Type D affine permutation with window [-2, -6, 5, 1]
sage: p.apply_simple_reflection_left(1)
Type D affine permutation with window [2, -6, 5, -1]
sage: p.apply_simple_reflection_left(4)
Type D affine permutation with window [1, -4, 3, -2]
apply_simple_reflection_right(i)

Applies the simple reflection indexed by \(i\) on positions.

EXAMPLES:

sage: D=AffinePermutationGroup(['D',4,1])
sage: p=D([1,-6,5,-2])
sage: p.apply_simple_reflection_right(0)
Type D affine permutation with window [6, -1, 5, -2]
sage: p.apply_simple_reflection_right(1)
Type D affine permutation with window [-6, 1, 5, -2]
sage: p.apply_simple_reflection_right(4)
Type D affine permutation with window [1, -6, 11, 4]
check()

Check that self is an affine permutation.

EXAMPLES:

sage: D=AffinePermutationGroup(['D',4,1])
sage: p=D([1,-6,5,-2])
sage: p
Type D affine permutation with window [1, -6, 5, -2]
has_left_descent(i)

Determines whether there is a descent at \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: D=AffinePermutationGroup(['D',4,1])
sage: p=D([1,-6,5,-2])
sage: [p.has_left_descent(i) for i in D.index_set()]
[True, True, False, True, True]
has_right_descent(i)

Determines whether there is a descent at index \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: D=AffinePermutationGroup(['D',4,1])
sage: p=D([1,-6,5,-2])
sage: [p.has_right_descent(i) for i in D.index_set()]
[True, True, False, True, False]
class sage.combinat.affine_permutation.AffinePermutationTypeG(parent, lst, check=True)

Bases: sage.combinat.affine_permutation.AffinePermutation

Initialize self

INPUT:

  • parent – The parent affine permutation group.
  • lst – List giving the base window of the affine permutation.
  • check– Chooses whether to test that the affine permutation is legit.

EXAMPLES:

sage: A=AffinePermutationGroup(['A',7,1])
sage: p=A([3, -1, 0, 6, 5, 4, 10, 9]) #indirect doctest
sage: p
Type A affine permutation with window [3, -1, 0, 6, 5, 4, 10, 9]
apply_simple_reflection_left(i)

Applies simple reflection indexed by \(i\) on values.

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: p.apply_simple_reflection_left(0)
Type G affine permutation with window [0, 10, -7, 14, -3, 7]
sage: p.apply_simple_reflection_left(1)
Type G affine permutation with window [1, 9, -4, 11, -2, 6]
sage: p.apply_simple_reflection_left(2)
Type G affine permutation with window [3, 11, -5, 12, -4, 4]
apply_simple_reflection_right(i)

Applies the simple reflection indexed by \(i\) on positions.

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: p.apply_simple_reflection_right(0)
Type G affine permutation with window [-9, -1, -5, 12, 8, 16]
sage: p.apply_simple_reflection_right(1)
Type G affine permutation with window [10, 2, 12, -5, 5, -3]
sage: p.apply_simple_reflection_right(2)
Type G affine permutation with window [2, -5, 10, -3, 12, 5]
check()

Check that self is an affine permutation.

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: p
Type G affine permutation with window [2, 10, -5, 12, -3, 5]
has_left_descent(i)

Determines whether there is a descent at \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: [p.has_left_descent(i) for i in G.index_set()]
[False, True, False]
has_right_descent(i)

Determines whether there is a descent at index \(i\).

INPUT:

  • i – an integer.

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: [p.has_right_descent(i) for i in G.index_set()]
[False, False, True]
position(i)

Find the position \(j\) such the self.value(j)=i

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: [p.position(i) for i in p._lst]
[1, 2, 3, 4, 5, 6]
to_type_a()

Returns an embedding of self into the affine permutation group of type A.

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: p.to_type_a()
Type A affine permutation with window [2, 10, -5, 12, -3, 5]
value(i, base_window=False)

Returns the image of the integer \(i\) under this permutation.

INPUT:

  • base_window – a Boolean indicating whether \(i\) is between 1 and \(k+1\). If True, will run a bit faster, but the method will screw up if \(i\) is not actually in the index set.

EXAMPLES:

sage: G=AffinePermutationGroup(['G',2,1])
sage: p=G([2, 10, -5, 12, -3, 5])
sage: [p.value(i) for i in [1..12]]
[2, 10, -5, 12, -3, 5, 8, 16, 1, 18, 3, 11]

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