Symmetric Group Algebra¶
-
sage.combinat.symmetric_group_algebra.
HeckeAlgebraSymmetricGroupT
(R, n, q=None)¶ Return the Hecke algebra of the symmetric group
on the T-basis with quantum parameter
q
over the ring.
If
is a commutative ring and
is an invertible element of
, and if
is a nonnegative integer, then the Hecke algebra of the symmetric group
over
with quantum parameter
is defined as the algebra generated by the generators
with relations
for all
(“braid relations”),
for all
and
such that
(“locality relations”), and
for all
(the “quadratic relations”, also known in the form
). (This is only one of several existing definitions in literature, not all of which are fully equivalent. We are following the conventions of [GS93].) For any permutation
, we can define an element
of this Hecke algebra by setting
, where
is a reduced word for
(with
meaning the transposition
, and the product of permutations being evaluated by first applying
, then
, etc.). This element is independent of the choice of the reduced decomposition, and can be computed in Sage by calling
H[w]
whereH
is the Hecke algebra andw
is the permutation.The Hecke algebra of the symmetric group
with quantum parameter
over
can be seen as a deformation of the group algebra
; indeed, it becomes
when
.
Warning
The multiplication on the Hecke algebra of the symmetric group does not follow the global option
mult
of thePermutations
class (seeglobal_options()
). It is always as defined above. It does not match the default option (mult=l2r
) of the symmetric group algebra!REFERENCES:
[GS93] David M. Goldschmidt. Group characters, symmetric functions, and the Hecke algebras. AMS 1993. EXAMPLES:
sage: HeckeAlgebraSymmetricGroupT(QQ, 3) Hecke algebra of the symmetric group of order 3 on the T basis over Univariate Polynomial Ring in q over Rational Field
sage: HeckeAlgebraSymmetricGroupT(QQ, 3, 2) Hecke algebra of the symmetric group of order 3 with q=2 on the T basis over Rational Field
The multiplication on the Hecke algebra follows a different convention than the one on the symmetric group algebra does by default:
sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3) sage: H3([1,3,2]) * H3([2,1,3]) T[3, 1, 2] sage: S3 = SymmetricGroupAlgebra(QQ, 3) sage: S3([1,3,2]) * S3([2,1,3]) [2, 3, 1] sage: TestSuite(H3).run()
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class
sage.combinat.symmetric_group_algebra.
HeckeAlgebraSymmetricGroup_generic
(R, n, q=None)¶ Bases:
sage.combinat.combinatorial_algebra.CombinatorialAlgebra
TESTS:
sage: HeckeAlgebraSymmetricGroupT(QQ, 3) Hecke algebra of the symmetric group of order 3 on the T basis over Univariate Polynomial Ring in q over Rational Field
sage: HeckeAlgebraSymmetricGroupT(QQ, 3, q=1) Hecke algebra of the symmetric group of order 3 with q=1 on the T basis over Rational Field
-
q
()¶ EXAMPLES:
sage: HeckeAlgebraSymmetricGroupT(QQ, 3).q() q sage: HeckeAlgebraSymmetricGroupT(QQ, 3, 2).q() 2
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-
class
sage.combinat.symmetric_group_algebra.
HeckeAlgebraSymmetricGroup_t
(R, n, q=None)¶ Bases:
sage.combinat.symmetric_group_algebra.HeckeAlgebraSymmetricGroup_generic
TESTS:
sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3) sage: H3 == loads(dumps(H3)) True
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algebra_generators
()¶ Return the generators of the algebra.
EXAMPLES:
sage: HeckeAlgebraSymmetricGroupT(QQ,3).algebra_generators() [T[2, 1, 3], T[1, 3, 2]]
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jucys_murphy
(k)¶ Return the Jucys-Murphy element
of the Hecke algebra.
These Jucys-Murphy elements are defined by
More explicitly,
For generic
, the
generate a maximal commutative sub-algebra of the Hecke algebra.
Warning
The specialization
does not map these elements
to the Young-Jucys-Murphy elements of the group algebra
. (Instead, it maps the “reduced” Jucys-Murphy elements
to the Young-Jucys-Murphy elements of
.)
EXAMPLES:
sage: H3 = HeckeAlgebraSymmetricGroupT(QQ,3) sage: j2 = H3.jucys_murphy(2); j2 q*T[1, 2, 3] + (q-1)*T[2, 1, 3] sage: j3 = H3.jucys_murphy(3); j3 q^2*T[1, 2, 3] + (q^2-q)*T[1, 3, 2] + (q-1)*T[3, 2, 1] sage: j2*j3 == j3*j2 True sage: j0 = H3.jucys_murphy(1); j0 == H3.one() True sage: H3.jucys_murphy(0) Traceback (most recent call last): ... ValueError: k (= 0) must be between 1 and n (= 3)
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t
(i)¶ Return the element
of the Hecke algebra
self
.EXAMPLES:
sage: H3 = HeckeAlgebraSymmetricGroupT(QQ,3) sage: H3.t(1) T[2, 1, 3] sage: H3.t(2) T[1, 3, 2] sage: H3.t(0) Traceback (most recent call last): ... ValueError: i (= 0) must be between 1 and n-1 (= 2)
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t_action
(a, i)¶ Return the product
.
EXAMPLES:
sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3) sage: a = H3([2,1,3])+2*H3([1,2,3]) sage: H3.t_action(a, 1) q*T[1, 2, 3] + (q+1)*T[2, 1, 3] sage: H3.t(1)*a q*T[1, 2, 3] + (q+1)*T[2, 1, 3]
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t_action_on_basis
(perm, i)¶ Return the product
, where
perm
is a permutation in the symmetric group.
EXAMPLES:
sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3) sage: H3.t_action_on_basis(Permutation([2,1,3]), 1) q*T[1, 2, 3] + (q-1)*T[2, 1, 3] sage: H3.t_action_on_basis(Permutation([1,2,3]), 1) T[2, 1, 3] sage: H3 = HeckeAlgebraSymmetricGroupT(QQ, 3, 1) sage: H3.t_action_on_basis(Permutation([2,1,3]), 1) T[1, 2, 3] sage: H3.t_action_on_basis(Permutation([1,3,2]), 2) T[1, 2, 3]
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-
sage.combinat.symmetric_group_algebra.
SymmetricGroupAlgebra
(R, W, category=None)¶ Return the symmetric group algebra of order
W
over the ringR
.INPUT:
W
– a symmetric group; alternatively an integercan be provided, as shorthand for
Permutations(n)
.R
– a base ringcategory
– a category (default: the category ofW
)
This supports several implementations of the symmetric group. At this point this has been tested with
W=Permutations(n)
andW=SymmetricGroup(n)
.Warning
Some features are failing in the latter case, in particular if the domain of the symmetric group is not
.
Note
The brave can also try setting
W=WeylGroup(['A',n-1])
, but little support for this currently exists.EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3); QS3 Symmetric group algebra of order 3 over Rational Field sage: QS3(1) [1, 2, 3] sage: QS3(2) 2*[1, 2, 3] sage: basis = [QS3(p) for p in Permutations(3)] sage: a = sum(basis); a [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1] sage: a^2 6*[1, 2, 3] + 6*[1, 3, 2] + 6*[2, 1, 3] + 6*[2, 3, 1] + 6*[3, 1, 2] + 6*[3, 2, 1] sage: a^2 == 6*a True sage: b = QS3([3, 1, 2]) sage: b [3, 1, 2] sage: b*a [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1] sage: b*a == a True
We now construct the symmetric group algebra by providing explicitly the underlying group:
sage: SGA = SymmetricGroupAlgebra(QQ, Permutations(4)); SGA Symmetric group algebra of order 4 over Rational Field sage: SGA.group() Standard permutations of 4 sage: SGA.an_element() [1, 2, 3, 4] + 2*[1, 2, 4, 3] + 3*[1, 3, 2, 4] + [4, 1, 2, 3] sage: SGA = SymmetricGroupAlgebra(QQ, SymmetricGroup(4)); SGA Symmetric group algebra of order 4 over Rational Field sage: SGA.group() Symmetric group of order 4! as a permutation group sage: SGA.an_element() () + 2*(1,2) + 4*(1,2,3,4) sage: SGA = SymmetricGroupAlgebra(QQ, WeylGroup(["A",3], prefix='s')); SGA Symmetric group algebra of order 4 over Rational Field sage: SGA.group() Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space) sage: SGA.an_element() 2*s1*s2*s3*s2*s1 + 3*s1*s2*s3*s1 + s1*s2*s3 + 1
The preferred way to construct the symmetric group algebra is to go through the usual
algebra
method:sage: SGA = Permutations(3).algebra(QQ); SGA Symmetric group algebra of order 3 over Rational Field sage: SGA.group() Standard permutations of 3 sage: SGA = SymmetricGroup(3).algebra(QQ); SGA Symmetric group algebra of order 3 over Rational Field sage: SGA.group() Symmetric group of order 3! as a permutation group
The canonical embedding from the symmetric group algebra of order
to the symmetric group algebra of order
is available as a coercion:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: QS4 = SymmetricGroupAlgebra(QQ, 4) sage: QS4.coerce_map_from(QS3) Generic morphism: From: Symmetric group algebra of order 3 over Rational Field To: Symmetric group algebra of order 4 over Rational Field sage: x3 = QS3([3,1,2]) + 2 * QS3([2,3,1]); x3 2*[2, 3, 1] + [3, 1, 2] sage: QS4(x3) 2*[2, 3, 1, 4] + [3, 1, 2, 4]
This allows for mixed expressions:
sage: x4 = 3*QS4([3, 1, 4, 2]) sage: x3 + x4 2*[2, 3, 1, 4] + [3, 1, 2, 4] + 3*[3, 1, 4, 2] sage: QS0 = SymmetricGroupAlgebra(QQ, 0) sage: QS1 = SymmetricGroupAlgebra(QQ, 1) sage: x0 = QS0([]) sage: x1 = QS1([1]) sage: x0 * x1 [1] sage: x3 - (2*x0 + x1) - x4 -3*[1, 2, 3, 4] + 2*[2, 3, 1, 4] + [3, 1, 2, 4] - 3*[3, 1, 4, 2]
Caveat: to achieve this, constructing
SymmetricGroupAlgebra(QQ, 10)
currently triggers the construction of all symmetric group algebras of smaller order. Is this a feature we really want to have?Warning
The semantics of multiplication in symmetric group algebras with index set
Permutations(n)
is determined by the order in which permutations are multiplied, which currently defaults to “in such a way that multiplication is associative with permutations acting on integers from the right”, but can be changed to the opposite order at runtime by setting the global variablePermutations.global_options['mult']
(seesage.combinat.permutation.Permutations.global_options()
). On the other hand, the semantics of multiplication in symmetric group algebras with index setSymmetricGroup(n)
does not depend on this global variable. (This has the awkward consequence that the coercions between these two sorts of symmetric group algebras do not respect multiplication when this global variable is set to'r2l'
.) In view of this, it is recommended that code not rely on the usual multiplication function, but rather use the methodsleft_action_product()
andright_action_product()
for multiplying permutations (these methods don’t depend on the setting). See trac ticket #14885 for more information.We conclude by constructing the algebra of the symmetric group as a monoid algebra:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3, category=Monoids()) sage: QS3.category() Category of finite dimensional monoid algebras over Rational Field sage: TestSuite(QS3).run()
TESTS:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: TestSuite(QS3).run() sage: QS3.group() Standard permutations of 3 sage: QS3.one_basis() [1, 2, 3] sage: p1 = Permutation([1,2,3]) sage: p2 = Permutation([2,1,3]) sage: QS3.product_on_basis(p1,p2) [2, 1, 3] sage: W = WeylGroup(["A",3]) sage: SGA = SymmetricGroupAlgebra(QQ, W) sage: SGA.group() is W True sage: TestSuite(SGA).run() sage: SG = SymmetricGroupAlgebra(ZZ, 3) sage: SG.group().conjugacy_classes_representatives() [[1, 2, 3], [2, 1, 3], [2, 3, 1]] sage: SGg = SymmetricGroup(3).algebra(ZZ) sage: SGg.group().conjugacy_classes_representatives() [(), (1,2), (1,2,3)]
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class
sage.combinat.symmetric_group_algebra.
SymmetricGroupAlgebra_n
(R, W, category)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
TESTS:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: TestSuite(QS3).run() sage: QS3 in GroupAlgebras(QQ) True sage: QS3 in FiniteDimensionalAlgebrasWithBasis(QQ) True
Check that trac ticket #16926 works:
sage: S = SymmetricGroup(4) sage: SGA = S.algebra(QQ) sage: TestSuite(SGA).run()
Checking that coercion works between equivalent indexing sets:
sage: G = SymmetricGroup(4).algebra(QQ) sage: S = SymmetricGroupAlgebra(QQ,4) sage: S(G.an_element()) [1, 2, 3, 4] + 2*[2, 1, 3, 4] + 4*[2, 3, 4, 1] sage: G(S.an_element()) () + 2*(3,4) + 3*(2,3) + (1,4,3,2)
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algebra_generators
()¶ Return generators of this group algebra (as algebra) as a list of permutations.
The generators used for the group algebra of
are the transposition
and the
-cycle
, unless
(in which case no generators are needed).
EXAMPLES:
sage: SymmetricGroupAlgebra(ZZ,5).algebra_generators() Family ([2, 1, 3, 4, 5], [2, 3, 4, 5, 1]) sage: SymmetricGroupAlgebra(QQ,0).algebra_generators() Family () sage: SymmetricGroupAlgebra(QQ,1).algebra_generators() Family ()
TESTS:
Check that trac ticket #15309 is fixed:
sage: S3 = SymmetricGroupAlgebra(QQ, 3) sage: S3.algebra_generators() Family ([2, 1, 3], [2, 3, 1]) sage: C = CombinatorialFreeModule(ZZ, ZZ) sage: M = C.module_morphism(lambda x: S3.zero(), codomain=S3) sage: M.register_as_coercion()
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antipode
(x)¶ Return the image of the element
x
ofself
under the antipode of the Hopf algebraself
(where the comultiplication is the usual one on a group algebra).Explicitly, this is obtained by replacing each permutation
by
in
x
while keeping all coefficients as they are.EXAMPLES:
sage: QS4 = SymmetricGroupAlgebra(QQ, 4) sage: QS4.antipode(2 * QS4([1, 3, 4, 2]) - 1/2 * QS4([1, 4, 2, 3])) -1/2*[1, 3, 4, 2] + 2*[1, 4, 2, 3] sage: all( QS4.antipode(QS4(p)) == QS4(p.inverse()) ....: for p in Permutations(4) ) True sage: ZS3 = SymmetricGroupAlgebra(ZZ, 3) sage: ZS3.antipode(ZS3.zero()) 0 sage: ZS3.antipode(-ZS3(Permutation([2, 3, 1]))) -[3, 1, 2]
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binary_unshuffle_sum
(k)¶ Return the
-th binary unshuffle sum in the group algebra
self
.The
-th binary unshuffle sum in the symmetric group algebra
over a ring
is defined as the sum of all permutations
satisfying
and
.
This element has the property that, if it is denoted by
, and if the
-th semi-RSW element (see
semi_rsw_element()
) is denoted by, then
and
both equal the
-th Reiner-Saliola-Welker shuffling element of
(see
rsw_shuffling_element()
).The
-th binary unshuffle sum is the image of the complete non-commutative symmetric function
in the ring of non-commutative symmetric functions under the canonical projection on the symmetric group algebra (through the descent algebra).
EXAMPLES:
The binary unshuffle sums on
:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: QS3.binary_unshuffle_sum(0) [1, 2, 3] sage: QS3.binary_unshuffle_sum(1) [1, 2, 3] + [2, 1, 3] + [3, 1, 2] sage: QS3.binary_unshuffle_sum(2) [1, 2, 3] + [1, 3, 2] + [2, 3, 1] sage: QS3.binary_unshuffle_sum(3) [1, 2, 3] sage: QS3.binary_unshuffle_sum(4) 0
Let us check the relation with the
-th Reiner-Saliola-Welker shuffling element stated in the docstring:
sage: def test_rsw(n): ....: ZSn = SymmetricGroupAlgebra(ZZ, n) ....: for k in range(1, n): ....: a = ZSn.semi_rsw_element(k) ....: b = ZSn.binary_unshuffle_sum(k) ....: c = ZSn.left_action_product(a, ZSn.antipode(b)) ....: d = ZSn.left_action_product(b, ZSn.antipode(a)) ....: e = ZSn.rsw_shuffling_element(k) ....: if c != e or d != e: ....: return False ....: return True sage: test_rsw(3) True sage: test_rsw(4) # long time True sage: test_rsw(5) # long time True
Let us also check the statement about the complete non-commutative symmetric function:
sage: def test_rsw_ncsf(n): ....: ZSn = SymmetricGroupAlgebra(ZZ, n) ....: NSym = NonCommutativeSymmetricFunctions(ZZ) ....: S = NSym.S() ....: for k in range(1, n): ....: a = S(Composition([k, n-k])).to_symmetric_group_algebra() ....: if a != ZSn.binary_unshuffle_sum(k): ....: return False ....: return True sage: test_rsw_ncsf(3) True sage: test_rsw_ncsf(4) True sage: test_rsw_ncsf(5) # long time True
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canonical_embedding
(other)¶ Return the canonical coercion of
self
into a symmetric group algebraother
.INPUT:
other
– a symmetric group algebra with ordersatisfying
, where
is the order of
self
, over a ground ring into which the ground ring ofself
coerces.
EXAMPLES:
sage: QS2 = SymmetricGroupAlgebra(QQ, 2) sage: QS4 = SymmetricGroupAlgebra(QQ, 4) sage: phi = QS2.canonical_embedding(QS4); phi Generic morphism: From: Symmetric group algebra of order 2 over Rational Field To: Symmetric group algebra of order 4 over Rational Field sage: x = QS2([2,1]) + 2 * QS2([1,2]) sage: phi(x) 2*[1, 2, 3, 4] + [2, 1, 3, 4] sage: loads(dumps(phi)) Generic morphism: From: Symmetric group algebra of order 2 over Rational Field To: Symmetric group algebra of order 4 over Rational Field sage: ZS2 = SymmetricGroupAlgebra(ZZ, 2) sage: phi = ZS2.canonical_embedding(QS4); phi Generic morphism: From: Symmetric group algebra of order 2 over Integer Ring To: Symmetric group algebra of order 4 over Rational Field sage: phi = ZS2.canonical_embedding(QS2); phi Generic morphism: From: Symmetric group algebra of order 2 over Integer Ring To: Symmetric group algebra of order 2 over Rational Field sage: QS4.canonical_embedding(QS2) Traceback (most recent call last): ... ValueError: There is no canonical embedding from Symmetric group algebra of order 2 over Rational Field to Symmetric group algebra of order 4 over Rational Field sage: QS4g = SymmetricGroup(4).algebra(QQ) sage: QS4.canonical_embedding(QS4g)(QS4([1,3,2,4])) (2,3) sage: QS4g.canonical_embedding(QS4)(QS4g((2,3))) [1, 3, 2, 4] sage: ZS2.canonical_embedding(QS4g)(ZS2([2,1])) (1,2) sage: ZS2g = SymmetricGroup(2).algebra(ZZ) sage: ZS2g.canonical_embedding(QS4)(ZS2g((1,2))) [2, 1, 3, 4]
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cpi
(p)¶ Return the centrally primitive idempotent for the symmetric group of order
corresponding to the irreducible representation indexed by the partition
p
.EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ,3) sage: QS3.cpi([2,1]) 2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2] sage: QS3.cpi([3]) 1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1] sage: QS3.cpi([1,1,1]) 1/6*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] - 1/6*[3, 2, 1] sage: QS0 = SymmetricGroupAlgebra(QQ, 0) sage: QS0.cpi(Partition([])) []
TESTS:
sage: QS3.cpi([2,2]) Traceback (most recent call last): ... TypeError: p (= [2, 2]) must be a partition of n (= 3)
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cpis
()¶ Return a list of the centrally primitive idempotents of
self
.EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ,3) sage: a = QS3.cpis() sage: a[0] # [3] 1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1] sage: a[1] # [2, 1] 2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2]
TESTS:
Check this works with other indexing sets:
sage: G = SymmetricGroup(3).algebra(QQ) sage: a = G.cpis() sage: a[0] 1/6*() + 1/6*(2,3) + 1/6*(1,2) + 1/6*(1,2,3) + 1/6*(1,3,2) + 1/6*(1,3) sage: a[1] 2/3*() - 1/3*(1,2,3) - 1/3*(1,3,2)
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dft
(form='seminormal', mult='l2r')¶ Return the discrete Fourier transform for
self
.INPUT:
mult
– string (default:). If set to
, this causes the method to use the antipodes (
antipode()
) of the seminormal basis instead of the seminormal basis.
EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: QS3.dft() [ 1 1 1 1 1 1] [ 1 1/2 -1 -1/2 -1/2 1/2] [ 0 3/4 0 3/4 -3/4 -3/4] [ 0 1 0 -1 1 -1] [ 1 -1/2 1 -1/2 -1/2 -1/2] [ 1 -1 -1 1 1 -1]
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epsilon_ik
(itab, ktab, star=0, mult='l2r')¶ Return the seminormal basis element of
self
corresponding to the pair of tableauxitab
andktab
(or restrictions of these tableaux, if the optional variablestar
is set).INPUT:
itab
,ktab
– two standard tableaux of size.
star
– integer (default:).
mult
– string (default:). If set to
, this causes the method to return the antipode (
antipode()
) ofinstead of
itself.
OUTPUT:
The element
, where
and
are the tableaux obtained by removing all entries higher than
from
itab
andktab
, respectively. Here, we are using the notations fromseminormal_basis()
.EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: a = QS3.epsilon_ik([[1,2,3]], [[1,2,3]]); a 1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1] sage: QS3.dft()*vector(a) (1, 0, 0, 0, 0, 0) sage: a = QS3.epsilon_ik([[1,2],[3]], [[1,2],[3]]); a 1/3*[1, 2, 3] - 1/6*[1, 3, 2] + 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] - 1/6*[3, 2, 1] sage: QS3.dft()*vector(a) (0, 0, 0, 0, 1, 0)
Let us take some properties of the seminormal basis listed in the docstring of
seminormal_basis()
, and verify them on the situation of.
First, check the formula
In fact:
sage: from sage.combinat.symmetric_group_algebra import e sage: def test_sn1(n): ....: QSn = SymmetricGroupAlgebra(QQ, n) ....: QSn1 = SymmetricGroupAlgebra(QQ, n - 1) ....: for T in StandardTableaux(n): ....: TT = T.restrict(n-1) ....: eTT = QSn1.epsilon_ik(TT, TT) ....: eT = QSn.epsilon_ik(T, T) ....: kT = prod(T.shape().hooks()) ....: if kT * eT != eTT * e(T) * eTT: ....: return False ....: return True sage: test_sn1(3) True sage: test_sn1(4) # long time True
Next, we check the identity
which we used to define
. In fact:
sage: from sage.combinat.symmetric_group_algebra import e sage: def test_sn2(n): ....: QSn = SymmetricGroupAlgebra(QQ, n) ....: mul = QSn.left_action_product ....: QSn1 = SymmetricGroupAlgebra(QQ, n - 1) ....: for lam in Partitions(n): ....: k = prod(lam.hooks()) ....: for T in StandardTableaux(lam): ....: for S in StandardTableaux(lam): ....: TT = T.restrict(n-1) ....: SS = S.restrict(n-1) ....: eTT = QSn1.epsilon_ik(TT, TT) ....: eSS = QSn1.epsilon_ik(SS, SS) ....: eTS = QSn.epsilon_ik(T, S) ....: piTS = [0] * n ....: for (i, j) in T.cells(): ....: piTS[T[i][j] - 1] = S[i][j] ....: piTS = QSn(Permutation(piTS)) ....: if k * eTS != mul(mul(eSS, piTS), mul(e(T), eTT)): ....: return False ....: return True sage: test_sn2(3) True sage: test_sn2(4) # long time True
Let us finally check the identity
In fact:
sage: def test_sn3(lam): ....: n = lam.size() ....: QSn = SymmetricGroupAlgebra(QQ, n) ....: mul = QSn.left_action_product ....: for T in StandardTableaux(lam): ....: for S in StandardTableaux(lam): ....: for U in StandardTableaux(lam): ....: for V in StandardTableaux(lam): ....: lhs = mul(QSn.epsilon_ik(T, S), QSn.epsilon_ik(U, V)) ....: if T == V: ....: rhs = QSn.epsilon_ik(U, S) ....: else: ....: rhs = QSn.zero() ....: if rhs != lhs: ....: return False ....: return True sage: all( test_sn3(lam) for lam in Partitions(3) ) True sage: all( test_sn3(lam) for lam in Partitions(4) ) # long time True
-
jucys_murphy
(k)¶ Return the Jucys-Murphy element
(also known as a Young-Jucys-Murphy element) for the symmetric group algebra
self
.The Jucys-Murphy element
in the symmetric group algebra
is defined for every
by
where the addends are transpositions in
(regarded as elements of
). We note that there is not a dependence on
, so it is often surpressed in the notation.
EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: QS3.jucys_murphy(1) 0 sage: QS3.jucys_murphy(2) [2, 1, 3] sage: QS3.jucys_murphy(3) [1, 3, 2] + [3, 2, 1] sage: QS4 = SymmetricGroupAlgebra(QQ, 4) sage: j3 = QS4.jucys_murphy(3); j3 [1, 3, 2, 4] + [3, 2, 1, 4] sage: j4 = QS4.jucys_murphy(4); j4 [1, 2, 4, 3] + [1, 4, 3, 2] + [4, 2, 3, 1] sage: j3*j4 == j4*j3 True sage: QS5 = SymmetricGroupAlgebra(QQ, 5) sage: QS5.jucys_murphy(4) [1, 2, 4, 3, 5] + [1, 4, 3, 2, 5] + [4, 2, 3, 1, 5]
TESTS:
sage: QS3.jucys_murphy(4) Traceback (most recent call last): ... ValueError: k (= 4) must be between 1 and n (= 3) (inclusive)
-
left_action_product
(left, right)¶ Return the product of two elements
left
andright
ofself
, where multiplication is defined in such a way that for two permutationsand
, the product
is the permutation obtained by first applying
and then applying
. This definition of multiplication is tailored to make multiplication of permutations associative with their action on numbers if permutations are to act on numbers from the left.
EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: p1 = Permutation([2, 1, 3]) sage: p2 = Permutation([3, 1, 2]) sage: QS3.left_action_product(QS3(p1), QS3(p2)) [3, 2, 1] sage: x = QS3([1, 2, 3]) - 2*QS3([1, 3, 2]) sage: y = 1/2 * QS3([3, 1, 2]) + 3*QS3([1, 2, 3]) sage: QS3.left_action_product(x, y) 3*[1, 2, 3] - 6*[1, 3, 2] - [2, 1, 3] + 1/2*[3, 1, 2] sage: QS3.left_action_product(0, x) 0
The method coerces its input into the algebra
self
:sage: QS4 = SymmetricGroupAlgebra(QQ, 4) sage: QS4.left_action_product(QS3([1, 2, 3]), QS3([2, 1, 3])) [2, 1, 3, 4] sage: QS4.left_action_product(1, Permutation([4, 1, 2, 3])) [4, 1, 2, 3]
TESTS:
sage: QS4 = SymmetricGroup(4).algebra(QQ) sage: QS4.left_action_product(QS4((1,2)), QS4((2,3))) (1,2,3) sage: QS4.left_action_product(1, QS4((1,2))) (1,2)
Warning
Note that coercion presently works from permutations of
n
into then
-th symmetric group algebra, and also from all smaller symmetric group algebras into then
-th symmetric group algebra, but not from permutations of integers smaller thann
into then
-th symmetric group algebra.
-
monomial_from_smaller_permutation
(permutation)¶ Convert
permutation
into a permutation, possibly extending it to the appropriate size, and return the corresponding basis element ofself
.EXAMPLES:
sage: QS5 = SymmetricGroupAlgebra(QQ, 5) sage: QS5.monomial_from_smaller_permutation([]) [1, 2, 3, 4, 5] sage: QS5.monomial_from_smaller_permutation(Permutation([3,1,2])) [3, 1, 2, 4, 5] sage: QS5.monomial_from_smaller_permutation([5,3,4,1,2]) [5, 3, 4, 1, 2] sage: QS5.monomial_from_smaller_permutation(SymmetricGroup(2)((1,2))) [2, 1, 3, 4, 5] sage: QS5g = SymmetricGroup(5).algebra(QQ) sage: QS5g.monomial_from_smaller_permutation([2,1]) (1,2)
TESTS:
sage: QS5.monomial_from_smaller_permutation([5,3,4,1,2]).parent() Symmetric group algebra of order 5 over Rational Field
-
retract_direct_product
(f, m)¶ Return the direct-product retract of the element
to
, where
(and where
is
self
).If
is a nonnegative integer less or equal to
, then the direct-product retract from
to
is defined as an
-linear map
which sends every permutation
to
Here
denotes the direct-product retract of the permutation
to
, which is defined in
retract_direct_product()
.EXAMPLES:
sage: SGA3 = SymmetricGroupAlgebra(QQ, 3) sage: SGA3.retract_direct_product(2*SGA3([1,2,3]) - 4*SGA3([2,1,3]) + 7*SGA3([1,3,2]), 2) 2*[1, 2] - 4*[2, 1] sage: SGA3.retract_direct_product(2*SGA3([1,3,2]) - 5*SGA3([2,3,1]), 2) 0 sage: SGA5 = SymmetricGroupAlgebra(QQ, 5) sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 4) 11*[3, 2, 1, 4] sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 3) -6*[1, 3, 2] + 11*[3, 2, 1] sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 2) 0 sage: SGA5.retract_direct_product(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 1) 2*[1] sage: SGA5.retract_direct_product(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 3) 8*[1, 2, 3] - 6*[1, 3, 2] sage: SGA5.retract_direct_product(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 1) 2*[1] sage: SGA5.retract_direct_product(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 0) 2*[]
TESTS:
Check this works with other indexing sets:
sage: G = SymmetricGroup(4).algebra(QQ) sage: G.retract_direct_product(G.an_element(), 3) () + 2*(1,2)
See also
-
retract_okounkov_vershik
(f, m)¶ Return the Okounkov-Vershik retract of the element
to
, where
(and where
is
self
).If
is a nonnegative integer less or equal to
, then the Okounkov-Vershik retract from
to
is defined as an
-linear map
which sends every permutation
to the Okounkov-Vershik retract of the permutation
to
, which is defined in
retract_okounkov_vershik()
.EXAMPLES:
sage: SGA3 = SymmetricGroupAlgebra(QQ, 3) sage: SGA3.retract_okounkov_vershik(2*SGA3([1,2,3]) - 4*SGA3([2,1,3]) + 7*SGA3([1,3,2]), 2) 9*[1, 2] - 4*[2, 1] sage: SGA3.retract_okounkov_vershik(2*SGA3([1,3,2]) - 5*SGA3([2,3,1]), 2) 2*[1, 2] - 5*[2, 1] sage: SGA5 = SymmetricGroupAlgebra(QQ, 5) sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 4) -6*[1, 3, 2, 4] + 8*[1, 4, 2, 3] + 11*[3, 2, 1, 4] sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 3) 2*[1, 3, 2] + 11*[3, 2, 1] sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 2) 13*[1, 2] sage: SGA5.retract_okounkov_vershik(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 1) 13*[1] sage: SGA5.retract_okounkov_vershik(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 3) 8*[1, 2, 3] - 6*[1, 3, 2] sage: SGA5.retract_okounkov_vershik(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 1) 2*[1] sage: SGA5.retract_okounkov_vershik(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 0) 2*[]
TESTS:
Check this works with other indexing sets:
sage: G = SymmetricGroup(4).algebra(QQ) sage: G.retract_okounkov_vershik(G.an_element(), 3) () + 2*(1,2) + 4*(1,2,3)
See also
-
retract_plain
(f, m)¶ Return the plain retract of the element
to
, where
(and where
is
self
).If
is a nonnegative integer less or equal to
, then the plain retract from
to
is defined as an
-linear map
which sends every permutation
to
Here
denotes the plain retract of the permutation
to
, which is defined in
retract_plain()
.EXAMPLES:
sage: SGA3 = SymmetricGroupAlgebra(QQ, 3) sage: SGA3.retract_plain(2*SGA3([1,2,3]) - 4*SGA3([2,1,3]) + 7*SGA3([1,3,2]), 2) 2*[1, 2] - 4*[2, 1] sage: SGA3.retract_plain(2*SGA3([1,3,2]) - 5*SGA3([2,3,1]), 2) 0 sage: SGA5 = SymmetricGroupAlgebra(QQ, 5) sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 4) 11*[3, 2, 1, 4] sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 3) 11*[3, 2, 1] sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 2) 0 sage: SGA5.retract_plain(8*SGA5([1,4,2,5,3]) - 6*SGA5([1,3,2,5,4]) + 11*SGA5([3,2,1,4,5]), 1) 0 sage: SGA5.retract_plain(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 3) 8*[1, 2, 3] - 6*[1, 3, 2] sage: SGA5.retract_plain(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 1) 8*[1] sage: SGA5.retract_plain(8*SGA5([1,2,3,4,5]) - 6*SGA5([1,3,2,4,5]), 0) 8*[]
TESTS:
Check this works with other indexing sets:
sage: G = SymmetricGroup(4).algebra(QQ) sage: G.retract_plain(G.an_element(), 3) () + 2*(1,2)
-
right_action_product
(left, right)¶ Return the product of two elements
left
andright
ofself
, where multiplication is defined in such a way that for two permutationsand
, the product
is the permutation obtained by first applying
and then applying
. This definition of multiplication is tailored to make multiplication of permutations associative with their action on numbers if permutations are to act on numbers from the right.
EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: p1 = Permutation([2, 1, 3]) sage: p2 = Permutation([3, 1, 2]) sage: QS3.right_action_product(QS3(p1), QS3(p2)) [1, 3, 2] sage: x = QS3([1, 2, 3]) - 2*QS3([1, 3, 2]) sage: y = 1/2 * QS3([3, 1, 2]) + 3*QS3([1, 2, 3]) sage: QS3.right_action_product(x, y) 3*[1, 2, 3] - 6*[1, 3, 2] + 1/2*[3, 1, 2] - [3, 2, 1] sage: QS3.right_action_product(0, x) 0
The method coerces its input into the algebra
self
:sage: QS4 = SymmetricGroupAlgebra(QQ, 4) sage: QS4.right_action_product(QS3([1, 2, 3]), QS3([2, 1, 3])) [2, 1, 3, 4] sage: QS4.right_action_product(1, Permutation([4, 1, 2, 3])) [4, 1, 2, 3]
TESTS:
sage: QS4 = SymmetricGroup(4).algebra(QQ) sage: QS4.right_action_product(QS4((1,2)), QS4((2,3))) (1,3,2) sage: QS4.right_action_product(1, QS4((1,2))) (1,2)
Warning
Note that coercion presently works from permutations of
n
into then
-th symmetric group algebra, and also from all smaller symmetric group algebras into then
-th symmetric group algebra, but not from permutations of integers smaller thann
into then
-th symmetric group algebra.
-
rsw_shuffling_element
(k)¶ Return the
-th Reiner-Saliola-Welker shuffling element in the group algebra
self
.The
-th Reiner-Saliola-Welker shuffling element in the symmetric group algebra
over a ring
is defined as the sum
, where for every permutation
, the number
is the number of all
-noninversions of
(that is, the number of all
-element subsets of
on which
restricts to a strictly increasing map). See
sage.combinat.permutation.number_of_noninversions()
for themap.
This element is more or less the operator
introduced in [RSW2011]; more precisely,
is the left multiplication by this element.
It is a nontrivial theorem (Theorem 1.1 in [RSW2011]) that the operators
(for fixed
and varying
) pairwise commute. It is a conjecture (Conjecture 1.2 in [RSW2011]) that all their eigenvalues are integers (which, in light of their commutativity and easily established symmetry, yields that they can be simultaneously diagonalized over
with only integer eigenvalues).
EXAMPLES:
The Reiner-Saliola-Welker shuffling elements on
:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: QS3.rsw_shuffling_element(0) [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1] sage: QS3.rsw_shuffling_element(1) 3*[1, 2, 3] + 3*[1, 3, 2] + 3*[2, 1, 3] + 3*[2, 3, 1] + 3*[3, 1, 2] + 3*[3, 2, 1] sage: QS3.rsw_shuffling_element(2) 3*[1, 2, 3] + 2*[1, 3, 2] + 2*[2, 1, 3] + [2, 3, 1] + [3, 1, 2] sage: QS3.rsw_shuffling_element(3) [1, 2, 3] sage: QS3.rsw_shuffling_element(4) 0
Checking the commutativity of Reiner-Saliola-Welker shuffling elements (we leave out the ones for which it is trivial):
sage: def test_rsw_comm(n): ....: QSn = SymmetricGroupAlgebra(QQ, n) ....: rsws = [QSn.rsw_shuffling_element(k) for k in range(2, n)] ....: return all( all( rsws[i] * rsws[j] == rsws[j] * rsws[i] ....: for j in range(i) ) ....: for i in range(len(rsws)) ) sage: test_rsw_comm(3) True sage: test_rsw_comm(4) True sage: test_rsw_comm(5) # long time True
Note
For large
k
(relative ton
), it might be faster to callQSn.left_action_product(QSn.semi_rsw_element(k), QSn.antipode(binary_unshuffle_sum(k)))
thanQSn.rsw_shuffling_element(n)
.See also
-
semi_rsw_element
(k)¶ Return the
-th semi-RSW element in the group algebra
self
.The
-th semi-RSW element in the symmetric group algebra
over a ring
is defined as the sum of all permutations
satisfying
.
This element has the property that, if it is denoted by
, then
is
times the
-th Reiner-Saliola-Welker shuffling element of
(see
rsw_shuffling_element()
). Here,denotes the antipode of the group algebra
.
The
-th semi-RSW element is the image of the complete non-commutative symmetric function
in the ring of non-commutative symmetric functions under the canonical projection on the symmetric group algebra (through the descent algebra).
EXAMPLES:
The semi-RSW elements on
:
sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: QS3.semi_rsw_element(0) [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1] sage: QS3.semi_rsw_element(1) [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1] sage: QS3.semi_rsw_element(2) [1, 2, 3] + [1, 3, 2] + [2, 3, 1] sage: QS3.semi_rsw_element(3) [1, 2, 3] sage: QS3.semi_rsw_element(4) 0
Let us check the relation with the
-th Reiner-Saliola-Welker shuffling element stated in the docstring:
sage: def test_rsw(n): ....: ZSn = SymmetricGroupAlgebra(ZZ, n) ....: for k in range(1, n): ....: a = ZSn.semi_rsw_element(k) ....: b = ZSn.left_action_product(a, ZSn.antipode(a)) ....: if factorial(n-k) * ZSn.rsw_shuffling_element(k) != b: ....: return False ....: return True sage: test_rsw(3) True sage: test_rsw(4) True sage: test_rsw(5) # long time True
Let us also check the statement about the complete non-commutative symmetric function:
sage: def test_rsw_ncsf(n): ....: ZSn = SymmetricGroupAlgebra(ZZ, n) ....: NSym = NonCommutativeSymmetricFunctions(ZZ) ....: S = NSym.S() ....: for k in range(1, n): ....: a = S(Composition([k] + [1]*(n-k))).to_symmetric_group_algebra() ....: if a != ZSn.semi_rsw_element(k): ....: return False ....: return True sage: test_rsw_ncsf(3) True sage: test_rsw_ncsf(4) True sage: test_rsw_ncsf(5) # long time True
-
seminormal_basis
(mult='l2r')¶ Return a list of the seminormal basis elements of
self
.The seminormal basis of a symmetric group algebra is defined as follows:
Let
be a nonnegative integer. Let
be a
-algebra. In the following, we will use the “left action” convention for multiplying permutations. This means that for all permutations
and
in
, the product
is defined in such a way that
for each
(this is the same convention as in
left_action_product()
, but not the default semantics of theoperator on permutations in Sage). Thus, for instance,
is the permutation obtained by first transposing
with
and then transposing
with
(where
).
For every partition
of
, let
where
is the number of standard Young tableaux of shape
. Note that
is an integer, namely the product of all hook lengths of
(by the hook length formula). In Sage, this integer can be computed by using
sage.combinat.symmetric_group_algebra.kappa()
.Let
be a standard tableau of size
.
Let
denote the formal sum (in
) of all permutations in
which stabilize the rows of
(as sets), i. e., which map each entry
of
to an entry in the same row as
. (See
sage.combinat.symmetric_group_algebra.a()
for an implementation of this.)Let
denote the signed formal sum (in
) of all permutations in
which stabilize the columns of
(as sets). Here, “signed” means that each permutation is multiplied with its sign. (This is implemented in
sage.combinat.symmetric_group_algebra.b()
.)Define an element
of
to be
. (This is implemented in
sage.combinat.symmetric_group_algebra.e()
for.)
Let
denote the shape of
. (See
shape()
.)Let
denote the standard tableau of size
obtained by removing the letter
(along with its cell) from
(if
).
Now, we define an element
of
. We define it by induction on the size
of
, so we set
and only need to define
for
, assuming that
is already defined. We do this by setting
This element
is implemented as
sage.combinat.symmetric_group_algebra.epsilon()
for, but it is also a particular case of the elements
defined below.
Now let
be a further tableau of the same shape as
(possibly equal to
). Let
denote the permutation in
such that applying this permutation to the entries of
yields the tableau
. Define an element
of
by
This element
is called Young’s seminormal unit corresponding to the bitableau `(T, S)`, and is the return value of
epsilon_ik()
applied toT
andS
. Note that.
If we let
run through all partitions of
, and
run through all pairs of tableaux of shape
, then the elements
form a basis of
. This basis is called Young’s seminormal basis and has the properties that
(where
stands for the Kronecker delta).
Warning
Because of our convention, we are multiplying our elements in reverse of those given in some papers, for example [Ram1997]. Using the other convention of multiplying permutations, we would instead have
.
In other words, Young’s seminormal basis consists of the matrix units in a (particular) Artin-Wedderburn decomposition of
into a direct product of matrix algebras over
.
The output of
seminormal_basis()
is a list of all elements of the seminormal basis ofself
.INPUT:
mult
– string (default:'l2r'
). If set to'r2l'
, this causes the method to return the list of the antipodes (antipode()
) of allinstead of the
themselves.
EXAMPLES:
sage: QS3 = SymmetricGroupAlgebra(QQ,3) sage: QS3.seminormal_basis() [1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1], 1/3*[1, 2, 3] + 1/6*[1, 3, 2] - 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] + 1/6*[3, 2, 1], 1/3*[1, 3, 2] + 1/3*[2, 3, 1] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1], 1/4*[1, 3, 2] - 1/4*[2, 3, 1] + 1/4*[3, 1, 2] - 1/4*[3, 2, 1], 1/3*[1, 2, 3] - 1/6*[1, 3, 2] + 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] - 1/6*[3, 2, 1], 1/6*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] - 1/6*[3, 2, 1]]
TESTS:
sage: QS3g = SymmetricGroup(3).algebra(QQ) sage: QS3g.seminormal_basis() [1/6*() + 1/6*(2,3) + 1/6*(1,2) + 1/6*(1,2,3) + 1/6*(1,3,2) + 1/6*(1,3), 1/3*() + 1/6*(2,3) - 1/3*(1,2) - 1/6*(1,2,3) - 1/6*(1,3,2) + 1/6*(1,3), 1/3*(2,3) + 1/3*(1,2,3) - 1/3*(1,3,2) - 1/3*(1,3), 1/4*(2,3) - 1/4*(1,2,3) + 1/4*(1,3,2) - 1/4*(1,3), 1/3*() - 1/6*(2,3) + 1/3*(1,2) - 1/6*(1,2,3) - 1/6*(1,3,2) - 1/6*(1,3), 1/6*() - 1/6*(2,3) - 1/6*(1,2) + 1/6*(1,2,3) + 1/6*(1,3,2) - 1/6*(1,3)]
REFERENCES:
[Ram1997] Arun Ram. Seminormal representations of Weyl groups and Iwahori-Hecke algebras. Proc. London Math. Soc. (3) 75 (1997). 99-133. Arxiv math/9511223v1. http://www.ms.unimelb.edu.au/~ram/Publications/1997PLMSv75p99.pdf
-
-
sage.combinat.symmetric_group_algebra.
a
(tableau, star=0, base_ring=Rational Field)¶ The row projection operator corresponding to the Young tableau
tableau
(which is supposed to contain every integer fromto its size precisely once, but may and may not be standard).
This is the sum (in the group algebra of the relevant symmetric group over
) of all the permutations which preserve the rows of
tableau
. It is calledin [EtRT], Section 4.2.
REFERENCES:
[EtRT] (1, 2) Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina, “Introduction to representation theory”, Arxiv 0901.0827v5. INPUT:
tableau
– Young tableau which contains every integer fromto its size precisely once.
star
– nonnegative integer (default:). When this optional variable is set, the method computes not the row projection operator of
tableau
, but the row projection operator of the restriction oftableau
to the entries1, 2, ..., tableau.size() - star
instead.base_ring
– commutative ring (default:QQ
). When this optional variable is set, the row projection operator is computed over a user-determined base ring instead of. (Note that symmetric group algebras currently don’t preserve coercion, so e. g. a symmetric group algebra over
does not coerce into the corresponding one over
; so convert manually or choose your base rings wisely!)
EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import a sage: a([[1,2]]) [1, 2] + [2, 1] sage: a([[1],[2]]) [1, 2] sage: a([]) [] sage: a([[1, 5], [2, 3], [4]]) [1, 2, 3, 4, 5] + [1, 3, 2, 4, 5] + [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1] sage: a([[1,4], [2,3]], base_ring=ZZ) [1, 2, 3, 4] + [1, 3, 2, 4] + [4, 2, 3, 1] + [4, 3, 2, 1]
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sage.combinat.symmetric_group_algebra.
b
(tableau, star=0, base_ring=Rational Field)¶ The column projection operator corresponding to the Young tableau
tableau
(which is supposed to contain every integer fromto its size precisely once, but may and may not be standard).
This is the signed sum (in the group algebra of the relevant symmetric group over
) of all the permutations which preserve the column of
tableau
(where the signs are the usual signs of the permutations). It is calledin [EtRT], Section 4.2.
INPUT:
tableau
– Young tableau which contains every integer fromto its size precisely once.
star
– nonnegative integer (default:). When this optional variable is set, the method computes not the column projection operator of
tableau
, but the column projection operator of the restriction oftableau
to the entries1, 2, ..., tableau.size() - star
instead.base_ring
– commutative ring (default:QQ
). When this optional variable is set, the column projection operator is computed over a user-determined base ring instead of. (Note that symmetric group algebras currently don’t preserve coercion, so e. g. a symmetric group algebra over
does not coerce into the corresponding one over
; so convert manually or choose your base rings wisely!)
EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import b sage: b([[1,2]]) [1, 2] sage: b([[1],[2]]) [1, 2] - [2, 1] sage: b([]) [] sage: b([[1, 2, 4], [5, 3]]) [1, 2, 3, 4, 5] - [1, 3, 2, 4, 5] - [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1] sage: b([[1, 4], [2, 3]], base_ring=ZZ) [1, 2, 3, 4] - [1, 2, 4, 3] - [2, 1, 3, 4] + [2, 1, 4, 3] sage: b([[1, 4], [2, 3]], base_ring=Integers(5)) [1, 2, 3, 4] + 4*[1, 2, 4, 3] + 4*[2, 1, 3, 4] + [2, 1, 4, 3]
With the
l2r
setting for multiplication, the unnormalized Young symmetrizere(tableau)
should be the productb(tableau) * a(tableau)
for everytableau
. Let us check this on the standard tableaux of size 5:sage: from sage.combinat.symmetric_group_algebra import a, b, e sage: all( e(t) == b(t) * a(t) for t in StandardTableaux(5) ) True
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sage.combinat.symmetric_group_algebra.
e
(tableau, star=0)¶ The unnormalized Young projection operator corresponding to the Young tableau
tableau
(which is supposed to contain every integer fromto its size precisely once, but may and may not be standard).
If
is a nonnegative integer, and
is a Young tableau containing every integer from
to
exactly once, then the unnormalized Young projection operator
is defined by
where
is the sum of all permutations in
which fix the rows of
(as sets), and
is the signed sum of all permutations in
which fix the columns of
(as sets). Here, “signed” means that each permutation is multiplied with its sign; and the product on the group
is defined in such a way that
for any permutations
and
and any
.
Note that the definition of
is not uniform across literature. Others define it as
instead, or include certain scalar factors (we do not, whence “unnormalized”).
EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import e sage: e([[1,2]]) [1, 2] + [2, 1] sage: e([[1],[2]]) [1, 2] - [2, 1] sage: e([]) []
There are differing conventions for the order of the symmetrizers and antisymmetrizers. This example illustrates our conventions:
sage: e([[1,2],[3]]) [1, 2, 3] + [2, 1, 3] - [3, 1, 2] - [3, 2, 1]
To obtain the product
, one has to take the antipode of this:
sage: QS3 = parent(e([[1,2],[3]])) sage: QS3.antipode(e([[1,2],[3]])) [1, 2, 3] + [2, 1, 3] - [2, 3, 1] - [3, 2, 1]
See also
-
sage.combinat.symmetric_group_algebra.
e_hat
(tab, star=0)¶ The Young projection operator corresponding to the Young tableau
tab
(which is supposed to contain every integer fromto its size precisely once, but may and may not be standard). This is an idempotent in the rational group algebra.
If
is a nonnegative integer, and
is a Young tableau containing every integer from
to
exactly once, then the Young projection operator
is defined by
where
is the shape of
, where
is
divided by the number of standard tableaux of shape
, where
is the sum of all permutations in
which fix the rows of
(as sets), and where
is the signed sum of all permutations in
which fix the columns of
(as sets). Here, “signed” means that each permutation is multiplied with its sign; and the product on the group
is defined in such a way that
for any permutations
and
and any
.
Note that the definition of
is not uniform across literature. Others define it as
instead.
EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import e_hat sage: e_hat([[1,2,3]]) 1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1] sage: e_hat([[1],[2]]) 1/2*[1, 2] - 1/2*[2, 1]
There are differing conventions for the order of the symmetrizers and antisymmetrizers. This example illustrates our conventions:
sage: e_hat([[1,2],[3]]) 1/3*[1, 2, 3] + 1/3*[2, 1, 3] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1]
See also
-
sage.combinat.symmetric_group_algebra.
e_ik
(itab, ktab, star=0)¶ EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import e_ik sage: e_ik([[1,2,3]], [[1,2,3]]) [1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1] sage: e_ik([[1,2,3]], [[1,2,3]], star=1) [1, 2] + [2, 1]
-
sage.combinat.symmetric_group_algebra.
epsilon
(tab, star=0)¶ The
-th element of the seminormal basis of the group algebra
, where
is the tableau
tab
(with itsstar
highest entries removed if the optional variablestar
is set).See the docstring of
seminormal_basis()
for the notation used herein.EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import epsilon sage: epsilon([[1,2]]) 1/2*[1, 2] + 1/2*[2, 1] sage: epsilon([[1],[2]]) 1/2*[1, 2] - 1/2*[2, 1]
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sage.combinat.symmetric_group_algebra.
epsilon_ik
(itab, ktab, star=0)¶ Return the seminormal basis element of the symmetric group algebra
corresponding to the pair of tableaux
itab
andktab
(or restrictions of these tableaux, if the optional variablestar
is set).INPUT:
itab
,ktab
– two standard tableaux of same size.star
– integer (default:).
OUTPUT:
The element
, where
and
are the tableaux obtained by removing all entries higher than
from
itab
andktab
, respectively (whereis the size of
itab
andktab
). Here, we are using the notations fromseminormal_basis()
.EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import epsilon_ik sage: epsilon_ik([[1,2],[3]], [[1,3],[2]]) 1/4*[1, 3, 2] - 1/4*[2, 3, 1] + 1/4*[3, 1, 2] - 1/4*[3, 2, 1] sage: epsilon_ik([[1,2],[3]], [[1,3],[2]], star=1) Traceback (most recent call last): ... ValueError: the two tableaux must be of the same shape
-
sage.combinat.symmetric_group_algebra.
kappa
(alpha)¶ Return
, which is
divided by the number of standard tableaux of shape
(where
is a partition of
).
INPUT:
alpha
– integer partition (can be encoded as a list).
OUTPUT:
The factorial of the size of
alpha
, divided by the number of standard tableaux of shapealpha
. Equivalently, the product of all hook lengths ofalpha
.EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import kappa sage: kappa(Partition([2,1])) 3 sage: kappa([2,1]) 3
-
sage.combinat.symmetric_group_algebra.
pi_ik
(itab, ktab)¶ Return the permutation
which sends every entry of the tableau
itab
to the respective entry of the tableauktab
, as an element of the corresponding symmetric group algebra.This assumes that
itab
andktab
are tableaux (possibly given just as lists of lists) of the same shape.EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import pi_ik sage: pi_ik([[1,3],[2]], [[1,2],[3]]) [1, 3, 2]
-
sage.combinat.symmetric_group_algebra.
seminormal_test
(n)¶ Run a variety of tests to verify that the construction of the seminormal basis works as desired. The numbers appearing are results in James and Kerber’s ‘Representation Theory of the Symmetric Group’ [JamesKerber].
EXAMPLES:
sage: from sage.combinat.symmetric_group_algebra import seminormal_test sage: seminormal_test(3) True