Partition tuples#
A PartitionTuple
is a tuple of partitions. That is, an ordered
\(k\)-tuple of partitions \(\mu=(\mu^{(1)},\mu^{(2)},...,\mu^{(k)})\). If
then we say that \(\mu\) is a \(k\)-partition of \(n\).
In representation theory partition tuples arise as the natural indexing set for the ordinary irreducible representations of:
the wreath products of cyclic groups with symmetric groups,
the Ariki-Koike algebras, or the cyclotomic Hecke algebras of the complex reflection groups of type \(G(r,1,n)\),
the degenerate cyclotomic Hecke algebras of type \(G(r,1,n)\).
When these algebras are not semisimple, partition tuples index an important class of modules for the algebras, which are generalisations of the Specht modules of the symmetric groups.
Tuples of partitions also index the standard basis of the higher level combinatorial Fock spaces. As a consequence, the combinatorics of partition tuples encapsulates the canonical bases of crystal graphs for the irreducible integrable highest weight modules of the (quantized) affine special linear groups and the (quantized) affine general linear groups. By the categorification theorems of Ariki, Varagnolo-Vasserot, Stroppel-Webster and others, in characteristic zero the degenerate and non-degenerate cyclotomic Hecke algebras, via their Khovanov-Lauda-Rouquier grading, categorify the canonical bases of the quantum affine special and general linear groups.
Partitions are naturally in bijection with 1-tuples of partitions. Most of the combinatorial operations defined on partitions extend to partition tuples in a meaningful way. For example, the semisimple branching rules for the Specht modules are described by adding and removing cells from partition tuples and the modular branching rules correspond to adding and removing good and cogood nodes, which is the underlying combinatorics for the associated crystal graphs.
A PartitionTuple
belongs to PartitionTuples
and its derived
classes. PartitionTuples
is the parent class for all partitions
tuples. Four different classes of tuples of partitions are currently supported:
PartitionTuples(level=k,size=n)
are \(k\)-tuple of partitions of \(n\).PartitionTuples(level=k)
are \(k\)-tuple of partitions.PartitionTuples(size=n)
are tuples of partitions of \(n\).PartitionTuples()
are tuples of partitions.
Note
As with Partitions
, in sage the cells, or nodes, of partition
tuples are 0-based. For example, the (lexicographically) first cell in
any non-empty partition tuple is \([0,0,0]\).
EXAMPLES:
sage: PartitionTuple([[2,2],[1,1],[2]]).cells()
[(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 1, 0), (2, 0, 0), (2, 0, 1)]
Note
Many PartitionTuple
methods take the individual coordinates \((k,r,c)\)
as their arguments, here \(k\) is the component, \(r\) is the row index and \(c\) is
the column index. If your coordinates are in the form (k,r,c)
then use
Python’s *-operator.
EXAMPLES:
sage: mu=PartitionTuple([[1,1],[2],[2,1]])
sage: [ mu.arm_length(*c) for c in mu.cells()]
[0, 0, 1, 0, 1, 0, 0]
Warning
In sage, if mu
is a partition tuple then mu[k]
most naturally refers
to the \(k\)-th component of mu
, so we use the convention of the
\((k,r,c)\)-th cell in a partition tuple refers to the cell in component \(k\),
row \(r\), and column \(c\). In the literature, the cells of a partition tuple
are usually written in the form \((r,c,k)\), where \(r\) is the row index, \(c\)
is the column index, and \(k\) is the component index.
REFERENCES:
AUTHORS:
Andrew Mathas (2012-06-01): Initial classes.
EXAMPLES:
First is a finite enumerated set and the remaining classes are infinite enumerated sets:
sage: PartitionTuples().an_element()
([1, 1, 1, 1], [2, 1, 1], [3, 1], [4])
sage: PartitionTuples(4).an_element()
([], [1], [2], [3])
sage: PartitionTuples(size=5).an_element()
([1], [1], [1], [1], [1])
sage: PartitionTuples(4,5).an_element()
([1], [], [], [4])
sage: PartitionTuples(3,2)[:]
[([2], [], []),
([1, 1], [], []),
([1], [1], []),
([1], [], [1]),
([], [2], []),
([], [1, 1], []),
([], [1], [1]),
([], [], [2]),
([], [], [1, 1])]
sage: PartitionTuples(2,3).list()
[([3], []),
([2, 1], []),
([1, 1, 1], []),
([2], [1]),
([1, 1], [1]),
([1], [2]),
([1], [1, 1]),
([], [3]),
([], [2, 1]),
([], [1, 1, 1])]
One tuples of partitions are naturally in bijection with partitions and, as far as possible, partition tuples attempts to identify one tuples with partitions:
sage: Partition([4,3]) == PartitionTuple([[4,3]])
True
sage: Partition([4,3]) == PartitionTuple([4,3])
True
sage: PartitionTuple([4,3])
[4, 3]
sage: Partition([4,3]) in PartitionTuples()
True
Partition tuples come equipped with many of the corresponding methods for partitions. For example, it is possible to add and remove cells, to conjugate partition tuples, to work with their diagrams, compare partition tuples in dominance and so:
sage: PartitionTuple([[4,1],[],[2,2,1],[3]]).pp()
**** - ** ***
* **
*
sage: PartitionTuple([[4,1],[],[2,2,1],[3]]).conjugate()
([1, 1, 1], [3, 2], [], [2, 1, 1, 1])
sage: PartitionTuple([[4,1],[],[2,2,1],[3]]).conjugate().pp()
* *** - **
* ** *
* *
*
sage: lam=PartitionTuples(3)([[3,2],[],[1,1,1,1]]); lam
([3, 2], [], [1, 1, 1, 1])
sage: lam.level()
3
sage: lam.size()
9
sage: lam.category()
Category of elements of Partition tuples of level 3
sage: lam.parent()
Partition tuples of level 3
sage: lam[0]
[3, 2]
sage: lam[1]
[]
sage: lam[2]
[1, 1, 1, 1]
sage: lam.pp()
*** - *
** *
*
*
sage: lam.removable_cells()
[(0, 0, 2), (0, 1, 1), (2, 3, 0)]
sage: lam.down_list()
[([2, 2], [], [1, 1, 1, 1]),
([3, 1], [], [1, 1, 1, 1]),
([3, 2], [], [1, 1, 1])]
sage: lam.addable_cells()
[(0, 0, 3), (0, 1, 2), (0, 2, 0), (1, 0, 0), (2, 0, 1), (2, 4, 0)]
sage: lam.up_list()
[([4, 2], [], [1, 1, 1, 1]),
([3, 3], [], [1, 1, 1, 1]),
([3, 2, 1], [], [1, 1, 1, 1]),
([3, 2], [1], [1, 1, 1, 1]),
([3, 2], [], [2, 1, 1, 1]),
([3, 2], [], [1, 1, 1, 1, 1])]
sage: lam.conjugate()
([4], [], [2, 2, 1])
sage: lam.dominates( PartitionTuple([[3],[1],[2,2,1]]) )
False
sage: lam.dominates( PartitionTuple([[3],[2],[1,1,1]]))
True
Every partition tuple behaves every much like a tuple of partitions:
sage: mu=PartitionTuple([[4,1],[],[2,2,1],[3]])
sage: [ nu for nu in mu ]
[[4, 1], [], [2, 2, 1], [3]]
sage: Set([ type(nu) for nu in mu ])
{<class 'sage.combinat.partition.Partitions_all_with_category.element_class'>}
sage: mu[2][2]
1
sage: mu[3]
[3]
sage: mu.components()
[[4, 1], [], [2, 2, 1], [3]]
sage: mu.components() == [ nu for nu in mu ]
True
sage: mu[0]
[4, 1]
sage: mu[1]
[]
sage: mu[2]
[2, 2, 1]
sage: mu[2][0]
2
sage: mu[2][1]
2
sage: mu.level()
4
sage: len(mu)
4
sage: mu.cells()
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 1, 0), (2, 0, 0), (2, 0, 1), (2, 1, 0), (2, 1, 1), (2, 2, 0), (3, 0, 0), (3, 0, 1), (3, 0, 2)]
sage: mu.addable_cells()
[(0, 0, 4), (0, 1, 1), (0, 2, 0), (1, 0, 0), (2, 0, 2), (2, 2, 1), (2, 3, 0), (3, 0, 3), (3, 1, 0)]
sage: mu.removable_cells()
[(0, 0, 3), (0, 1, 0), (2, 1, 1), (2, 2, 0), (3, 0, 2)]
Attached to a partition tuple is the corresponding Young, or parabolic, subgroup:
sage: mu.young_subgroup()
Permutation Group with generators [(), (12,13), (11,12), (8,9), (6,7), (3,4), (2,3), (1,2)]
sage: mu.young_subgroup_generators()
[1, 2, 3, 6, 8, 11, 12]
- class sage.combinat.partition_tuple.PartitionTuple(parent, mu)#
Bases:
CombinatorialElement
A tuple of
Partition
.A tuple of partition comes equipped with many of methods available to partitions. The
level
of the PartitionTuple is the length of the tuple.This is an ordered \(k\)-tuple of partitions \(\mu=(\mu^{(1)},\mu^{(2)},...,\mu^{(k)})\). If
\[n = \lvert \mu \rvert = \lvert \mu^{(1)} \rvert + \lvert \mu^{(2)} \rvert + \cdots + \lvert \mu^{(k)} \rvert\]then \(\mu\) is a \(k\)-partition of \(n\).
In representation theory PartitionTuples arise as the natural indexing set for the ordinary irreducible representations of:
the wreath products of cyclic groups with symmetric groups
the Ariki-Koike algebras, or the cyclotomic Hecke algebras of the complex reflection groups of type \(G(r,1,n)\)
the degenerate cyclotomic Hecke algebras of type \(G(r,1,n)\)
When these algebras are not semisimple, partition tuples index an important class of modules for the algebras which are generalisations of the Specht modules of the symmetric groups.
Tuples of partitions also index the standard basis of the higher level combinatorial Fock spaces. As a consequence, the combinatorics of partition tuples encapsulates the canonical bases of crystal graphs for the irreducible integrable highest weight modules of the (quantized) affine special linear groups and the (quantized) affine general linear groups. By the categorification theorems of Ariki, Varagnolo-Vasserot, Stroppel-Webster and others, in characteristic zero the degenerate and non-degenerate cyclotomic Hecke algebras, via their Khovanov-Lauda-Rouquier grading, categorify the canonical bases of the quantum affine special and general linear groups.
Partitions are naturally in bijection with 1-tuples of partitions. Most of the combinatorial operations defined on partitions extend to PartitionTuples in a meaningful way. For example, the semisimple branching rules for the Specht modules are described by adding and removing cells from partition tuples and the modular branching rules correspond to adding and removing good and cogood nodes, which is the underlying combinatorics for the associated crystal graphs.
Warning
In the literature, the cells of a partition tuple are usually written in the form \((r,c,k)\), where \(r\) is the row index, \(c\) is the column index, and \(k\) is the component index. In sage, if
mu
is a partition tuple thenmu[k]
most naturally refers to the \(k\)-th component ofmu
, so we use the convention of the \((k,r,c)\)-th cell in a partition tuple refers to the cell in component \(k\), row \(r\), and column \(c\).INPUT:
Anything which can reasonably be interpreted as a tuple of partitions. That is, a list or tuple of partitions or valid input to
Partition
.EXAMPLES:
sage: mu=PartitionTuple( [[3,2],[2,1],[],[1,1,1,1]] ); mu ([3, 2], [2, 1], [], [1, 1, 1, 1]) sage: nu=PartitionTuple( ([3,2],[2,1],[],[1,1,1,1]) ); nu ([3, 2], [2, 1], [], [1, 1, 1, 1]) sage: mu == nu True sage: mu is nu False sage: mu in PartitionTuples() True sage: mu.parent() Partition tuples sage: lam=PartitionTuples(3)([[3,2],[],[1,1,1,1]]); lam ([3, 2], [], [1, 1, 1, 1]) sage: lam.level() 3 sage: lam.size() 9 sage: lam.category() Category of elements of Partition tuples of level 3 sage: lam.parent() Partition tuples of level 3 sage: lam[0] [3, 2] sage: lam[1] [] sage: lam[2] [1, 1, 1, 1] sage: lam.pp() *** - * ** * * * sage: lam.removable_cells() [(0, 0, 2), (0, 1, 1), (2, 3, 0)] sage: lam.down_list() [([2, 2], [], [1, 1, 1, 1]), ([3, 1], [], [1, 1, 1, 1]), ([3, 2], [], [1, 1, 1])] sage: lam.addable_cells() [(0, 0, 3), (0, 1, 2), (0, 2, 0), (1, 0, 0), (2, 0, 1), (2, 4, 0)] sage: lam.up_list() [([4, 2], [], [1, 1, 1, 1]), ([3, 3], [], [1, 1, 1, 1]), ([3, 2, 1], [], [1, 1, 1, 1]), ([3, 2], [1], [1, 1, 1, 1]), ([3, 2], [], [2, 1, 1, 1]), ([3, 2], [], [1, 1, 1, 1, 1])] sage: lam.conjugate() ([4], [], [2, 2, 1]) sage: lam.dominates( PartitionTuple([[3],[1],[2,2,1]]) ) False sage: lam.dominates( PartitionTuple([[3],[2],[1,1,1]])) True
See also
- add_cell(k, r, c)#
Return the partition tuple obtained by adding a cell in row
r
, columnc
, and componentk
.This does not change
self
.EXAMPLES:
sage: PartitionTuple([[1,1],[4,3],[2,1,1]]).add_cell(0,0,1) ([2, 1], [4, 3], [2, 1, 1])
- addable_cells()#
Return a list of the removable cells of this partition tuple.
All indices are of the form
(k, r, c)
, wherer
is the row-index,c
is the column index andk
is the component.EXAMPLES:
sage: PartitionTuple([[1,1],[2],[2,1]]).addable_cells() [(0, 0, 1), (0, 2, 0), (1, 0, 2), (1, 1, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0)] sage: PartitionTuple([[1,1],[4,3],[2,1,1]]).addable_cells() [(0, 0, 1), (0, 2, 0), (1, 0, 4), (1, 1, 3), (1, 2, 0), (2, 0, 2), (2, 1, 1), (2, 3, 0)]
- arm_length(k, r, c)#
Return the length of the arm of cell
(k, r, c)
inself
.INPUT:
k
– The componentr
– The rowc
– The cell
OUTPUT:
The arm length as an integer
The arm of cell
(k, r, c)
is the number of cells in thek
-th component which are to the right of the cell in rowr
and columnc
.EXAMPLES:
sage: PartitionTuple([[],[2,1],[2,2,1],[3]]).arm_length(2,0,0) 1 sage: PartitionTuple([[],[2,1],[2,2,1],[3]]).arm_length(2,0,1) 0 sage: PartitionTuple([[],[2,1],[2,2,1],[3]]).arm_length(2,2,0) 0
- block(e, multicharge)#
Return a dictionary \(\beta\) that determines the block associated to the partition
self
and thequantum_characteristic()
e
.INPUT:
e
– the quantum characteristicmulticharge
– the multicharge (default \((0,)\))
OUTPUT:
a dictionary giving the multiplicities of the residues in the partition tuple
self
In more detail, the value
beta[i]
is equal to the number of nodes of residuei
. This corresponds to the positive root\[\sum_{i\in I} \beta_i \alpha_i \in Q^+,\]a element of the positive root lattice of the corresponding Kac-Moody algebra. See [DJM1998] and [BK2009] for more details.
This is a useful statistics because two Specht modules for a cyclotomic Hecke algebra of type \(A\) belong to the same block if and only if they correspond to same element \(\beta\) of the root lattice, given above.
We return a dictionary because when the quantum characteristic is \(0\), the Cartan type is \(A_{\infty}\), in which case the simple roots are indexed by the integers.
EXAMPLES:
sage: PartitionTuple([[2,2],[2,2]]).block(0,(0,0)) {-1: 2, 0: 4, 1: 2} sage: PartitionTuple([[2,2],[2,2]]).block(2,(0,0)) {0: 4, 1: 4} sage: PartitionTuple([[2,2],[2,2]]).block(2,(0,1)) {0: 4, 1: 4} sage: PartitionTuple([[2,2],[2,2]]).block(3,(0,2)) {0: 3, 1: 2, 2: 3} sage: PartitionTuple([[2,2],[2,2]]).block(3,(0,2)) {0: 3, 1: 2, 2: 3} sage: PartitionTuple([[2,2],[2,2]]).block(3,(3,2)) {0: 3, 1: 2, 2: 3} sage: PartitionTuple([[2,2],[2,2]]).block(4,(0,0)) {0: 4, 1: 2, 3: 2}
- cells()#
Return the coordinates of the cells of
self
. Coordinates are given as (component index, row index, column index) and are 0 based.EXAMPLES:
sage: PartitionTuple([[2,1],[1],[1,1,1]]).cells() [(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0), (2, 0, 0), (2, 1, 0), (2, 2, 0)]
- components()#
Return a list containing the shape of this partition.
This function exists in order to give a uniform way of iterating over the "components" of partition tuples of level 1 (partitions) and for higher levels.
EXAMPLES:
sage: for t in PartitionTuple([[2,1],[3,2],[3]]).components(): ....: print('%s\n' % t.ferrers_diagram()) ** * *** ** *** sage: for t in PartitionTuple([3,2]).components(): ....: print('%s\n' % t.ferrers_diagram()) *** **
- conjugate()#
Return the conjugate partition tuple of
self
.The conjugate partition tuple is obtained by reversing the order of the components and then swapping the rows and columns in each component.
EXAMPLES:
sage: PartitionTuple([[2,1],[1],[1,1,1]]).conjugate() ([3], [1], [2, 1])
- contains(mu)#
Return
True
if this partition tuple contains \(\mu\).If \(\lambda=(\lambda^{(1)}, \ldots, \lambda^{(l)})\) and \(\mu=(\mu^{(1)}, \ldots, \mu^{(m)})\) are two partition tuples then \(\lambda\) contains \(\mu\) if \(m \leq l\) and \(\mu^{(i)}_r \leq \lambda^{(i)}_r\) for \(1 \leq i \leq m\) and \(r \geq 0\).
EXAMPLES:
sage: PartitionTuple([[1,1],[2],[2,1]]).contains( PartitionTuple([[1,1],[2],[2,1]]) ) True
- content(k, r, c, multicharge)#
Return the content of the cell.
Let \(m_k =\)
multicharge[k]
, then the content of a cell is \(m_k + c - r\).If the
multicharge
is a list of integers then it simply offsets the values of the contents in each component. On the other hand, if themulticharge
belongs to \(\ZZ/e\ZZ\) then the corresponding \(e\)-residue is returned (that is, the content mod \(e\)).As with the content method for partitions, the content of a cell does not technically depend on the partition tuple, but this method is included because it is often useful.
EXAMPLES:
sage: PartitionTuple([[2,1],[2],[1,1,1]]).content(0,1,0, [0,0,0]) -1 sage: PartitionTuple([[2,1],[2],[1,1,1]]).content(0,1,0, [1,0,0]) 0 sage: PartitionTuple([[2,1],[2],[1,1,1]]).content(2,1,0, [0,0,0]) -1
and now we return the 3-residue of a cell:
sage: multicharge = [IntegerModRing(3)(c) for c in [0,0,0]] sage: PartitionTuple([[2,1],[2],[1,1,1]]).content(0,1,0, multicharge) 2
- content_tableau(multicharge)#
Return the tableau which has (k,r,c)th entry equal to the content
multicharge[k]-r+c
of this cell.As with the content function, by setting the
multicharge
appropriately the tableau containing the residues is returned.EXAMPLES:
sage: PartitionTuple([[2,1],[2],[1,1,1]]).content_tableau([0,0,0]) ([[0, 1], [-1]], [[0, 1]], [[0], [-1], [-2]]) sage: PartitionTuple([[2,1],[2],[1,1,1]]).content_tableau([0,0,1]).pp() 0 1 0 1 1 -1 0 -1
as with the content function the multicharge can be used to return the tableau containing the residues of the cells:
sage: multicharge=[ IntegerModRing(3)(c) for c in [0,0,1] ] sage: PartitionTuple([[2,1],[2],[1,1,1]]).content_tableau(multicharge).pp() 0 1 0 1 1 2 0 2
- corners()#
Return a list of the removable cells of this partition tuple.
All indices are of the form
(k, r, c)
, wherer
is the row-index,c
is the column index andk
is the component.EXAMPLES:
sage: PartitionTuple([[1,1],[2],[2,1]]).removable_cells() [(0, 1, 0), (1, 0, 1), (2, 0, 1), (2, 1, 0)] sage: PartitionTuple([[1,1],[4,3],[2,1,1]]).removable_cells() [(0, 1, 0), (1, 0, 3), (1, 1, 2), (2, 0, 1), (2, 2, 0)]
- defect(e, multicharge)#
Return the
e
-defect or thee
-weightself
.The \(e\)-defect is the number of (connected) \(e\)-rim hooks that can be removed from the partition.
The defect of a partition tuple is given by
\[\text{defect}(\beta) = (\Lambda, \beta) - \tfrac12(\beta, \beta),\]where \(\Lambda = \sum_r \Lambda_{\kappa_r}\) for the multicharge \((\kappa_1, \ldots, \kappa_{\ell})\) and \(\beta = \sum_{(r,c)} \alpha_{(c-r) \pmod e}\), with the sum being over the cells in the partition.
INPUT:
e
– the quantum characteristicmulticharge
– the multicharge (default \((0,)\))
OUTPUT:
a non-negative integer, which is the defect of the block containing the partition tuple
self
EXAMPLES:
sage: PartitionTuple([[2,2],[2,2]]).defect(0,(0,0)) 0 sage: PartitionTuple([[2,2],[2,2]]).defect(2,(0,0)) 8 sage: PartitionTuple([[2,2],[2,2]]).defect(2,(0,1)) 8 sage: PartitionTuple([[2,2],[2,2]]).defect(3,(0,2)) 5 sage: PartitionTuple([[2,2],[2,2]]).defect(3,(0,2)) 5 sage: PartitionTuple([[2,2],[2,2]]).defect(3,(3,2)) 2 sage: PartitionTuple([[2,2],[2,2]]).defect(4,(0,0)) 0
- degree(e)#
Return the
e
-th degree ofself
.The \(e\)-th degree is the sum of the degrees of the standard tableaux of shape \(\lambda\). The \(e\)-th degree is the exponent of \(\Phi_e(q)\) in the Gram determinant of the Specht module for a semisimple cyclotomic Hecke algebra of type \(A\) with parameter \(q\).
For this calculation the multicharge \((\kappa_1, \ldots, \kappa_l)\) is chosen so that \(\kappa_{r+1} - \kappa_r > n\), where \(n\) is the
size()
of \(\lambda\) as this ensures that the Hecke algebra is semisimple.INPUT:
e
– an integer \(e > 1\)
OUTPUT:
A non-negative integer.
EXAMPLES:
sage: PartitionTuple([[2,1],[2,2]]).degree(2) 532 sage: PartitionTuple([[2,1],[2,2]]).degree(3) 259 sage: PartitionTuple([[2,1],[2,2]]).degree(4) 196 sage: PartitionTuple([[2,1],[2,2]]).degree(5) 105 sage: PartitionTuple([[2,1],[2,2]]).degree(6) 105 sage: PartitionTuple([[2,1],[2,2]]).degree(7) 0
Therefore, the Gram determinant of \(S(2,1|2,2)\) when the Hecke parameter \(q\) is “generic” is
\[q^N \Phi_2(q)^{532}\Phi_3(q)^{259}\Phi_4(q)^{196}\Phi_5(q)^{105}\Phi_6(q)^{105}\]for some integer \(N\). Compare with
prime_degree()
.
- diagram()#
Return a string for the Ferrers diagram of
self
.EXAMPLES:
sage: print(PartitionTuple([[2,1],[3,2],[1,1,1]]).diagram()) ** *** * * ** * * sage: print(PartitionTuple([[3,2],[2,1],[],[1,1,1,1]]).diagram()) *** ** - * ** * * * * sage: PartitionTuples.options(convention="french") sage: print(PartitionTuple([[3,2],[2,1],[],[1,1,1,1]]).diagram()) * * ** * * *** ** - * sage: PartitionTuples.options._reset()
- dominates(mu)#
Return
True
if the PartitionTuple dominates or equals \(\mu\) andFalse
otherwise.Given partition tuples \(\mu=(\mu^{(1)},...,\mu^{(m)})\) and \(\nu=(\nu^{(1)},...,\nu^{(n)})\) then \(\mu\) dominates \(\nu\) if
\[\sum_{k=1}^{l-1} |\mu^{(k)}| +\sum_{r \geq 1} \mu^{(l)}_r \geq \sum_{k=1}^{l-1} |\nu^{(k)}| + \sum_{r \geq 1} \nu^{(l)}_r\]EXAMPLES:
sage: mu=PartitionTuple([[1,1],[2],[2,1]]) sage: nu=PartitionTuple([[1,1],[1,1],[2,1]]) sage: mu.dominates(mu) True sage: mu.dominates(nu) True sage: nu.dominates(mu) False sage: tau=PartitionTuple([[],[2,1],[]]) sage: tau.dominates([[2,1],[],[]]) False sage: tau.dominates([[],[],[2,1]]) True
- down()#
Generator (iterator) for the partition tuples that are obtained from
self
by removing a cell.EXAMPLES:
sage: [mu for mu in PartitionTuple([[],[3,1],[1,1]]).down()] [([], [2, 1], [1, 1]), ([], [3], [1, 1]), ([], [3, 1], [1])] sage: [mu for mu in PartitionTuple([[],[],[]]).down()] []
- down_list()#
Return a list of the partition tuples that can be formed from
self
by removing a cell.EXAMPLES:
sage: PartitionTuple([[],[3,1],[1,1]]).down_list() [([], [2, 1], [1, 1]), ([], [3], [1, 1]), ([], [3, 1], [1])] sage: PartitionTuple([[],[],[]]).down_list() []
- ferrers_diagram()#
Return a string for the Ferrers diagram of
self
.EXAMPLES:
sage: print(PartitionTuple([[2,1],[3,2],[1,1,1]]).diagram()) ** *** * * ** * * sage: print(PartitionTuple([[3,2],[2,1],[],[1,1,1,1]]).diagram()) *** ** - * ** * * * * sage: PartitionTuples.options(convention="french") sage: print(PartitionTuple([[3,2],[2,1],[],[1,1,1,1]]).diagram()) * * ** * * *** ** - * sage: PartitionTuples.options._reset()
- garnir_tableau(*cell)#
Return the Garnir tableau of shape
self
corresponding to the cellcell
.If
cell
\(= (k,a,c)\) then \((k,a+1,c)\) must belong to the diagram of thePartitionTuple
. If this is not the case then we returnFalse
.Note
The function also sets
g._garnir_cell
equal tocell
which is used by some other functions.The Garnir tableaux play an important role in integral and non-semisimple representation theory because they determine the “straightening” rules for the Specht modules over an arbitrary ring.
The Garnir tableau are the “first” non-standard tableaux which arise when you act by simple transpositions. If \((k,a,c)\) is a cell in the Young diagram of a partition, which is not at the bottom of its column, then the corresponding Garnir tableau has the integers \(1, 2, \ldots, n\) entered in order from left to right along the rows of the diagram up to the cell \((k,a,c-1)\), then along the cells \((k,a+1,1)\) to \((k,a+1,c)\), then \((k,a,c)\) until the end of row \(a\) and then continuing from left to right in the remaining positions. The examples below probably make this clearer!
EXAMPLES:
sage: PartitionTuple([[5,3],[2,2],[4,3]]).garnir_tableau((0,0,2)).pp() 1 2 6 7 8 9 10 13 14 15 16 3 4 5 11 12 17 18 19 sage: PartitionTuple([[5,3,3],[2,2],[4,3]]).garnir_tableau((0,0,2)).pp() 1 2 6 7 8 12 13 16 17 18 19 3 4 5 14 15 20 21 22 9 10 11 sage: PartitionTuple([[5,3,3],[2,2],[4,3]]).garnir_tableau((0,1,2)).pp() 1 2 3 4 5 12 13 16 17 18 19 6 7 11 14 15 20 21 22 8 9 10 sage: PartitionTuple([[5,3,3],[2,2],[4,3]]).garnir_tableau((1,0,0)).pp() 1 2 3 4 5 13 14 16 17 18 19 6 7 8 12 15 20 21 22 9 10 11 sage: PartitionTuple([[5,3,3],[2,2],[4,3]]).garnir_tableau((1,0,1)).pp() 1 2 3 4 5 12 15 16 17 18 19 6 7 8 13 14 20 21 22 9 10 11 sage: PartitionTuple([[5,3,3],[2,2],[4,3]]).garnir_tableau((2,0,1)).pp() 1 2 3 4 5 12 13 16 19 20 21 6 7 8 14 15 17 18 22 9 10 11 sage: PartitionTuple([[5,3,3],[2,2],[4,3]]).garnir_tableau((2,1,1)).pp() Traceback (most recent call last): ... ValueError: (comp, row+1, col) must be inside the diagram
See also
- hook_length(k, r, c)#
Return the length of the hook of cell
(k, r, c)
in the partition.The hook of cell
(k, r, c)
is defined as the cells to the right or below (in the English convention). If your coordinates are in the form(k,r,c)
, use Python’s *-operator.EXAMPLES:
sage: mu=PartitionTuple([[1,1],[2],[2,1]]) sage: [ mu.hook_length(*c) for c in mu.cells()] [2, 1, 2, 1, 3, 1, 1]
- initial_column_tableau()#
Return the initial column tableau of shape
self
.The initial column tableau of shape \(\lambda\) is the standard tableau that has the numbers \(1\) to \(n\), where \(n\) is the
size()
of \(\lambda\), entered in order from top to bottom, and then left to right, down the columns of each component, starting from the rightmost component and working to the left.EXAMPLES:
sage: PartitionTuple([ [3,1],[3,2] ]).initial_column_tableau() ([[6, 8, 9], [7]], [[1, 3, 5], [2, 4]])
- initial_tableau()#
Return the
StandardTableauTuple
which has the numbers \(1, 2, \ldots, n\), where \(n\) is thesize()
ofself
, entered in order from left to right along the rows of each component, where the components are ordered from left to right.EXAMPLES:
sage: PartitionTuple([ [2,1],[3,2] ]).initial_tableau() ([[1, 2], [3]], [[4, 5, 6], [7, 8]])
- leg_length(k, r, c)#
Return the length of the leg of cell
(k, r, c)
inself
.INPUT:
k
– The componentr
– The rowc
– The cell
OUTPUT:
The leg length as an integer
The leg of cell
(k, r, c)
is the number of cells in thek
-th component which are below the node in rowr
and columnc
.EXAMPLES:
sage: PartitionTuple([[],[2,1],[2,2,1],[3]]).leg_length(2,0,0) 2 sage: PartitionTuple([[],[2,1],[2,2,1],[3]]).leg_length(2,0,1) 1 sage: PartitionTuple([[],[2,1],[2,2,1],[3]]).leg_length(2,2,0) 0
- level()#
Return the level of this partition tuple.
The level is the length of the tuple.
EXAMPLES:
sage: PartitionTuple([[2,1,1,0],[2,1]]).level() 2 sage: PartitionTuple([[],[],[2,1,1]]).level() 3
- outside_corners()#
Return a list of the removable cells of this partition tuple.
All indices are of the form
(k, r, c)
, wherer
is the row-index,c
is the column index andk
is the component.EXAMPLES:
sage: PartitionTuple([[1,1],[2],[2,1]]).addable_cells() [(0, 0, 1), (0, 2, 0), (1, 0, 2), (1, 1, 0), (2, 0, 2), (2, 1, 1), (2, 2, 0)] sage: PartitionTuple([[1,1],[4,3],[2,1,1]]).addable_cells() [(0, 0, 1), (0, 2, 0), (1, 0, 4), (1, 1, 3), (1, 2, 0), (2, 0, 2), (2, 1, 1), (2, 3, 0)]
- pp()#
Pretty prints this partition tuple. See
diagram()
.EXAMPLES:
sage: PartitionTuple([[5,5,2,1],[3,2]]).pp() ***** *** ***** ** ** *
- prime_degree(p)#
Return the
p
-th prime degree ofself
.The degree of a partition \(\lambda\) is the sum of the \(e\)-degrees` of the standard tableaux of shape \(\lambda\) (see
degree()
), for \(e\) a power of the prime \(p\). The prime degree gives the exponent of \(p\) in the Gram determinant of the integral Specht module of the symmetric group.The \(p\)-th degree is the sum of the degrees of the standard tableaux of shape \(\lambda\). The \(p\)-th degree is the exponent of \(p\) in the Gram determinant of a semisimple cyclotomic Hecke algebra of type \(A\) with parameter \(q = 1\).
As with
degree()
, for this calculation the multicharge \((\kappa_1, \ldots, \kappa_l)\) is chosen so that \(\kappa_{r+1} - \kappa_r > n\), where \(n\) is thesize()
of \(\lambda\) as this ensures that the Hecke algebra is semisimple.INPUT:
e
– an integer \(e > 1\)multicharge
– an \(l\)-tuple of integers, where \(l\) is thelevel()
ofself
OUTPUT:
A non-negative integer
EXAMPLES:
sage: PartitionTuple([[2,1],[2,2]]).prime_degree(2) 728 sage: PartitionTuple([[2,1],[2,2]]).prime_degree(3) 259 sage: PartitionTuple([[2,1],[2,2]]).prime_degree(5) 105 sage: PartitionTuple([[2,1],[2,2]]).prime_degree(7) 0
Therefore, the Gram determinant of \(S(2,1|2,2)\) when \(q=1\) is \(2^{728} 3^{259}5^{105}\). Compare with
degree()
.
- removable_cells()#
Return a list of the removable cells of this partition tuple.
All indices are of the form
(k, r, c)
, wherer
is the row-index,c
is the column index andk
is the component.EXAMPLES:
sage: PartitionTuple([[1,1],[2],[2,1]]).removable_cells() [(0, 1, 0), (1, 0, 1), (2, 0, 1), (2, 1, 0)] sage: PartitionTuple([[1,1],[4,3],[2,1,1]]).removable_cells() [(0, 1, 0), (1, 0, 3), (1, 1, 2), (2, 0, 1), (2, 2, 0)]
- remove_cell(k, r, c)#
Return the partition tuple obtained by removing a cell in row
r
, columnc
, and componentk
.This does not change
self
.EXAMPLES:
sage: PartitionTuple([[1,1],[4,3],[2,1,1]]).remove_cell(0,1,0) ([1], [4, 3], [2, 1, 1])
- row_standard_tableaux()#
Return the
row standard tableau tuples
of shapeself
.EXAMPLES:
sage: PartitionTuple([[],[3,2,2,1],[2,2,1],[3]]).row_standard_tableaux() Row standard tableau tuples of shape ([], [3, 2, 2, 1], [2, 2, 1], [3])
- size()#
Return the size of a partition tuple.
EXAMPLES:
sage: PartitionTuple([[2,1],[],[2,2]]).size() 7 sage: PartitionTuple([[],[],[1],[3,2,1]]).size() 7
- standard_tableaux()#
Return the
standard tableau tuples
of shapeself
.EXAMPLES:
sage: PartitionTuple([[],[3,2,2,1],[2,2,1],[3]]).standard_tableaux() Standard tableau tuples of shape ([], [3, 2, 2, 1], [2, 2, 1], [3])
- to_exp(k=0)#
Return a tuple of the multiplicities of the parts of a partition.
Use the optional parameter
k
to get a return list of length at leastk
.EXAMPLES:
sage: PartitionTuple([[1,1],[2],[2,1]]).to_exp() ([2], [0, 1], [1, 1]) sage: PartitionTuple([[1,1],[2,2,2,2],[2,1]]).to_exp() ([2], [0, 4], [1, 1])
- to_list()#
Return
self
as a list of lists.EXAMPLES:
sage: PartitionTuple([[1,1],[4,3],[2,1,1]]).to_list() [[1, 1], [4, 3], [2, 1, 1]]
- top_garnir_tableau(e, cell)#
Return the most dominant standard tableau which dominates the corresponding Garnir tableau and has the same residue that has shape
self
and is determined bye
andcell
.The Garnir tableau play an important role in integral and non-semisimple representation theory because they determine the “straightening” rules for the Specht modules over an arbitrary ring. The top Garnir tableaux arise in the graded representation theory of the symmetric groups and higher level Hecke algebras. They were introduced in [KMR2012].
If the Garnir node is
cell=(k,r,c)
and \(m\) and \(M\) are the entries in the cells(k,r,c)
and(k,r+1,c)
, respectively, in the initial tableau then the tope
-Garnir tableau is obtained by inserting the numbers \(m, m+1, \ldots, M\) in order from left to right first in the cells in rowr+1
which are not in thee
-Garnir belt, then in the cell in rowsr
andr+1
which are in the Garnir belt and then, finally, in the remaining cells in rowr
which are not in the Garnir belt. All other entries in the tableau remain unchanged.If
e = 0
, or if there are noe
-bricks in either rowr
orr+1
, then the top Garnir tableau is the corresponding Garnir tableau.EXAMPLES:
sage: PartitionTuple([[3,3,2],[5,4,3,2]]).top_garnir_tableau(2,(1,0,2)).pp() 1 2 3 9 10 12 13 16 4 5 6 11 14 15 17 7 8 18 19 20 21 22 sage: PartitionTuple([[3,3,2],[5,4,3,2]]).top_garnir_tableau(2,(1,0,1)).pp() 1 2 3 9 10 11 12 13 4 5 6 14 15 16 17 7 8 18 19 20 21 22 sage: PartitionTuple([[3,3,2],[5,4,3,2]]).top_garnir_tableau(3,(1,0,1)).pp() 1 2 3 9 12 13 14 15 4 5 6 10 11 16 17 7 8 18 19 20 21 22 sage: PartitionTuple([[3,3,2],[5,4,3,2]]).top_garnir_tableau(3,(3,0,1)).pp() Traceback (most recent call last): ... ValueError: (comp, row+1, col) must be inside the diagram
See also
garnir_tableau()
- up()#
Generator (iterator) for the partition tuples that are obtained from
self
by adding a cell.EXAMPLES:
sage: [mu for mu in PartitionTuple([[],[3,1],[1,1]]).up()] [([1], [3, 1], [1, 1]), ([], [4, 1], [1, 1]), ([], [3, 2], [1, 1]), ([], [3, 1, 1], [1, 1]), ([], [3, 1], [2, 1]), ([], [3, 1], [1, 1, 1])] sage: [mu for mu in PartitionTuple([[],[],[],[]]).up()] [([1], [], [], []), ([], [1], [], []), ([], [], [1], []), ([], [], [], [1])]
- up_list()#
Return a list of the partition tuples that can be formed from
self
by adding a cell.EXAMPLES:
sage: PartitionTuple([[],[3,1],[1,1]]).up_list() [([1], [3, 1], [1, 1]), ([], [4, 1], [1, 1]), ([], [3, 2], [1, 1]), ([], [3, 1, 1], [1, 1]), ([], [3, 1], [2, 1]), ([], [3, 1], [1, 1, 1])] sage: PartitionTuple([[],[],[],[]]).up_list() [([1], [], [], []), ([], [1], [], []), ([], [], [1], []), ([], [], [], [1])]
- young_subgroup()#
Return the corresponding Young, or parabolic, subgroup of the symmetric group.
EXAMPLES:
sage: PartitionTuple([[2,1],[4,2],[1]]).young_subgroup() Permutation Group with generators [(), (8,9), (6,7), (5,6), (4,5), (1,2)]
- young_subgroup_generators()#
Return an indexing set for the generators of the corresponding Young subgroup.
EXAMPLES:
sage: PartitionTuple([[2,1],[4,2],[1]]).young_subgroup_generators() [1, 4, 5, 6, 8]
- class sage.combinat.partition_tuple.PartitionTuples#
Bases:
UniqueRepresentation
,Parent
Class of all partition tuples.
For more information about partition tuples, see
PartitionTuple
.This is a factory class which returns the appropriate parent based on the values of
level
,size
, andregular
INPUT:
level
– the length of the tuplesize
– the total number of cellsregular
– a positive integer or a tuple of non-negative integers; if an integer, the highest multiplicity an entry may have in a component plus \(1\)
If a level \(k\) is specified and
regular
is a tuple of integers \(\ell_1, \ldots, \ell_k\), then this specifies partition tuples \(\mu\) such that \(\mu_i\) is \(\ell_i\)-regular, where \(0\) here represents \(\infty\)-regular partitions (equivalently, partitions without restrictions). Ifregular
is an integer \(\ell\), then we set \(\ell_i = \ell\) for all \(i\).- Element#
alias of
PartitionTuple
- level()#
Return the level or
None
if it is not defined.EXAMPLES:
sage: PartitionTuples().level() is None True sage: PartitionTuples(7).level() 7
- options = Current options for Partitions - convention: English - diagram_str: * - display: list - latex: young_diagram - latex_diagram_str: \ast#
- size()#
Return the size or
None
if it is not defined.EXAMPLES:
sage: PartitionTuples().size() is None True sage: PartitionTuples(size=7).size() 7
- class sage.combinat.partition_tuple.PartitionTuples_all#
Bases:
PartitionTuples
Class of partition tuples of a arbitrary level and arbitrary sum.
- class sage.combinat.partition_tuple.PartitionTuples_level(level, category=None)#
Bases:
PartitionTuples
Class of partition tuples of a fixed level, but summing to an arbitrary integer.
- class sage.combinat.partition_tuple.PartitionTuples_level_size(level, size)#
Bases:
PartitionTuples
Class of partition tuples with a fixed level and a fixed size.
- class sage.combinat.partition_tuple.PartitionTuples_size(size)#
Bases:
PartitionTuples
Class of partition tuples of a fixed size, but arbitrary level.
- class sage.combinat.partition_tuple.RegularPartitionTuples(regular, **kwds)#
Bases:
PartitionTuples
Abstract base class for \(\ell\)-regular partition tuples.
- class sage.combinat.partition_tuple.RegularPartitionTuples_all(regular)#
Bases:
RegularPartitionTuples
Class of \(\ell\)-regular partition tuples.
- class sage.combinat.partition_tuple.RegularPartitionTuples_level(level, regular)#
Bases:
PartitionTuples_level
Regular Partition tuples of a fixed level.
INPUT:
level
– a non-negative Integer; the levelregular
– a positive integer or a tuple of non-negative integers; if an integer, the highest multiplicity an entry may have in a component plus \(1\) with \(0\) representing \(\infty\)-regular (equivalently, partitions without restrictions)
regular
is a tuple of integers \((\ell_1, \ldots, \ell_k)\) that specifies partition tuples \(\mu\) such that \(\mu_i\) is \(\ell_i\)-regular. Ifregular
is an integer \(\ell\), then we set \(\ell_i = \ell\) for all \(i\).EXAMPLES:
sage: RPT = PartitionTuples(level=4, regular=(2,3,0,2)) sage: RPT[:24] [([], [], [], []), ([1], [], [], []), ([], [1], [], []), ([], [], [1], []), ([], [], [], [1]), ([2], [], [], []), ([1], [1], [], []), ([1], [], [1], []), ([1], [], [], [1]), ([], [2], [], []), ([], [1, 1], [], []), ([], [1], [1], []), ([], [1], [], [1]), ([], [], [2], []), ([], [], [1, 1], []), ([], [], [1], [1]), ([], [], [], [2]), ([3], [], [], []), ([2, 1], [], [], []), ([2], [1], [], []), ([2], [], [1], []), ([2], [], [], [1]), ([1], [2], [], []), ([1], [1, 1], [], [])] sage: [[1,1],[3],[5,5,5],[7,2]] in RPT False sage: [[3,1],[3],[5,5,5],[7,2]] in RPT True sage: [[3,1],[3],[5,5,5]] in RPT False
- class sage.combinat.partition_tuple.RegularPartitionTuples_level_size(level, size, regular)#
Bases:
PartitionTuples_level_size
Class of \(\ell\)-regular partition tuples with a fixed level and a fixed size.
INPUT:
level
– a non-negative Integer; the levelsize
– a non-negative Integer; the sizeregular
– a positive integer or a tuple of non-negative integers; if an integer, the highest multiplicity an entry may have in a component plus \(1\) with \(0\) representing \(\infty\)-regular (equivalently, partitions without restrictions)
regular
is a tuple of integers \((\ell_1, \ldots, \ell_k)\) that specifies partition tuples \(\mu\) such that \(\mu_i\) is \(\ell_i\)-regular. Ifregular
is an integer \(\ell\), then we set \(\ell_i = \ell\) for all \(i\).EXAMPLES:
sage: PartitionTuples(level=3, size=7, regular=(2,1,3))[0:24] [([7], [], []), ([6, 1], [], []), ([5, 2], [], []), ([4, 3], [], []), ([4, 2, 1], [], []), ([6], [], [1]), ([5, 1], [], [1]), ([4, 2], [], [1]), ([3, 2, 1], [], [1]), ([5], [], [2]), ([5], [], [1, 1]), ([4, 1], [], [2]), ([4, 1], [], [1, 1]), ([3, 2], [], [2]), ([3, 2], [], [1, 1]), ([4], [], [3]), ([4], [], [2, 1]), ([3, 1], [], [3]), ([3, 1], [], [2, 1]), ([3], [], [4]), ([3], [], [3, 1]), ([3], [], [2, 2]), ([3], [], [2, 1, 1]), ([2, 1], [], [4])]
- class sage.combinat.partition_tuple.RegularPartitionTuples_size(size, regular)#
Bases:
RegularPartitionTuples
Class of \(\ell\)-regular partition tuples with a fixed size.