Shifted primed tableaux

AUTHORS:

  • Kirill Paramonov (2017-08-18): initial implementation
sage.combinat.shifted_primed_tableau.CrystalElementShiftedPrimedTableau

Class for elements of crystals.ShiftedPrimedTableau.

class sage.combinat.shifted_primed_tableau.PrimedEntry(entry=None, double=None)

Bases: sage.structure.sage_object.SageObject

The class of entries in shifted primed tableaux.

An entry in a shifted primed tableau is an element in the alphabet \(\{1' < 1 < 2' < 2 < \cdots < n' < n\}\). The difference between two elements \(i\) and \(i-1\) counts as a whole unit, whereas the difference between \(i\) and \(i'\) counts as half a unit. Internally, we represent an unprimed element \(x\) as \(2x\) and the primed elements as the corresponding odd integer that respects the total order.

INPUT:

  • entry – a half integer or a string of an integer possibly ending in p or '
  • double – the doubled value
decrease_half()

Decrease self by half a unit.

decrease_one()

Decrease self by one unit.

increase_half()

Increase self by half a unit.

increase_one()

Increase self by one unit.

integer()

Return the corresponding integer \(i\) for primed entries of the form \(i\) or \(i'\).

is_primed()

Checks if self is a primed element.

is_unprimed()

Checks if self is an unprimed element.

primed()

Prime self if it is an unprimed element.

unprimed()

Unprime self if it is a primed element.

sage.combinat.shifted_primed_tableau.ShiftedPrimedTableau

A shifted primed tableau.

A primed tableau is a tableau of shifted shape in the alphabet \(X' = \{1' < 1 < 2' < 2 < \cdots < n' < n\}\) such that

  1. the entries are weakly increasing along rows and columns;
  2. a row cannot have two repeated primed elements, and a column cannot have two repeated non-primed elements;
  3. there are only non-primed elements on the main diagonal.

Skew shape of the shifted primed tableaux is specified either with an optional argument skew or with None entries.

EXAMPLES:

sage: T = ShiftedPrimedTableaux([4,2])
sage: T([[1,"2'","3'",3],[2,"3'"]])[1]
(2, 3')
sage: t = ShiftedPrimedTableau([[1,"2p",2.5,3],[2,2.5]])
sage: t[1]
(2, 3')
sage: ShiftedPrimedTableau([["2p",2,3],["2p","3p"],[2]], skew=[2,1])
[(None, None, 2', 2, 3), (None, 2', 3'), (2,)]
sage: ShiftedPrimedTableau([[None,None,"2p"],[None,"2p"]])
[(None, None, 2'), (None, 2')]
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux

Returns the combinatorial class of shifted primed tableaux subject to the constraints given by the arguments.

A primed tableau is a tableau of shifted shape on the alphabet \(X' = \{1' < 1 < 2' < 2 < \cdots < n' < n\}\) such that

  1. the entries are weakly increasing along rows and columns
  2. a row cannot have two repeated primed entries, and a column cannot have two repeated non-primed entries
  3. there are only non-primed entries along the main diagonal

INPUT:

Valid optional keywords:

  • shape – the (outer skew) shape of tableaux
  • weight – the weight of tableaux
  • max_entry – the maximum entry of tableaux
  • skew – the inner skew shape of tableaux

The weight of a tableau is defined to be the vector with \(i\)-th component equal to the number of entries \(i\) and \(i'\) in the tableau. The sum of the coordinates in the weight vector must be equal to the number of entries in the partition.

The shape and skew must be strictly decreasing partitions.

EXAMPLES:

sage: SPT = ShiftedPrimedTableaux(weight=(1,2,2), shape=[3,2]); SPT
Shifted Primed Tableaux of weight (1, 2, 2) and shape [3, 2]
sage: SPT.list()
[[(1, 2, 2), (3, 3)],
 [(1, 2', 3'), (2, 3)],
 [(1, 2', 3'), (2, 3')],
 [(1, 2', 2), (3, 3)]]
sage: SPT = ShiftedPrimedTableaux(weight=(1,2)); SPT
Shifted Primed Tableaux of weight (1, 2)
sage: list(SPT)
[[(1, 2, 2)], [(1, 2', 2)], [(1, 2'), (2,)]]
sage: SPT = ShiftedPrimedTableaux([3,2], max_entry = 2); SPT
Shifted Primed Tableaux of shape [3, 2] and maximum entry 2
sage: list(SPT)
[[(1, 1, 1), (2, 2)], [(1, 1, 2'), (2, 2)]]
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_all

The class of all shifted primed tableaux.

sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_shape

Shifted primed tableaux of a fixed shape.

Shifted primed tableaux admit a type \(A_n\) classical crystal structure with highest weights corresponding to a given shape.

The list of module generators consists of all elements of the crystal with nonincreasing weight entries.

The crystal is constructed following operations described in [HPS2017].

EXAMPLES:

sage: ShiftedPrimedTableaux([4,3,1], max_entry=4)
Shifted Primed Tableaux of shape [4, 3, 1] and maximum entry 4
sage: ShiftedPrimedTableaux([4,3,1], max_entry=4).cardinality()
384

We compute some of the crystal structure:

sage: SPTC = crystals.ShiftedPrimedTableaux([3,2], 3)
sage: T = SPTC.module_generators[-1]
sage: T
[(1, 1, 2'), (2, 3')]
sage: T.f(2)
[(1, 1, 3'), (2, 3')]
sage: len(SPTC.module_generators)
7
sage: SPTC[0]
[(1, 1, 1), (2, 2)]
sage: SPTC.cardinality()
24
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_weight

Shifted primed tableaux of fixed weight.

EXAMPLES:

sage: ShiftedPrimedTableaux(weight=(2,3,1))
Shifted Primed Tableaux of weight (2, 3, 1)
sage: ShiftedPrimedTableaux(weight=(2,3,1)).cardinality()
17
sage.combinat.shifted_primed_tableau.ShiftedPrimedTableaux_weight_shape

Shifted primed tableaux of the fixed weight and shape.

EXAMPLES:

sage: ShiftedPrimedTableaux([4,2,1], weight=(2,3,2))
Shifted Primed Tableaux of weight (2, 3, 2) and shape [4, 2, 1]
sage: ShiftedPrimedTableaux([4,2,1], weight=(2,3,2)).cardinality()
4