#20587: Residues and degrees for tableaux
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Reporter: andrew.mathas | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.3
Component: combinatorics | Resolution:
Keywords: tableaux, rsidue | Merged in:
sequences, tableaux degrees | Reviewers:
Authors: Andrew Mathas | Work issues:
Report Upstream: N/A | Commit:
Branch: | 634765704f924f8e4a8c732298325f6ee0aac18d
u/andrew.mathas/residues_and_degrees_for_tableaux| Stopgaps:
Dependencies: |
-------------------------------------+-------------------------------------
Changes (by andrew.mathas):
* status: needs_info => needs_review
* dependencies: #20564 =>
Old description:
> Residue sequences of tableaux are an important statistic, particularly in
> the non-semisimple and graded representation theories. This ticket
> implements two classes `ResidueSequence` and its parent class
> `ResidueSequences` for these sequences, In addition, it implement the
> Brundan-Kleshchev-Wang degree function on tableaux and tableaux tuples.
> In practice, these are all implemented as methods on tableaux and
> tableaux tuples:
> {{{#!python
> sage:
> StandardTableau([[1,2,3],[4,5]]).residue_sequence(3).standard_tableaux()
> Standard tableaux of residue (0,1,2,2,0)
> Tableau([[1,2,5],[3,4]]).degree(3)
> 0
> sage:
> StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue_sequence(3,[0,0])
> 3-residue sequence (0,1,2,0,0) with multicharge (0,0)
> sage: StandardTableauTuple([[[2,8],[7]],[[1,4,6],[3,5]]]).degree(0,(0,1))
> 1
> }}}
> As an added bonus, we convert `TableauTuples` to use `ClonableArray`.
New description:
Residue sequences of tableaux are an important statistic, particularly in
the non-semisimple and graded representation theories. This ticket
implements two classes `ResidueSequence` and its parent class
`ResidueSequences` for these sequences, In addition, it implements the
Brundan-Kleshchev-Wang degree function on tableaux and tableaux tuples. In
practice, these are all implemented as methods on tableaux and tableaux
tuples:
{{{#!python
sage:
StandardTableau([[1,2,3],[4,5]]).residue_sequence(3).standard_tableaux()
Standard tableaux of residue (0,1,2,2,0)
Tableau([[1,2,5],[3,4]]).degree(3)
0
sage:
StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue_sequence(3,[0,0])
3-residue sequence (0,1,2,0,0) with multicharge (0,0)
sage: StandardTableauTuple([[[2,8],[7]],[[1,4,6],[3,5]]]).degree(0,(0,1))
1
}}}
As an added bonus, we convert `TableauTuples` to use `ClonableArray`.
**New classes**
- ResidueSequence: a residue sequence
- ResidueSequences: parent class for residue sequence
- StandardTableaux_residue: residue class standard tableaux (tuples) of
given residue
- StandardTableaux_residue_shape: residue class of given residue and shape
** New partition (tuple) methods**
- defect: the Lie theoretic defect of the partition considered as a weight
- degree: the exponent of the e-th cyclotomic polynomial in the Gram
determinant of the corresponding q-Specht module
- p_Degree: the exponent of p in the Gram determinant of the corresponding
Specht module
- standard_tableaux: returns the parent for the standard tableaux of this
shape
** New tableau (tuple) methods**
- codegree: The BKW codegree function
- content: returns the content of an integer in a tableau
- degree: The BKW degree function
- first_column_descent: first descent in tableau using row reading
- first_row_descent: first descent in tableau using column reading
- residue: returns the residue of an integer in a tableau
- residue_sequence: the residue sequence of the tableau
- reduced_row_word: lexicographically minimal reduced expression for
**Main residue sequence methods**
- block: returns an invariant that determines the block associated to a
residue sequence
- negative: returns the "negative" residue sequence, corresponding to
Mullineux conjugation
- restrict: returns a restriction of the residue sequence
- standard_tableaux: a generator for the standard tabelau with this
residue sequence
- swap_residues: returns the residue sequence obtained by swapping two
residues
--
--
Ticket URL: <http://trac.sagemath.org/ticket/20587#comment:8>
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