Strong Marked Tableau

Sage has have a builtin StrongTableau object. This module contains useful functions pertaining to Standard Marked Tableau (SMT), the most standard form of a StrongTableau (a strong tableau with standard weight).

AUTHORS:

• George Seelinger (2018): Initial version
• Matthew Lancellotti (2018): End core to strong marked tableaux

REFERENCES:

strong_marked_tableau.end_core_to_marked_core_sequences(end_core, k, row_markings)

Return the set of core sequences marked by row_markings ending in end_core.

INPUTS:

• end_core – a \(k+1\)-core
• k – All of the cores in the core sequences are \(k+1\)-cores.
• row_markings – vector of row-indices indicating the rows of the markings for each core sequence. (Note that “markings” are row-col-coordinates, while “row_markings” are merely row-coordinates.)

OUTPUTS:

• A set of all possible core sequences that end in end_core and can be marked by the row marking vector row_markings.

EXAMPLES:

sage: end_core_to_marked_core_sequences([5, 3, 1], 2, [0, 1, 2, 0, 1])
{([], [1], [1, 1], [2, 1, 1], [3, 1, 1], [5, 3, 1])}

sage: end_core_to_marked_core_sequences([5, 3, 1], 2, [1])
{([3, 1, 1], [5, 3, 1]), ([4, 2], [5, 3, 1])}

See also

end_core_to_strong_marked_tableaux()

strong_marked_tableau.end_core_to_strong_marked_tableaux(end_core, k, row_markings)

Return the set of strong marked tableaux marked by row_markings ending in end_core.

INPUTS:

• end_core – a \(k+1\)-core
• k – All of the cores in the core sequences are \(k+1\)-cores.
• row_markings – vector of row-indices indicating the rows of the markings for each core sequence. (Note that “markings” are row-col-coordinates, while “row_markings” are merely row-coordinates.)

OUTPUTS:

• A set of all possible strong marked tableau that end in end_core and can be marked by the row marking vector row_markings.

EXAMPLES:

sage: end_core_to_strong_marked_tableaux([5, 3, 1], 2, [0, 1, 2, 0, 1])
{[[-1, 3, -4, 5, 5], [-2, 5, -5], [-3]]}

sage: end_core_to_strong_marked_tableaux([5, 3, 1], 2, [1])
{[[None, None, None, 1, 1], [None, 1, -1], [None]],
 [[None, None, None, None, 1], [None, None, -1], [1]]}

See also

end_core_to_marked_core_sequences()

strong_marked_tableau.is_row_markable(outer_core, inner_core, row_marking)

Given two cores (typically consecutive cores in a core sequence), see if row_marking is a possible row_marking of outer_core / inner_core.

EXAMPLES:

Consider the skew shape [3, 2, 2] / [2, 1] depicted

It has a marking in row 0 and row 1, but not in row 2:

sage: is_row_markable([3, 2, 2], [2, 1], 0)
True
sage: is_row_markable([3, 2, 2], [2, 1], 1)
True
sage: is_row_markable([3, 2, 2], [2, 1], 2)
False

See also

strong_tableau_has_row_marking()

strong_marked_tableau.k_coverees(core, k, algorithm=1)

Given a \(k+1\)-core, find all sub-\(k+1\)-cores that have \(k\)-boundary 1 less than the given.

strong_marked_tableau.k_marked_coverees(core, k, row_marking)

Given a k+1-core, find all sub-k+1-cores that have k-boundary 1 less than the given with the given row_marking.

INPUTS:

• core – a \(k+1\)-core
• k – an integer
• row_marking – The desired row marking for the covers

OUTPUTS:

The set of all \(k+1\)-cores \(\lambda\) that satify:

• core is a cover of \(\lambda\)
• The skew shape core / \(\lambda\) has a marking in the row indexed row_marking.

EXAMPLES:

sage: k_marked_coverees([6, 4, 2, 2, 1], 5, 0)
{[5, 4, 2, 2, 1]}
sage: k_marked_coverees([6, 4, 2, 2, 1], 5, 1)
{[6, 2, 2, 2, 1], [6, 3, 2, 2]}
sage: k_marked_coverees([6, 4, 2, 2, 1], 5, 2)
{}
sage: k_marked_coverees([6, 4, 2, 2, 1], 5, 3)
{[6, 4, 2, 1, 1]}
sage: k_marked_coverees([6, 4, 2, 2, 1], 5, 4)
{[6, 3, 2, 2]}

strong_marked_tableau.row_marking_to_marking(outer_core, inner_core, row_marking)

Convert a row marking to a marking.

Given the skew shape defined by outer_core and inner_core, and the row index row_marking where a marking exists, return the marking \((\text{marking row index}, \text{marking column index})\).

Note that \(\text{marking row index}\) mentioned above is simply row_marking. Therefore, the real usefulness of this function is that it finds the column index of the marking.

EXAMPLES:

The skew shape [3, 2, 2] / [2, 1] depicted by

has the marking \((0, 2)\) in row \(0\). Hence:

sage: row_marking_to_marking([3, 2, 2], [2, 1], 0)
(0, 2)

It also has the marking \((1, 1)\) in row \(1\). Therefore:

sage: row_marking_to_marking([3, 2, 2], [2, 1], 1)
(1, 1)

It has no marking in row \(2\), so:

sage: row_marking_to_marking([3, 2, 2], [2, 1], 2)
Traceback (most recent call last):
...
ValueError: no such row marking

See also

row_markings_to_markings()

strong_marked_tableau.row_markings_to_markings(core_sequence, row_markings)

Given a core_sequence and corresponding row_markings for each cover of the sequence, convert the row markings to markings and return them.

Each row marking in row_markings is merely the row index of where the marking occurs. The purpose of this function is to convert each row marking to a “marking” which includes the column index. In order to understand this function, it is best to understand row_marking_to_marking() first.

EXAMPLES:

Consider the core sequence ([], [1], [1, 1], [2, 1, 1]) which corresponds to the sequence of skew shapes

Now consider the row markings 0, 1, and 2, respectively. Pair each skew shape up with its respective row marking, and then convert the row marking to a marking. Row marking 0 corresponds to marking \((0, 0)\), row marking 1 corresponds to marking \((1, 0)\), and row marking 2 corresponds to marking \((2, 0)\):

sage: row_markings_to_markings(([], [1], [1, 1], [2, 1, 1]), [0, 1, 2])
[(0, 0), (1, 0), (2, 0)]

Similarly:

sage: row_markings_to_markings(([], [1], [1, 1], [2, 1, 1]), [0, 1, 0])
[(0, 0), (1, 0), (0, 1)]

See also

row_marking_to_marking()

strong_marked_tableau.strong_tableau_has_row_marking(tab, row_index)

Returns whether tab has a row marking in row row_index.

Checks if a strong tableau tab has a row marking in row row_index (row indexing starts at 0) and returns True if row_marking is present; False otherwise.

See also

is_row_markable()