Littlewood-Richardson tableaux

A semistandard tableau is Littlewood-Richardson with respect to the sequence of partitions $({\mu }^{(1)},\dots ,{\mu }^{(k)})$ if, when restricted to each alphabet $\{|{\mu }^{(1)}|+\cdots +|{\mu }^{(i-1)}|+1,\dots ,|{\mu }^{(1)}|+\cdots +|{\mu }^{(i)}|-1\}$, is Yamanouchi.

AUTHORS:

• Maria Gillespie, Jake Levinson, Anne Schilling (2016): initial version

class sage.combinat.lr_tableau.LittlewoodRichardsonTableau(parent, t)

Bases: SemistandardTableau

A semistandard tableau is Littlewood-Richardson with respect to the sequence of partitions $({\mu }^{(1)},\dots ,{\mu }^{(k)})$ if, when restricted to each alphabet $\{|{\mu }^{(1)}|+\cdots +|{\mu }^{(i-1)}|+1,\dots ,|{\mu }^{(1)}|+\cdots +|{\mu }^{(i)}|-1\}$, is Yamanouchi.

INPUT:

• t – Littlewood-Richardson tableau; the input is supposed to be a list of lists specifying the rows of the tableau

EXAMPLES:

Sage

sage: from sage.combinat.lr_tableau import LittlewoodRichardsonTableau
sage: LittlewoodRichardsonTableau([[1,1,3],[2,3],[4]], [[2,1],[2,1]])
[[1, 1, 3], [2, 3], [4]]

Python

>>> from sage.all import *
>>> from sage.combinat.lr_tableau import LittlewoodRichardsonTableau
>>> LittlewoodRichardsonTableau([[Integer(1),Integer(1),Integer(3)],[Integer(2),Integer(3)],[Integer(4)]], [[Integer(2),Integer(1)],[Integer(2),Integer(1)]])
[[1, 1, 3], [2, 3], [4]]

Sage Live

from sage.combinat.lr_tableau import LittlewoodRichardsonTableau
LittlewoodRichardsonTableau([[1,1,3],[2,3],[4]], [[2,1],[2,1]])

check()

Check that self is a valid Littlewood-Richardson tableau.

EXAMPLES:

Sage

sage: from sage.combinat.lr_tableau import LittlewoodRichardsonTableau
sage: t = LittlewoodRichardsonTableau([[1,1,3],[2,3],[4]], [[2,1],[2,1]])
sage: t.check()

Python

>>> from sage.all import *
>>> from sage.combinat.lr_tableau import LittlewoodRichardsonTableau
>>> t = LittlewoodRichardsonTableau([[Integer(1),Integer(1),Integer(3)],[Integer(2),Integer(3)],[Integer(4)]], [[Integer(2),Integer(1)],[Integer(2),Integer(1)]])
>>> t.check()

Sage Live

from sage.combinat.lr_tableau import LittlewoodRichardsonTableau
t = LittlewoodRichardsonTableau([[1,1,3],[2,3],[4]], [[2,1],[2,1]])
t.check()

class sage.combinat.lr_tableau.LittlewoodRichardsonTableaux(shape, weight)

Bases: SemistandardTableaux

Littlewood-Richardson tableaux.

A semistandard tableau $t$ is Littlewood-Richardson with respect to the sequence of partitions $({\mu }^{(1)},\dots ,{\mu }^{(k)})$ (called the weight) if $t$ is Yamanouchi when restricted to each alphabet $\{|{\mu }^{(1)}|+\cdots +|{\mu }^{(i-1)}|+1,\dots ,|{\mu }^{(1)}|+\cdots +|{\mu }^{(i)}|-1\}$.

INPUT:

• shape – the shape of the Littlewood-Richardson tableaux

• weight – the weight is a sequence of partitions

EXAMPLES:

Sage

sage: LittlewoodRichardsonTableaux([3,2,1],[[2,1],[2,1]])
Littlewood-Richardson Tableaux of shape [3, 2, 1] and weight ([2, 1], [2, 1])

Python

>>> from sage.all import *
>>> LittlewoodRichardsonTableaux([Integer(3),Integer(2),Integer(1)],[[Integer(2),Integer(1)],[Integer(2),Integer(1)]])
Littlewood-Richardson Tableaux of shape [3, 2, 1] and weight ([2, 1], [2, 1])

Sage Live

LittlewoodRichardsonTableaux([3,2,1],[[2,1],[2,1]])

Element

alias of LittlewoodRichardsonTableau

sage.combinat.lr_tableau.is_littlewood_richardson(t, heights)

Return whether semistandard tableau t is Littleword-Richardson with respect to heights.

A tableau is Littlewood-Richardson with respect to heights given by $(h_1,h_2,\dots )$ if each subtableau with respect to the alphabets $\{1,2,\dots ,h_1\}$, $\{h_1+1,\dots ,h_1+h_2\}$, etc. is Yamanouchi.

EXAMPLES:

Sage

sage: from sage.combinat.lr_tableau import is_littlewood_richardson
sage: t = Tableau([[1,1,2,3,4],[2,3,3],[3]])
sage: is_littlewood_richardson(t,[2,2])
False
sage: t = Tableau([[1,1,3],[2,3],[4,4]])
sage: is_littlewood_richardson(t,[2,2])
True
sage: t = Tableau([[7],[8]])
sage: is_littlewood_richardson(t,[2,3,3])
False
sage: is_littlewood_richardson([[2],[3]],[3,3])
False

Python

>>> from sage.all import *
>>> from sage.combinat.lr_tableau import is_littlewood_richardson
>>> t = Tableau([[Integer(1),Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(2),Integer(3),Integer(3)],[Integer(3)]])
>>> is_littlewood_richardson(t,[Integer(2),Integer(2)])
False
>>> t = Tableau([[Integer(1),Integer(1),Integer(3)],[Integer(2),Integer(3)],[Integer(4),Integer(4)]])
>>> is_littlewood_richardson(t,[Integer(2),Integer(2)])
True
>>> t = Tableau([[Integer(7)],[Integer(8)]])
>>> is_littlewood_richardson(t,[Integer(2),Integer(3),Integer(3)])
False
>>> is_littlewood_richardson([[Integer(2)],[Integer(3)]],[Integer(3),Integer(3)])
False

Sage Live

from sage.combinat.lr_tableau import is_littlewood_richardson
t = Tableau([[1,1,2,3,4],[2,3,3],[3]])
is_littlewood_richardson(t,[2,2])
t = Tableau([[1,1,3],[2,3],[4,4]])
is_littlewood_richardson(t,[2,2])
t = Tableau([[7],[8]])
is_littlewood_richardson(t,[2,3,3])
is_littlewood_richardson([[2],[3]],[3,3])

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