Sage: Ticket #24659: Make semistandard extension of growth diagrams generic.
https://trac.sagemath.org/ticket/24659
<p>
Growth diagrams as introduced by Fomin work for partial fillings of skew Ferrers shapes: in every row and every column at most one cross (possibly coloured, depending on the type of growth diagram) is allowed.
</p>
<p>
However, for Robinson-Schensted insertion, one can accommodate several entries in rows and columns, and arbitrary nonnegative integer entries instead of crosses as follows:
</p>
<ul><li>for each row with sum of entries equal to e, subdivide the row into e rows,
</li><li>same for the columns
</li><li>arrange crosses in the new grid according to one of the rules below, whenever they correspond to entries in the same row or column of the original growth diagram.
</li></ul><ul><li>Rule RSK ("Knuth"): arrange crosses increasing in rows and columns
</li><li>Rule RSK' ("Burge"): arrange crosses decreasing in rows and columns
</li></ul><p>
and for 0-1 fillings:
</p>
<ul><li>Rule RSK*: increasing in columns, decreasing in rows
</li><li>Rule RSK'*: increasing in rows, decreasing in columns
</li></ul><p>
This ticket aims at adding supporting code.
</p>
<p>
For shifted insertion, there is a different standardisation procedure. This is described in Section 8 of Sagan's "Shifted Tableaux, Schur Q-Functions, and a Conjecture of R. Stanley". A naive implementation to standardise words is attached.
</p>
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https://trac.sagemath.org/ticket/24659
Trac 1.1.6mantepseTue, 06 Feb 2018 09:55:05 GMTattachment set
https://trac.sagemath.org/ticket/24659
https://trac.sagemath.org/ticket/24659
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<li><strong>attachment</strong>
set to <em>shifted.sage</em>
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TicketmantepseTue, 06 Feb 2018 09:56:03 GMTdescription changed
https://trac.sagemath.org/ticket/24659#comment:1
https://trac.sagemath.org/ticket/24659#comment:1
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<li><strong>description</strong>
modified (<a href="/ticket/24659?action=diff&version=1">diff</a>)
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TicketmantepseTue, 06 Feb 2018 10:05:21 GMT
https://trac.sagemath.org/ticket/24659#comment:2
https://trac.sagemath.org/ticket/24659#comment:2
<p>
Anne, which version of shifted semistandard insertion do you need precisely? Bruce Sagan's is a bijection between matrices with nonnegative integer entries, possibly primed, and pairs <code>(T,U)</code> of generalized shifted tableaux of the same shape, such that <code>U</code> has no primed entries on the diagonal.
</p>
<p>
A generalized shifted tableau is a filling that is weakly increasing in rows and columns and primed entries are not repeated in rows and unprimed entries are not repeated in columns.
</p>
<p>
Thus, <code>T</code> is <strong>not</strong> a <code>ShiftedPrimedTableau</code>, because it can have primed entries on the diagonal.
</p>
<p>
I'm not sure how this is dealt with best. Subclassing?
</p>
TicketmantepseTue, 06 Feb 2018 13:18:47 GMTdescription changed
https://trac.sagemath.org/ticket/24659#comment:3
https://trac.sagemath.org/ticket/24659#comment:3
<ul>
<li><strong>description</strong>
modified (<a href="/ticket/24659?action=diff&version=3">diff</a>)
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TicketaschillingFri, 09 Feb 2018 01:49:57 GMT
https://trac.sagemath.org/ticket/24659#comment:4
https://trac.sagemath.org/ticket/24659#comment:4
<p>
Replying to <a class="ticket" href="https://trac.sagemath.org/ticket/24659#comment:2" title="Comment 2">mantepse</a>:
</p>
<blockquote class="citation">
<p>
Anne, which version of shifted semistandard insertion do you need precisely? Bruce Sagan's is a bijection between matrices with nonnegative integer entries, possibly primed, and pairs <code>(T,U)</code> of generalized shifted tableaux of the same shape, such that <code>U</code> has no primed entries on the diagonal.
</p>
</blockquote>
<p>
I am interested in the Haiman mixed insertion, see for example Section 3.2 in <a class="ext-link" href="https://arxiv.org/pdf/1704.00889.pdf"><span class="icon"></span>https://arxiv.org/pdf/1704.00889.pdf</a>.
</p>
TicketmantepseFri, 09 Feb 2018 06:07:47 GMT
https://trac.sagemath.org/ticket/24659#comment:5
https://trac.sagemath.org/ticket/24659#comment:5
<p>
OK, that's (I believe) the same as Sagan's extension for unprimed words.
</p>
<p>
With the attached file, you can in principle do this, except that I haven't implemented "destandardisation". In fact, I need some help there or do some thinking myself - the question is what is the precise analogue of the labels between the letters in <code>to_chain</code>.
</p>
<p>
I now know how I want to add this to the growth diagram framework, I hope to do it over the weekend.
</p>
Ticketgh-MareoRaftMon, 12 Mar 2018 16:23:01 GMT
https://trac.sagemath.org/ticket/24659#comment:6
https://trac.sagemath.org/ticket/24659#comment:6
<p>
Take note that the addition of the superRSK algorithm (<a class="closed ticket" href="https://trac.sagemath.org/ticket/24894" title="enhancement: add super RSK algorithm to sage (closed: fixed)">#24894</a>) is similar in many ways to Haiman mixed insertion, but not the same.
</p>
<p>
It seems that the tableau in Haiman mixed insertion in the paper linked above don't follow the standard tableau shape. Also, Haiman mixed insertion behaves differently on diagonals. superRSK does not treat diagonals in any special way.
</p>
<p>
Still, I'm confident that these algorithms can share code. For example, you can probably reuse my <code>_bump_row</code> and <code>_bump_col</code> functions.
</p>
Ticketgh-MareoRaftMon, 12 Mar 2018 21:05:26 GMT
https://trac.sagemath.org/ticket/24659#comment:7
https://trac.sagemath.org/ticket/24659#comment:7
<p>
In fact, in Rob Muth's paper (<a class="ext-link" href="https://arxiv.org/pdf/1711.00420.pdf"><span class="icon"></span>https://arxiv.org/pdf/1711.00420.pdf</a>), Remark 3.2, he says "If X = N, where < is the usual order on integers and every element is of even parity, then 0-insertion is Schensted insertion [S]. For general X and
standard tableaux, 0-insertion is mixed insertion as defined by Haiman (where odd
letters are referred to as circled) [H]."
</p>
<p>
He might be mistaken though, since again when I read your reference, it says something about diagonals.
</p>
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