The Unapologetic Mathematician

Mathematics for the interested outsider

The Maximality of Standard Tableaux

Standard tableaux have a certain maximality property with respect to the dominance order on tabloids. Specifically, if t is standard and \{s\} appears as a summand in the polytabloid e_t, then \{t\}\trianglerighteq\{s\}.

Any such \{s\} comes from s=\pi t, where \pi\in C_t. We will make our induction on the number of “column inversions” in s. That is, the number of pairs of entries k<l that are in the same column of s, but which are “out of order”, in the sense that k is in a lower row than l.

Given any such pair, the dominance lemma tells us that \{s\}\triangleleft(k\,l)\{s\}. That is, by “untwisting” the column inversion, we can move up the dominance order while preserving the columns. It should also be clear that (k\,l)\{s\} has fewer column inversions than \{s\} does. But if we undo all the column inversions, the tableau we’re left with must be standard. That is, it must be \{t\} itself.

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January 13, 2011 - Posted by | Algebra, Representation Theory, Representations of Symmetric Groups

2 Comments »

  1. […] we consider for our the collection of polytabloids where is a standard tableau. In this case, we know that is the maximum of all the tabloids showing up as summands in . And these standard tabloids […]

    Pingback by Standard Polytabloids are Independent « The Unapologetic Mathematician | January 13, 2011 | Reply

  2. […] Adres URL: The Maximality of Standard Tableaux […]

    Pingback by The Maximality of Standard Tableaux : : standardy | January 16, 2011 | Reply


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