The Maximality of Standard Tableaux
Standard tableaux have a certain maximality property with respect to the dominance order on tabloids. Specifically, if 
is standard and 
appears as a summand in the polytabloid 
, then 
.
Any such comes from 
, where 
. We will make our induction on the number of “column inversions” in 
. That is, the number of pairs of entries 
that are in the same column of , but which are “out of order”, in the sense that 
is in a lower row than 
.
Given any such pair, the dominance lemma tells us that 
. That is, by “untwisting” the column inversion, we can move up the dominance order while preserving the columns. It should also be clear that 
has fewer column inversions than does. But if we undo all the column inversions, the tableau we’re left with must be standard. That is, it must be 
itself.
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January 13, 2011 -
Posted by John Armstrong |
Algebra, Representation Theory, Representations of Symmetric Groups
[...] 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 [...]
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