So we’ve described the Specht modules, and we’ve shown that they give us a complete set of irreducible representations for the symmetric groups. But we haven’t described them very explicitly, and we certainl can’t say much about them. There’s still work to be done.
We say that a Young tableau 
is “standard” if its rows and columns are all increasing sequences. In this case, we also say that the Young tabloid 
and the polytabloid 
are standard.
Recall that we had a canonical Young tableau for each shape 
that listed the numbers from 
to 
in each row from top to bottom, as in
It should be clear that this canonical tableau is standard, so there is always at least one standard tableau for each shape. There may be more, of course. For example:
Clearly, any two distinct standard tableaux 
and 
give rise to distinct tabloids 
and . Indeed, if 
, then 
and would have to be row-equivalent. But only one Young tableau in any row-equivalence class has increasing rows, and only that one even has a chance to be standard. Thus if and are row-equivalent standard tableaux, they must be equal.
What’s not immediately clear is that the standard polytabloids 
and 
are distinct. Further, it turns out that the collection of standard polytabloids of shape is actually independent, and furnishes a basis for the Specht module 
. This is our next major goal.
January 5, 2011
Posted by John Armstrong |
Algebra, Representation Theory, Representations of Symmetric Groups |
10 Comments
