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.
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.