## Properties of Garnir Elements from Tableaux 1

Pick a Young tableau , and sets and as we did last time. If there are more entries in than there are in the th column of — the one containing — then . In particular, if we pick and by selecting a row descent, letting be the entries below the left entry, and letting be the entries above the right entry, then this situation will hold.

As a first step, I say that . That is, if we allow all the permutations of entries in these two sets (along with signs) then everything cancels out. Indeed, let be any column-stabilizing permutation. Our hypothesis on the number of entries in tells us that we must have some pair of and in the same row of . Thus the swap . The sign lemma then tells us that . Since this is true for every summand of , it is true for itself.

Now, our assertion is not that this is true for all of , but rather that it holds for our transversal . We use the decomposition

This gives us a factorization

And so we conclude that .

But now we note that . So if we use the sign lemma to conclude

Thus , and so

which can only happen if , as asserted.

This result will allow us to pick out a row descent in and write down a linear combination of polytabloids that lets us rewrite in terms of other polytabloids. And it will turn out that all the other polytabloids will be “more standard” than .

[…] for the last couple posts I’ve talked about using Garnir elements to rewrite nonstandard polytabloids — those […]

Pingback by The Column Dominance Order « The Unapologetic Mathematician | January 20, 2011 |

[…] must be a row descent — we’ve ruled out column descents already — and so we can pick our Garnir element to write as the sum of a bunch of other polytabloids , where in the column dominance order. But […]

Pingback by Standard Polytabloids Span Specht Modules « The Unapologetic Mathematician | January 21, 2011 |

There seems to be a typo at the end of the 1st sentence—shouldn’t it be there?

Comment by Dima | April 3, 2011 |

I’m not sure I understand your question.

Comment by John Armstrong | April 3, 2011 |

oops, I meant “at the end of the 2nd sentence”

Comment by Dima | April 3, 2011 |

oh, I dropped the “=0”, sorry.

Comment by John Armstrong | April 3, 2011 |