We want to “categorify” the relation we came up with last time:
That is, we want to replace these numbers with objects of a category, replace the sum with a direct sum, and replace the equation with a natural isomorphism.
It should be clear that an obvious choice for the objects is to replace with the Specht module , since we’ve seen that . But what category are they in? On the left side, is an -module, but on the right side all the are -modules. Our solution is to restrict , suggesting the isomorphism
This tells us what happens to any of the Specht modules as we restrict it to a smaller symmetric group. As a side note, it doesn’t really matter which we use, since they’re all conjugate to each other inside . So we’ll just use the one that permutes all the numbers but .
Anyway, say the inner corners of occur in the rows , and of course they must occur at the ends of these rows. For each one, we’ll write for the partition that comes from removing that inner corner. Similarly, if is a standard tableau with in the th row, we write for the (standard) tableau with removed. And the same goes for the standard tabloids and .
Our method will be to find a tower of subspaces
so that at each step we have as -modules. Then we can see that
And similarly we find , and step by step we go until we find the proposed isomorphism. The construction itself will be presented next time.
The next thing we need to take note of it the idea of an “inner corner” and an “outer corner” of a Ferrers diagram, and thus of a partition.
An inner corner of a Ferrers diagram is a cell that, if it’s removed, the rest of the diagram is still the Ferrers diagram of a partition. It must be the rightmost cell in its row, and the bottommost cell in its column. Similarly, an outer corner is one that, if it’s added to the diagram, the result is still the Ferrers diagram of a partition. This is a little more subtle: it must be just to the right of the end of a row, and just below the bottom of a column.
As an example, consider the partition , with Ferrers diagram
We highlight the inner corners by shrinking them, and mark the outer corners with circles:
That is, there are three ways we could remove a cell and still have the Ferrers diagram of a partition:
And there are four ways that we could add a cell and still have the Ferrers diagram of a partition:
If the first partition is , we write a generic partition that comes from removing a single inner corner by . Similarly, we write a generic partition that comes from adding a single outer corner by . In our case, if , then the three possible partitions are , , and , while the four possible partitions are , , , and .
Now, as a quick use of this concept, think about how to fill a Ferrers diagram to make a standard Young tableau. It should be clear that since is the largest entry in the tableau, it must be in the rightmost cell of its row and the bottommost cell of its column in order for the tableau to be standard. Thus must occur in an inner corner. This means that we can describe any standard tableau by picking which inner corner contains , removing that corner, and filling the rest with a standard tableau with entries. Thus, the number of standard -tableaux is the sum of all the standard -tableaux:
Now that we have a canonical basis for our Specht modules composed of standard polytabloids it gives us a matrix representation of for each . We really only need to come up with matrices for the swaps , for , since these generate the whole symmetric group.
When we calculate the action of the swap on a polytabloid associated with a standard Young tableau , there are three possibilities. Either and are in the same column of , they’re in the same row of , or they’re not in the same row or column of .
The first case is easy. If and are in the same column of , then , and thus .
The third case isn’t much harder, although it’s subtler. I say that if and share neither a row nor a column, then is again standard. Indeed, swapping the two can’t introduce either a row or a column descent. The entries to the left of and above are all less than , and none of them are , so they’re all less than as well. Similarly, all the entries to the right of and below are greater than , and none of them are , so they’re all greater than as well.
Where things get complicated is when and share a row. But then they have to be next to each other, and the swap introduces a row descent between them. We can then use our Garnir elements to write this polytabloid in terms of standard ones.
Let’s work this out explicitly for the Specht module , which should give us our well-known two-dimensional representation of . The basis consists of the polytabloids associated to these two tableaux:
We need to come up with matrices for the two swaps and . And the second one is easy: it just swaps these two tableaux! Thus we get the matrix
The action of on the second standard tableau is similarly easy. Since and are in the same column, the swap acts by multiplying by . Thus we can write down a column of the matrix
As for the action on the first tableau, the swap induces a row descent. We use a Garnir element to straighten it out. With the same abuse of notation as last time, we write
and so we can fill in the other column:
From here we can write all the other matrices in the representation as products of these two.
We’ll start with one we’ve already partially worked out:
Now, it’s slightly abusive to the notation, but we’ll just write a tableau and know that we actually mean the polytabloid in our linear combinations. Using this, we’ve seen that we can write
Now, by the way we’ve selected our Garnir elements, we know that none of these can have any column descents. And we also know that they can’t have a row descent in the same place did. Indeed, the only three that have a row descent all have it between the second and third entries of the first row. So now let’s look at
We can write down another table, just like before:
which lets us write
Similarly we can write
Putting these all together, we conclude that
All of these tabloids are standard, and so we see that our original — nonstandard — is in the span of the standard polytabloids.
We defined the Specht module as the subspace of the Young tabloid module spanned by polytabloids of shape . But these polytabloids are not independent. We’ve seen that standard polytabloids are independent, and it turns out that they also span. That is, they provide an explicit basis for the Specht module .
Anyway, first off we can take care of all the where the columns of don’t increase down the column. Indeed, if stabilizes the columns of , then
where we’ve used the sign lemma. So any two polytabloids coming from tableaux in the same column equivalence class are scalar multiples of each other. We’ve just cut our spanning set down from one element for each tableau to one for each column equivalence class of tableaux.
To deal with column equivalence classes, start with the tableau that we get by filling in the shape with the numbers to in order down the first column, then the second, and so on. This is the maximum element in the column dominance order, and it’s standard. Given any other tableau , we assume that every tableau with is already in the span of the standard polytabloids. This is an inductive hypothesis, and the base case is taken care of by the maximum tabloid .
If is itself standard, we’re done, since it’s obviously in the span of the standard polytabloids. If not, there 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 by our inductive hypothesis, all these are in the span of the standard polytabloids, and thus is as well.
As an immediate consequence, we conclude that , where is the number of standard tableaux of shape . Further, since we know from our decomposition of the left regular representation that each irreducible representation of shows up in with a multiplicity equal to its dimension, we can write
Taking dimensions on both sides we find
When we pick a tableau with a certain row descent and use it to pick sets and , as we’ve done, the resulting Garnir element is a sum of a bunch of tabloids coming from a bunch of tableaux. I say that the column tabloid corresponding to the original tableau is dominated by all the other tabloids, using the column dominance order.
Indeed, when considering column tabloids we can rearrange the entries within columns freely, so we may assume that they’re always increasing down the columns. If we have our row descent in row , we can label the entries in the left column by s and those in the right column by s. Our tabloid then looks — in these two columns, at least — something like
We see our sets and . The permutations in the transversal that we use to construct our Garnir element work by moving swapping some of the s with some of the s. But since all that s are smaller than all the s, while they occur in a row further to the right, the dominance lemma for column tabloids tells us that any such swap can only move the tabloid up in the dominance order.
It is in this sense that the Garnir element lets us replace a tabloid with a linear combination of other tabloids that are “more standard”. And it puts us within striking distance of our goal.
Okay, for the last couple posts I’ve talked about using Garnir elements to rewrite nonstandard polytabloids — those coming from tableaux containing “row descents” — in terms of “more standard” polytabloids. Finally, we’re going to define another partial order that will give some meaning to this language.
For example, consider the tabloid
from which we get the column tabloid
And now we can define the dominance order on column tabloids just like the dominance order for row tabloids. Of course, in doing so we have to alter our definition of the dominance order on Ferrers diagrams to take columns into account instead of rows.
But one thing at least will make our life simpler: it should be clear that we still have a dominance lemma for column dominance. To be explicit: if , and appears in a column to the right of in the column tabloid , then dominates in the column dominance order.
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 .
Given a tableau , let be a subset of the entries in the th column of , and let be a subset of the entries in the st column. We can come up with Garnir elements associated to this choice of and , but — as we pointed out last time — we need some way of picking which particular transversal elements to use. For each summand in , we separate into a pair of sets , but we have flexibility in how we order the elements of and . Our answer in this case is to always pick the permutation that puts the elements of into increasing order as we move down the columns of .
For example, consider the tableau
This tableau has a “row descent” in the second row: a pair of adjacent entries in the row where the larger entry is on the left instead of the right. Let be the entry on the left along with all the entries below it in its column — — and let be the entry on the right along with all the entries above it in its column — . We look at all six ways of rearranging the collection into two subsets of two elements each (we listed then last time, actually) and choose permutations that keep entries in increasing order as we move down the columns.
Notice that we’ve picked different permutations this time, and so we get a different Garnir element:
Also, note that only the first of these tableaux has the descent in the second row, although some now have descents in the first row. Slightly less obvious is the fact that , and so we can write
Thus we can rewrite this polytabloid that has a row descent in terms of a bunch of other polytabloids that don’t have it and are “more standard”, in a sense we’ll define later.
Let and be two disjoint sets of positive integers. We’re mostly interested in the symmetric group , which shuffles around all the integers in both sets. But a particularly interesting subgroup is , which shuffles around the integers in and , but doesn’t mix the two together. Clearly, is a subgroup of .
Now, let be a transversal collection of permutations for this subgroup. That is, we can decompose the group into cosets
Then a “Garnir element” is
Now, the problem here is that we’ve written as if it only depends on the sets and , when it clearly depends on the choice of the transversal . But we’ll leave this alone for the moment.
How can we come up with an explicit transversal in the first place? Well, consider the set of pairs of sets so that , , and . That is, each is another way of breaking the same collection of integers up into two parts of the same sizes as and .
Any permutation acts on the collection of such pairs of sets in the obvious way, sending to , which is another such pair. In fact, it’s transitive, since we can always find some with and . If for each we make just such a choice of , then this collection of permutations gives us a transversal!
We can check this by first making sure we have the right number of elements. A pair is determined by taking all integers and picking to go into . That is, we have
pairs. But this is also the number of cosets of in !
Do we accidentally get two representatives and for the same coset? If we did, then we’d have to have . But then , and thus . But we only picked one permutation sending to a given pair, so .
As an example, let and . Then we have six pairs of sets to consider
where in each case I’ve picked a that sends to . This will give us the following Garnir element:
But, again, this is far from the only possible choice for this and .