The Jacobi-Trudi identities

Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let be a partition. A semistandard Young tableau of shape is a filling of the Young diagram of with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau is defined as where is the total number of times appears in the tableau.

Definition 4:

where the sum is taken over all semistandard Young tableaux of shape .

As before we can readily verify that . This definition will allow us to deduce the Jacobi-Trudi identities for the Schur functions, which describe among other things the action of the fundamental involution . Since I’m trying to emphasize how many different ways there are to define the Schur functions, I’ll call these definitions instead of propositions.

Definition 5: .

Definition 6: .

The proof

The following proof is one of the standard applications of the Lindstrom-Gessel-Viennot lemma, and like much of the other material in these posts it can be found (with accompanying diagrams) in both Sagan and Stanley.

We’ll work with , the acyclic plane, with edges going northward and eastward only. We’ll also add some points which can be reached going northward from any finite point, and we won’t weight northward steps. But we will weight eastward steps as follows:

• In the e-weighting, each eastward step is weighted where is the sum of the coordinates of the destination vertex of the edge minus the sum of the coordinates of the initial vertex of the path. The weight of a path is the product of the weights of its eastward edges.
• In the h-weighting, each eastward step is weighted where is the difference between the -coordinate of the destination vertex of the edge and the -coordinate of the initial vertex of the path. Again the weight of a path is the product of the weight of its eastward edges.

Choose sources and sinks . Then every path from to determines a monomial in in the e-weighting and a monomial in in the h-weighting, and conversely every monomial arises in this way. So by the lemma it follows that the Jacobi-Trudi determinants count non-intersecting -paths .

Given such a non-intersecting -path we may construct two tableaux from it: one whose columns can be read off from the e-weighting of each path which has shape , and one whose rows can be read off from the h-weighting of each path which has shape . The non-intersecting condition is then precisely the condition that rows are weakly increasing and columns are strictly increasing; intersections would correspond to a violation of one or both of those constraints. (This is much easier to see with the corresponding diagrams; my apologies.) So in fact one obtains SSYTs, and moreover it’s not hard to show that all SSYTs of the correct shapes are obtained in this way.

The fundamental involution

Since the fundamental involution flips and , it follows that . There is a representation-theoretic interpretation of this. Recall that

and that

.

Applying the fundamental involution it follows that . We should interpret as the sign of an -cycle so that . Now the identity

gives

which has an obvious interpretation: the representation must come from by tensoring with the sign representation!

Relationship to a previous definition

We are now in a position to prove that Definition 4 is equivalent to one of the other definitions we gave earlier, namely Definition 3. We follow the proof given in Sagan, p. 164. First define to be with all the terms containing removed.

Lemma: Let be a composition and define the matrices . Then .

Proof. This is equivalent to the collection of identities

which is in turn equivalent to

which is obvious.

The equivalence of Definition 3 and Definition 4 is a simple corollary; one just picks so that is first and then , and then applies the first Jacobi-Trudi identity.