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## The Jacobi-Trudi identities

Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let $\lambda \vdash n$ be a partition. A semistandard Young tableau $T$ of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau $T$ is defined as $\mathbf{x}^T = x_1^{T_1} x_2^{T_2} ...$ where $T_i$ is the total number of times $i$ appears in the tableau.

Definition 4: $\displaystyle s_{\lambda}(x_1, x_2, ...) = \sum_T \mathbf{x}^T$

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

As before we can readily verify that $s_{(k)} = h_k, s_{(1^k)} = e_k$. 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 $\omega$. 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: $\displaystyle s_{\lambda}= \det(h_{\lambda_i+j-i})_{1 \le i, j \le n}$.

Definition 6: $\displaystyle s_{\lambda'} = \det(e_{\lambda_i+j-i})_{1 \le i, j \le n}$.

## Young’s lattice

As we saw last time, a sequence of nested groups gives rise to a graded “poset” in which objects can be related by more than one arrow. The poset we want to construct now is given by the sequence $S_0 \le S_1 < S_2 < S_3$ of symmetric groups (where $S_0$ and $S_1$ are both the trivial group). Since irreducible representations of the symmetric groups are parameterized by Young diagrams, this gives the set of Young diagrams a graded structure where the elements of rank $n$ are the Young diagrams with $n$ boxes.

It turns out that this “poset” is a genuine poset; in other words, in the restriction of an irreducible representation of $S_n$ to an irreducible representation of $S_{n-1}$, the resulting representation contains each irreducible representation at most once. In fact, much more is true. In a poset we write $\lambda \triangleright \mu$ if $\lambda > \mu$ and there is no $\lambda'$ with $\lambda > \lambda' > \mu$; this relation is called covering. So elements of rank $n$ cover elements of rank $n-1$.

Theorem (Branching Rule): $\lambda \triangleright \mu$ if and only if $\mu$ is obtained from $\lambda$ by removing a box.

This is the remarkable theorem that really justifies the choice of Young diagrams as a natural description of the irreducible representations of the symmetric group. Today we’ll explore some of the consequences of this theorem.

## Standard Young tableaux and Robinson-Schensted-Knuth

Earlier I mentioned that the theory of Young tableaux is the source of one of my favorite proofs. Today I’d like to present, again from the theory of Young tableaux, one of my favorite pairs of proofs.

A standard Young tableau is a chain in Young’s lattice; equivalently, it is a sequence of Young diagrams, each of which has exactly one more box than the previous. One can succinctly write down such a chain by taking the last Young diagram in the chain and recording the “path” traced out by the rest of the chain: write down a 1 in the box corresponding to the first Young diagram, a 2 in the box added by the next Young diagram, and so forth. In other words, a standard Young tableau is a Young tableau filled with positive integers $1, 2, ... n$ which is strictly increasing across rows and columns.

Given a Young diagram $\lambda$, let $f^{\lambda}$ denote the number of standard Young tableaux of shape $\lambda$; equivalently, one can define $L(\lambda)$ to be the poset of Young diagrams fitting inside $\lambda$, and then $f^{\lambda}$ is equal to the number of maximal chains of $L(\lambda)$.

Finally, one last bit of notation. If $\lambda$ is a Young diagram for a partition of $n$, we will write either $|\lambda| = n$ or $\lambda \vdash n$. Then we have the following beautiful result.

Theorem: $\displaystyle \sum_{\lambda \vdash n} (f^{\lambda})^2 = n!$.

## GILA V: The Polya enumeration theorem and applications

I ended the last post by asking whether the proof of baby Polya extends to the multi-parameter setting where we want to keep track of how many of each color we use. In fact, it does. First, we should specify what exactly we’re trying to compute. Recall the setup: we have colors $1, 2, ... n$ (represented by variables $r_1, r_2, ... r_n$), and we have a set of slots $S$ with $|S| = m$ acted on by a group $G$ where each slot will be assigned a color. Define $f_G(t_1, ... t_n)$ to be the number of orbits of functions $S \to \{ 1, 2, ... n \}$ under the action of $G$ where color $i$ is used $t_i$ times. (Since the action of $G$ only permutes slots, it doesn’t change the multiset of colors used.) What we want to compute is the generating function

$\displaystyle F_G(r_1, r_2, ... r_n) = \sum_{t_1 + ... + t_n = m} f_G(t_1, t_2, ... t_n) r_1^{t_1} r_2^{t_2} ... r_n^{t_n}$.

Note that setting $r_1 = r_2 = .... = r_n = 1$ we recover $F_G(n)$, which doesn’t contain information about particular color combinations. By the orbit-counting lemma, this is equivalent to computing

$\displaystyle \frac{1}{|G|} \sum_{g \in G} | \text{Fix}(g) |(r_1, r_2, ... r_n)$

where we must now count fixed points in a weighted manner, recording the multiset of colors in each fixed point. How do we go about doing this?

## Young diagrams, q-analogues, and one of my favorite proofs

I’ve decided to start blogging a little more about the algebraic combinatorics I’ve learned over the past year. In particular, I’d like to present one of my favorite proofs from Stanley’s Enumerative Combinatorics I.

The theory of Young tableaux is a great example of the richness of modern mathematics: although they can be defined in an elementary combinatorial fashion, interest in the theory is primarily driven (as I understand it) by their applications to representation theory and other fields of mathematics. The standard application is in describing the irreducible representations of the symmetric group in characteristic zero. I’ll be describing a more elementary aspect of the theory: the relationship between Young diagrams and $q$-binomial coefficients.

Young diagrams can be defined as a visual representations of partitions. A partition of $k$ is a weakly decreasing sequence $\lambda_1 \ge \lambda_2 \ge ... \ge \lambda_i$ such that $\lambda_1 + ... + \lambda_i = k$. Partitions uniquely describe cycle decompositions, hence conjugacy classes, of the symmetric group, which is a hint at the connection to representation theory. A Young diagram of the partition $\lambda$ consists of $i$ rows of boxes where the $j^{th}$ row has $\lambda_j$ boxes. Let $L(m, n)$ denote the poset of Young diagrams that fit into an $m \times n$ box, ordered by inclusion: equivalently, one can define the full Young lattice and define $L(m, n)$ to be the elements less than or equal to the $m \times n$ partition. (One can then define a standard Young tableau to be a chain in the Young lattice, but we will not need this notion.) One can easily check that the following is true.

Proposition: $|L(m, n)| = {m+n \choose m}$.

The rest of this post explains the $q$-analogue of this result.