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## Four flavors of Schur-Weyl duality

If $V$ is a finite-dimensional complex vector space, then the symmetric group $S_n$ naturally acts on the tensor power $V^{\otimes n}$ by permuting the factors. This action of $S_n$ commutes with the action of $\text{GL}(V)$, so all permutations $\sigma : V^{\otimes n} \to V^{\otimes n}$ are morphisms of $\text{GL}(V)$-representations. This defines a morphism $\mathbb{C}[S_n] \to \text{End}_{\text{GL}(V)}(V^{\otimes n})$, and a natural question to ask is whether this map is surjective.

Part of Schur-Weyl duality asserts that the answer is yes. The double commutant theorem plays an important role in the proof and also highlights an important corollary, namely that $V^{\otimes n}$ admits a canonical decomposition

$\displaystyle V^{\otimes n} = \bigoplus_{\lambda} V_{\lambda} \otimes S_{\lambda}$

where $\lambda$ runs over partitions, $V_{\lambda}$ are some irreducible representations of $\text{GL}(V)$, and $S_{\lambda}$ are the Specht modules, which describe all irreducible representations of $S_n$. This gives a fundamental relationship between the representation theories of the general linear and symmetric groups; in particular, the assignment $V \mapsto V_{\lambda}$ can be upgraded to a functor called a Schur functor, generalizing the construction of the exterior and symmetric products.

The proof below is more or less from Etingof’s notes on representation theory (Section 4.18). We will prove four versions of Schur-Weyl duality involving $\mathfrak{gl}(V), \text{GL}(V)$, and (in the special case that $V$ is a complex inner product space) $\mathfrak{u}(V), \text{U}(V)$.

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## The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation by hand, but it is conceptually cleaner to use the symmetries of the situation and representation theory. Deducing these results using representation theory has the added benefit that it identifies which parts of the situation depend only on symmetry and which parts depend on the particular form of the Hamiltonian. This is nicely explained in Singer’s Linearity, symmetry, and prediction in the hydrogen atom.

For awhile now I’ve been interested in finding a toy model to study the basic structure of the arguments involved, as well as more generally to get a hang for quantum mechanics, while avoiding some of the mathematical difficulties. Today I’d like to describe one such model involving finite graphs, which replaces the infinite-dimensional Hilbert spaces and Lie groups occurring in the analysis of the hydrogen atom with finite-dimensional Hilbert spaces and finite groups. This model will, among other things, allow us to think of representations of finite groups as particles moving around on graphs.

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## Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on graphs and tensor products of representations which at least partially explains one of the results I’d been wondering about and also unites several other related computations that have been on my mind recently.

Let $G$ be a compact group and let $\text{Rep}(G)$ denote the category of finite-dimensional unitary representations of $G$. As in the finite case, due to the existence of Haar measure, $\text{Rep}(G)$ is semisimple (i.e. every unitary representation decomposes uniquely into a sum of irreducible representations), and via the diagonal action it comes equipped with a tensor product with the property that the character of the tensor product is the product of the characters of the factors.

Question: Fix a representation $V \in \text{Rep}(G)$. What is the multiplicity of the trivial representation in $V^{\otimes n}$?

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

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## The many faces of Schur functions

The last time we talked about symmetric functions, I asked whether the vector space $\mathcal{R}$ could be turned into an algebra, i.e. equipped with a nice product. It turns out that the induced representation allows us to construct such a product as follows:

Given representations $\rho_1, \rho_2$ of $S_n, S_m$, their tensor product $\rho_1 \otimes \rho_2$ is a representation of the direct product $S_n \times S_m$ in a natural way. Now, $S_n \times S_m$ injects naturally into $S_{n+m}$, which gives a new representation

$\rho = \text{Ind}_{S_n \times S_m}^{S_{n+m}} \rho_1 \otimes \rho_2$.

The character of this representation is called the induction product $\rho_1 * \rho_2$ of the characters of $\rho_1, \rho_2$, and with this product $\mathcal{R}$ becomes a graded commutative algebra. (Commutativity and associativity are fairly straightforward to verify.) It now remains to answer the first question: does there exist an algebra homomorphism $\phi : \Lambda \to \mathcal{R}$? And can we describe the inner product on $\Lambda$ coming from the inner product on $\mathcal{R}$?

To answer this question we’ll introduce perhaps the most important class of symmetric functions, the Schur functions $s_{\lambda}$.

N.B. I’ll be skipping even more proofs than usual today, partly because they require the development of a lot of machinery I haven’t described and partly because I don’t understand them all that well. Again, good references are Sagan or EC2.

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## 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.

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## Introduction to symmetric functions

The theory of symmetric functions, which generalizes some ideas that came up in the previous discussion of Polya theory, can be motivated by thinking about polynomial functions of the roots of a monic polynomial $P(x) \in \mathbb{Z}[x]$. Problems on high school competitions often ask for the sum of the squares or the cubes of the roots of a given polynomial, and those not familiar with symmetric functions will often be surprised that these quantities end up being integers even though the roots themselves aren’t integers. The key is that the sums being asked for are always symmetric in the roots, i.e. invariant under an arbitrary permutation. The coefficients of a monic polynomial are the elementary symmetric polynomials of its roots, which we are given are integers. It follows that any symmetric polynomial that can be written using integer coefficients in terms of the elementary symmetric polynomials must in fact be an integer, and as we will see, every symmetric polynomial with integer coefficients has this property.

These ideas lead naturally to the use of symmetric polynomials in Galois theory, but combinatorialists are interested in symmetric functions for a very different reason involving the representation theory of $S_n$. This post is a brief overview of the basic results of symmetric function theory relevant to combinatorialists. I’m mostly following Sagan, who recommends Macdonald as a more comprehensive reference.

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

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