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# Monthly Archives: November 2013

## Everything you wanted to know about symmetric polynomials, part III

Let’s talk about Schur polynomials. These were defined as:

$\displaystyle{ s_\lambda = \sum_{T \in SSYT(\lambda)} x^{w(T)}. }$

Here $\lambda$ is a partition, $T$ is a semistandard Young tableau of shape $\lambda$, and $x^{w(T)}$ is the monomial $x_1^{\mu_1} \cdots x_n^{\mu_n}$, where $\mu$ is the weight of $T$ (so $\mu_i$ is the number of $i$‘s in $T$.)

First things first: it’s not obvious from the definition that these are symmetric!

Claim. The Schur polynomials are symmetric polynomials.

Proof. It suffices to show that there as many tableaux of weight $\mu$ as there are of weight $s_i \cdot \mu$, where $s_i$ is the transposition that swaps $i$ and $i+1$. We’ll exhibit a bijection between these two sets, $SSYT(\lambda,\mu)$ and $SSYT(\lambda,s_i\cdot\mu)$ directly, called the Bender-Knuth involution.

(Personal note: this was one of the first proofs I ever learned in algebraic combinatorics, and remains one of my favourites, for its elegant use of the defining properties of SSYTs.)

## Everything you wanted to know about symmetric polynomials, part II

Last post, I introduced the ring $\Lambda$ of symmetric polynomials in infinitely-many variables. Its elements are infinite sums of monomials (of bounded degree), symmetric in all the variables. I also defined the five most common bases for it: the monomial, elementary, (complete) homogeneous, power and Schur polynomials, $m,e,h,p,s$ for short. All of them are indexed by partitions $\lambda$, and in each case the basis element is homogeneous of (total) degree $|\lambda|$.

Now, I’m going to prove that each of these is a basis for $\Lambda$.

## Everything you wanted to know about symmetric polynomials, part I

A symmetric polynomial $f(x_1, \ldots, x_n)$ is a polynomial which doesn’t change under permutations of the variables. So, $f(x,y,z) = f(x,z,y) = f(y,z,x)$ and so on. Because of the simplicity of the definition (and the ubiquity of polynomials in math), symmetric polynomials show up in a variety of fields: algebraic geometry and combinatorics, Galois theory, birational geometry and intersection theory, linear algebra, and representation theory — to say the least.

Perhaps the most well-known are the elementary symmetric polynomials $e_k$:

$\displaystyle{ e_k = \sum (\text{ all squarefree monomials of total degree k }) }$

## The RSK Correspondence

This is another tableau combinatorics post. The Robinson-Schensted-Knuth correspondence, or RSK for short, is another important theorem and algorithm in tableau combinatorics; I’ll discuss it here, though I won’t try to motivate it. For now, it will be something of a curiosity. As always, my reference is Fulton’s book Young Tableaux.

## Knuth Equivalence and Tableau Products

I found (and still find) the ideas and algorithms around Young Tableaux to be fairly unintuitive: jeux de taquin (JDT), Knuth equivalence and rectification; the RSK correspondence; the Littlewood-Richardson rule(s); promotion and evacuation.

In this post, I want to state (without proof) the basic ideas and facts surrounding the first algorithm above, JDT, and its connection to Knuth equivalence. The reference I am using is Fulton’s Young Tableaux textbook, which covers almost all of the above-mentioned algorithms. For this post, I’m essentially restating section 3.1, and parts of sections 1 and 2.

Let’s begin.

## Hello, world

For now, the main purpose of this blog is for me to have somewhere to sound out my thoughts as I study for my prelim exam. Perhaps it’ll last longer than that. (Certainly, I’m the kind of mathematician who especially likes talking about math with others. I don’t know whether that will translate to the blogging medium.)