The hidden basis

In the last few posts (see here and here), I’ve been talking about various bases for the symmetric functions: the monomial symmetric functions $m_\lambda$, the elementary symmetric functions $e_\lambda$, the power sum symmetric functions $p_\lambda$, and the homogeneous symmetric functions $h_\lambda$. As some of you aptly pointed out in the comments, there is one more important basis to discuss: the Schur functions!

When I first came across the Schur functions, I had no idea why they were what they were, why every symmetric function can be expressed in terms of them, or why they were useful or interesting. I first saw them defined using a simple, but rather arbitrary-sounding, combinatorial approach:

Theme and variations: the Newton-Girard identities

Time for another gemstone from symmetric function theory! (I am studying for my Ph.D. qualifying exam at the moment, and as a consequence, the next several posts will feature yet more gemstones from symmetric function theory. You can refer back to this post for the basic definitions.)

Start with a polynomial $p(x)$ that factors as \[p(x)=(x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_n).\] The coefficients of $p(x)$ are symmetric functions in $\alpha_1,\ldots,\alpha_n$ - in fact, they are, up to sign, the elementary symmetric functions in $\alpha_1,\ldots,\alpha_n$.

Addendum: An alternate proof of the FTSFT

Last week I posted about the Fundamental Theorem of Symmetric Function Theory. Zarathustra Brady pointed me to the following alternate proof in Serge Lang’s book Algebra. While not as direct or useful in terms of changing basis from the $e_\lambda$`s to the $m_\lambda$`s, it is a nice, clean inductive proof that I thought was worth sharing:

The Fundamental Theorem of Symmetric Function Theory

This little gemstone is hidden within the folds of algebraic combinatorics, but certainly deserves its name. Just as the Fundamental Theorem of Arithmetic gives us a way of writing common objects (numbers) in a canonical form (prime factorization), the Fundamental Theorem of Symmetric Function Theory allows us to express any symmetric function in a useful canonical form.

First, some background on symmetric functions. Like generating functions, symmetric functions arise naturally in combinatorics, and the coefficients are often the point of interest. In the poetic words of Herbert Wilf in his book generatingfunctionology:

A generating function is a clothesline on which we hang up a sequence of numbers for display.

I claim, and aim to demonstrate here, that

A symmetric function is a web of clotheslines on which we hang up a collection of numbers that are indexed by partitions.

Okay, so the clothesline might be a bit more complicated in this case. But it is much more useful for areas of mathematics in which partitions play an important role.

Circles of Apollonius... and magnetism!

These two concepts go together in a very natural way. Prepare for a breathtaking real-world application of Euclidean geometry!

Suppose you have two identical long wires side by side, parallel to each other and connected to each other at one end, and a current is flowing through one wire and back through the other in the other direction. As in this video, each wire generates a magnetic field, and the magnetic field forces the two wires towards each other. The question is, just before the wires start to move towards each other, what does the magnetic field look like?