Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Consider the general polynomial $P_1(t) = \prod_{j=1}^n (t+x_j)$. Construct $P_k(t) = \prod_{\sigma \subset [n], |\sigma|=k} (t+x_{\sigma_1}x_{\sigma_2}\cdots x_{\sigma_k})$ where the product is over all subsets of size $k$ of the numbers $1,2,\dots,n.$ The coefficients of $P_1(t)$ will be the elementary symmetric polynomials in $x_1,\dots,x_n$ and it is easy to argue that the coefficients in $P_k(t)$ are polynomials in the coefficients of $P_1.$

Now, my question is rather vague but I seek references to areas where these types of polynomials appear. I suspect they are related somehow to Schur-polynomials, representation theory, and determinants of band-matrices.

share|improve this question
    
I'm interested in what these have to do with determinants of band matrices! –  darij grinberg Feb 18 '12 at 20:08
    
@darij grindberg: It is related via Jacobi-Trudi identity. –  Per Alexandersson May 27 '12 at 18:18
add comment

2 Answers

up vote 6 down vote accepted

In the language of symmetric functions, you are computing the plethysm $e_j[e_k]$ (also denoted $e_k\circ e_j$) of elementary symmetric functions. In terms of $\mathrm{GL}(n,\mathbb{C})$ representations, you are looking at the representation $\Lambda^j(\Lambda^k(\mathbb{C}^n))$. Some information is at What is known about this plethysm?.

share|improve this answer
    
Thank you Prof. Stanley! –  Per Alexandersson Feb 20 '12 at 7:05
add comment

They appear in the definition of (special) $\lambda$-rings, more precisely in the formula

$\lambda^k\left(\lambda^j\left(x\right)\right) = P_{k,j}\left(\lambda^1\left(x\right),\lambda^2\left(x\right),...,\lambda^{kj}\left(x\right)\right)$,

where $P_{k,j}\in \mathbb Z\left[\alpha_1,\alpha_2,...,\alpha_{kj}\right]$ is the polynomial which is, more or less, the $k$-th coefficient of what you call $P_j\left(t\right)$ written out as a polynomial in the coefficients of $P_1\left(t\right)$ (not in the $x_i$). Okay, not exactly your $P_j\left(t\right)$, but instead $\prod\limits_{j=1}^n\left(1+x_jt\right)$ (this is your $P_j\left(t\right)$ "written upside down").

There are several texts on $\lambda$-rings nowadays; here are two:

Donald Knutson, $\lambda$-Rings and the Representation Theory of the Symmetric Group, New York 1973.

Donald Yau, Lambda-rings, WS 2010. (Here you can find the first chapter which contains the definitions.)

You can probably pick any text (as long as it's not Fulton-Lang...), but keep in mind that what some texts call $\lambda$-ring is what others call special $\lambda$-ring. Also, your polynomials $P_k$ are not what is traditionally called $P_k$ in $\lambda$-ring theory.

share|improve this answer
    
Thank you! I have not heard of such rings before, they seem to be very nice objects! –  Per Alexandersson Feb 20 '12 at 7:06
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.