[Paper Review] Boolean product polynomials and Schur-positivity

Louis J. Billera, Sara Billey+1 more|arXiv (Cornell University)|Jun 8, 2018

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper establishes the Schur-positivity of Boolean product polynomials—symmetric polynomials defined as products of subset sums over 0–1 vectors—by linking them to total Chern classes of vector bundles via Fulton-Lazarsfeld positivity. The key result is that all such polynomials, including multivariate extensions with two alphabets, expand into Schur functions with nonnegative integer coefficients, resolving a long-standing problem in symmetric function theory and hyperplane arrangements.

ABSTRACT

We study a family of symmetric polynomials that we refer to as the Boolean product polynomials. The motivation for studying these polynomials stems from the computation of the characteristic polynomial of the real matroid spanned by the nonzero vectors in $\mathbb{R}^n$ all of whose coordinates are either $0$ or $1$. To this end, one approach is to compute the zeros of the Boolean product polynomials over finite fields. The zero loci of these polynomials cut out hyperplane arrangements known as resonance arrangements, which show up in the context of double Hurwitz polynomials. By relating the Boolean product polynomials to certain total Chern classes of vector bundles, we establish their Schur-positivity by appealing to a result of Pragacz relying on earlier work on numerical positivity by Fulton-Lazarsfeld. Subsequently, we study a two-alphabet version of these polynomials from the viewpoint of Schur-positivity. As a special case of these polynomials, we recover symmetric functions first studied by Désarménien and Wachs in the context of descents in derangements.

Motivation & Objective

• To establish the Schur-positivity of the (n,k)-th Boolean product polynomial $B_{n,k}(X)$, defined as the product of all subset sums $X_S = \sum_{i \in S} x_i$ over $k$-element subsets $S \subseteq [n]$.
• To extend this result to a two-alphabet version $\mathcal{P}_{j,k}(X,Y)$, where polynomials are formed from sums $X_S + Y_T$ over $j$-subsets of $X$ and $k$-subsets of $Y$, and prove their Schur-positivity in both alphabets.
• To connect these polynomials to geometric and combinatorial objects such as the resonance arrangement, minimal balanced collections, and regions of hyperplane arrangements.
• To explore connections with representation theory, including the Frobenius characteristic of $\mathfrak{S}_n$-modules and higher Lie modules.
• To investigate combinatorial interpretations, such as the link to positroids and the sequence $\sum_{k=0}^n \frac{n!}{k!}$, and to suggest a potential generalization of the Robinson-Schensted algorithm.

Proposed method

• Use vector bundle methods: interpret $B_{n,k}(X)$ as the total Chern class of the exterior power $\bigwedge^k \mathcal{E}$ for a rank-$n$ vector bundle $\mathcal{E}$ with Chern roots $x_1, \dots, x_n$.
• Apply Pragacz’s theorem on numerical positivity, which ensures that total Chern classes of globally generated vector bundles are Schur-positive.
• Generalize to two alphabets by defining $\mathcal{P}_{j,k}(X,Y) = \prod_{|S|=j, |T|=k} (X_S + Y_T)$, which equals the elementary symmetric function $e_p(\bigwedge^j \mathcal{E} \otimes \bigwedge^k \mathcal{F})$.
• Leverage the fact that $\mathcal{P}_{j,k}(X,Y)$ is the Frobenius characteristic of a $\mathfrak{S}_n \times \mathfrak{S}_m$-module, and use Schur-positivity theorems to deduce its expansion in products of Schur functions.
• Connect to known symmetric function identities, such as the dual Cauchy identity for $\mathcal{P}_{1,1}(X,Y) = \prod_{i,j} (x_i + y_j)$, to motivate combinatorial interpretations.
• Use the finite field method to relate the zero locus of $B_n(X)$ to the resonance arrangement and its regions, which count maximal unbalanced collections and generalized retarded functions.

Experimental results

Research questions

• RQ1 Are the Boolean product polynomials $B_{n,k}(X)$ Schur-positive for all $n$ and $k \leq n$?
• RQ2 Can the Schur-positivity of $B_{n,k}(X)$ be established using geometric methods involving vector bundles and Chern classes?
• RQ3 Does the two-alphabet generalization $\mathcal{P}_{j,k}(X,Y)$ also yield Schur-positive symmetric functions in both sets of variables?
• RQ4 What is the combinatorial or representation-theoretic meaning of the Frobenius characteristic $B_{n,n-1}(X;1)$, which has dimension $\sum_{k=0}^n \frac{n!}{k!}$?
• RQ5 Is there a combinatorial rule, such as a generalized Robinson-Schensted insertion, that explains the Schur-positivity of $\mathcal{P}_{j,k}(X,Y)$?

Key findings

• The $(n,k)$-th Boolean product polynomial $B_{n,k}(X)$ is Schur-positive: $B_{n,k}(X) = \sum_{\lambda} \kappa_{\lambda}^{(n,k)} s_\lambda(X)$ with $\kappa_{\lambda}^{(n,k)} \in \mathbb{Z}_{\geq 0}$.
• The total Boolean product polynomial $B_n(X) = \prod_{k=1}^n B_{n,k}(X)$ is also Schur-positive, extending the result to the full resonance arrangement.
• The two-alphabet polynomial $\mathcal{P}_{j,k}(X,Y)$ is Schur-positive in both alphabets: $\mathcal{P}_{j,k}(X,Y) = \sum_{\lambda,\mu} a_{\lambda\mu} s_\lambda(X)s_\mu(Y)$ with $a_{\lambda\mu} \in \mathbb{Z}_{\geq 0}$.
• When $q=1$, the Frobenius characteristic $B_{n,n-1}(X;1)$ corresponds to a $\mathbb{C}\mathfrak{S}_n$-module of dimension $\sum_{k=0}^n \frac{n!}{k!}$, which counts positroids on $[n]$.
• The polynomial $\mathcal{P}_{1,1}(X,Y) = \prod_{i=1}^n \prod_{j=1}^m (x_i + y_j)$ is Schur-positive via the dual Cauchy identity, providing a known base case for generalization.
• The work suggests a potential link between Boolean product polynomials and higher Lie modules, as $B_{n,n-1}(X)$ matches the sum of Frobenius characteristics of $\mathrm{Lie}_\lambda(V)$ for partitions $\lambda$ without parts of size 1.