The symmetric functions catalog

An overview of symmetric functions and related topics

2019-05-20

Boolean product polynomials

The boolean product polynomials is a family of polynomials defined as follows:

$\booleanProduct_{k}(\xvec_n) \coloneqq \prod_{ 1 \leq i_1 \lt i_2 \lt \dotsb \lt i_k \leq n } (x_{i_1} + x_{i_2}+\dotsb + x_{i_k}).$

These were introduced by L. Billera, S. Billey, and V. Tewari, in an FPSAC abstract, and inspired by certain applications in hyperplane arrangement, among other things.

Schur positivity

In [BRT19] it is proved that the functions $\booleanProduct_{k}(\xvec_n)$ are Schur-positive. The authors use a new method for proving Schur positivity, called Chern plethysm.

Problem.

Find a combinatorial interpretation for the coefficients in the expansion

$\booleanProduct_{k}(\xvec_n) = \sum_{\lambda} \delta^{n,k}_{\lambda} \schurS_\lambda(\xvec_n).$

Find natural $\symS_n$-modules whose Frobenius characteristic is $\booleanProduct_{n,k}(\xvec).$

Bivariate extension

There is a bivariate extension,

$\booleanProduct_{k,\ell}(\xvec_n;\yvec_m) = \coloneqq \prod_{ \substack{ 1 \leq i_1 \lt i_2 \lt \dotsb \lt i_k \leq n \\ 1 \leq j_1 \lt j_2 \lt \dotsb \lt j_\ell \leq m }} (x_{i_1} + x_{i_2}+\dotsb + x_{i_k} + y_{j_1} + y_{j_2}+\dotsb + y_{j_\ell} ).$

One can show using representation theory (see [BRT19]) that the $a_{\lambda,\mu}$ in

$\booleanProduct_{k,\ell}(\xvec_n;\yvec_m) = \sum_{\lambda,\mu} a_{\lambda,\mu} \schurS_\lambda(\xvec_n)\schurS_\mu(\yvec_m).$

are non-negative integers. When $k=\ell = 1,$ this reduces to the dual Cauchy identity.