#Boolean product polynomials
The boolean product polynomials are 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 arrangements, among other things.
Example
The first nontrivial case is already Schur-positive: \[\begin{aligned} \booleanProduct_2(\xvec_3) &=(x_1+x_2)(x_1+x_3)(x_2+x_3)\\ &=\monomial_{21}+2\monomial_{111} =\schurS_{21}. \end{aligned}\]
#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 \(\booleanProduct_{n,k}(\xvec).\)
N. Libman and G. Orelowitz give combinatorial interpretations for Schur expansions of special cases of Chern plethysm [LO23]. They also construct a symmetric-group module whose Frobenius characteristic realizes one of these symmetric-function analogues.
#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.
Bibliography
- [BRT19]Sara C. Billey, Brendon Rhoades and Vasu Tewari. Boolean product polynomials, Schur positivity, and Chern plethysm. arXiv:1902.11165, 2019.
.bib
@article{BilleyRhoadesTewari2019, Author = {Sara C. Billey and Brendon Rhoades and Vasu Tewari}, Title = {Boolean product polynomials, {S}chur positivity, and {C}hern plethysm}, Year = {2019}, Eprint = {1902.11165}, url = {https://arxiv.org/abs/1902.11165}, journal = {arXiv e-prints} } - [LO23]Nathaniel Libman and Gidon Orelowitz. Combinatorics and Representation Theory of Special Cases of Chern Plethysm. arXiv:2310.01786, 2023.
.bib
@article{LibmanOrelowitz2023x, author = {Nathaniel Libman and Gidon Orelowitz}, title = {Combinatorics and {R}epresentation {T}heory of {S}pecial {C}ases of {C}hern {P}lethysm}, year = {2023}, eprint = {2310.01786}, url = {https://arxiv.org/abs/2310.01786}, journal = {arXiv e-prints} }