Lattice polytopes, finite abelian subgroups in $\SL(n,\C)$ and coding theory

Abstract

We consider $d$-dimensional lattice polytopes $\Delta$ with $h^{\ast }$-polynomial $h_{\Delta }^{\ast }=1+h_k^{\ast }{t}^k$ for $1<k<(d+1)/2$ and relate them to some abelian subgroups of $\SL_{d+1}(\C)$ of order $1+h_k^{\ast }=p^r$ where $p$ is a prime number. These subgroups can be investigate by means of coding theory as special linear constant weight codes in $\F_p^{d+1}$. If $p=2$, then the classication of these codes and corresponding lattice polytopes can be obtained using a theorem of Bonisoli. If $p>2$, the main technical tool in the classification of these linear codes is the non-vanishing theorem for generalized Bernoulli numbers $B_{1,\chi }^{(r)}$ associated with odd characters $χ:\F_q^*\to\C^*$ where $q=p^r$. Our result implies a complete classification of all lattice polytopes whose $h^{\ast }$-polynomial is a binomial.

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