The authors of this package used a combination of high-throughput and high-performance computing and sparse numerical linear algebra to compute the syzygies of $\mathbb{P}^{2}$ under the $d$-fold Veronese embedding for various values of $d$. See the paper ``Conjectures and Computations about Veronese Syzygies'' by Bruce, Erman, Goldstein and Yang, which we refer to as [BEGY] (see arXiv:1711.03513) throughout the documentation for this package. In addition, much of the data generated from these computations (graded Betti numbers, multigraded Betti numbers, Schur functor decompositions, etc.) is currently available online via syzygydata.com. The goal of this package is to make this data more accessible and easy to use by providing a way to access it via Macaulay2.
Most functions have been implemented with three parameters $(d,n,b)$, where the goal is to compute the syzygies of the pushforward of the line bundle $\mathcal{O}(b)$ under the $d$-fold embedding. However, we have produced data for $n=1$ and $n=2$, for $b$ between $0$ and $d$ and for a limited range of values of $d$. Other inputs will produce an error message. Our hope is that as we (or others) are able to compute new data, we will be able to update the package.
One of the main functions is totalBettiTally, which produces the standard graded Betti table of the corresponding standard graded Veronese module. Other main functions refine the data in the Betti table by providing the multigraded Betti number or the Schur functor decompositions, or by computing statistics related to the Betti table (e.g., the BoijSoederberg coefficients) or related to the SchurFunctor decomposition.
A number of functions in this package produce individual entries of a Betti table. There are two common notations for referring to Betti numbers in the literature, and it will be useful to reference these notations throughout the documentation, similar to how they are referenced in the corresponding paper. For a graded module $M$ we will write $\beta_{i,j}(M)$ for $\dim Tor_i(M,k)_j$ and we write $K_{p,q}(M)$ for the vector space $Tor_p(M,k)_{p+q}$.
Version 1.1 of this package was accepted for publication in volume 11 of The Journal of Software for Algebra and Geometry on 5 May 2021, in the article The Schur–Veronese package in Macaulay2 (DOI: 10.2140/jsag.2021.11.83). That version can be obtained from the journal or from the Macaulay2 source code repository.
This documentation describes version 1.1 of SchurVeronese.
The source code from which this documentation is derived is in the file SchurVeronese.m2. The auxiliary files accompanying it are in the directory SchurVeronese/.
The object SchurVeronese is a package.