Fix \(n>0\) and let \(S=\mathbb{C}[x_0,\ldots,x_n]\) be the standard graded polynomial ring. Rmk We denote the \(k\)th graded piece of \(S\) by \(S_k\). For any \(b\in \mathbb{Z}\) and \(d\in\mathbb{N}\) we let \[ S(b;d):= \bigoplus_{k\in \mathbb{Z}}S_{dk+b}\subset S\] which we think of as an \(R=\text{Sym}(S_d)\)-module. Notice that \(S(b;d)\) is the graded module corresponding to the pushforward of the line bundle \(\mathcal{O}_{\mathbb{P}^n}(b)\) by the \(d\)'uple embedding of \(\mathbb{P}^n\).
The Betti numbers of \(S(b;d)\) are given by the dimensions of the graded components of \(\text{Tor}_{p}^{R}(S(b;d),\mathbb{C})\). We are interested in the graded, multigraded, and \(\textbf{GL}_{n+1}(\mathbb{C})\)-equivariant structure of these \(\text{Tor}\) modules, described below:The standard grading on \(S\) induces a \(\mathbb{Z}\)-grading on \(\text{Tor}^R(S(b;d),\mathbb{C})\). The standard graded Betti number of \(S(b;d)\) is \[\beta_{p,q}(S(b;d)):=\dim_{\mathbb{C}}\text{Tor}_{p}^{R}(S(b;d),\ mathbb{C})_{q}.\] While this notation is standard, there is another standard notation that we will often use: \[K_{p,q}(S(b;d))=\text{Tor}_{p}^{R}(S(b;d),\mathbb{C})_{p+q}.\] The vector space \(K_{p,q}(S(b;d))\) is a Koszul cohomology group, and \(\dim K_{p,q}(n,b;d)\) is equal to \(\beta_{p,p+q}(n,b;d)\). The (graded) Betti table of \(S(b;d)\) is the matrix who's \((p,q)\) entry is \(\dim K_{p,q}(n,b;d)\). Rmk
The polynomial ring \(S\) carries an action of the general linear group \(\textbf{GL}_{n+1}(\mathbb{C})\). Rmk This descends to \(\text{Tor}^{R}(S(b;d),\mathbb{C})\), turning the Koszul cohomology groups into \(\textbf{GL}_{n+1}(\mathbb{C})\)-representations. We call the resulting decomposition the Schur decomposition of \(K_{p,q}(n,b;d\). Since the irreducible representations of \(\textbf{GL}_{n+1}(\mathbb{C})\) are given (up to isomorphisms) by Schur functors, we can write the Schur decomposition of \(K_{p,q}(n,b;d)\) as: \[K_{p,q}(n,b;d)=\bigoplus_{\lambda\in\mathbb{Z}^3}\textbf{S}_{\lambda }(\mathbb{C}^{n+1})^{\oplus \beta_{p,\lambda}(n,b;d)},\] where \(\lambda\) is a partition of \(d(p+q)+b\) and \(\textbf{S}_{\lambda}\) is the corresponding Schur functor. Rmk
The final grading we will make use of is the \(\mathbb{Z}^{n+1}\)-multigrading on \(S\) that arises either by letting \(\text{deg}(x_i)=\textbf{e}_i\) where \(\textbf{e}_0,\ldots,\textbf{e}_n\)\) is the standard basis or by specializing the \(\textbf{GL}_{n+1}(\mathbb{C})\)-action to an action of the subtorus \((\mathbb{C}^*)^{n+1}\). This also descends to the Koszul cohomology groups and thus \[K_{p,q}(n,b;d)=\bigoplus_{\textbf{a}\in\mathbb{Z}^{n+1}}\textbf{k}(\textbf{a})^{\oplus\beta_{p,\textbf{a}}(b;d)},\] where \(\textbf{k}(\textbf{a})\) is the \((\mathbb{C}^*)^{n+1}\)-representation given by \((z_0,\cdots,z_{n+1})\cdot w=(z_0^{a_0}\cdots z_{n+1}^{a_{n+1}})w.\) We call this the multigraded decomposition of \(K_{p,q}(n,b;d)\) and refer to \(\beta_{p,\textbf{a}}(n,d;d)\) as multigraded Betti numbers. Rmk
The multigraded Betti numbers determine both the standard and Schur graded Betti numbers. For example, we can recover the standard graded Betti numbers from the multigraded via the following formula: \[ \beta_{p,q}(n,b;d)=\sum_{|\textbf{a}|=d(p+q)+b}\beta_{p,\textbf{a}}(n,b;d). \] Recovering the Schur graded Betti numbers is more complicated, as it relies on a highest weight decomposition algorithm (see Section 5.1 of our accompanying paper.). Hence the bulk of our computation is focused on the multigraded Betti numbers. See Approach page for a discussion of how we go about this.
All of what follows holds over any field of characteristic zero. Our choice of working over \(\mathbb{C}\) is simply one of convenience.
For notational hygiene, we often write \(K_{p,q}(n,b;d)\) for \(K_{p,q}(S(b;d))\). We likewise write \(\beta_{p,q}(n,b;d)\) for \(\beta_{p,q}(S(b;d))\).
One can think of a grading of a ring/module as a decomposition into irreducible representations arising from some group action. For example, the standard grading on \(S\) can be thought of coming from the natural \(\mathbb{C}^*\).
The weight of an element \(\lambda\in \mathbb{Z}^{n+1}\) is defined to be \(|\lambda|=\lambda_0+\lambda_1+\cdots+ \lambda_n\).
Notice that if \(\beta_{p,\textbf{a}}(n,b;d)\neq0\) then \(\textbf{a}\) must have weight \(d(p+q)+b\).