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:**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

**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

**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 \(d(p+q)+b\).