In the present note the authors study the asymptotic behaviour of Veronese syzygies as representations of the general linear group. In order to formulate the main result let us now recall some notions.
Let \(\mathbb{P} = \mathbb{P}(V)\) be the projective plane. Recall that the \(d\)-th Veronese embedding of \(\mathbb{P}\) is \[ \mathbb{P} \hookrightarrow X_{d} \subseteq \mathbb{P}(\mathrm{Sym}^{d}(V)) \] defined by all monomials of degree \(d\). Denote by \[ S = \bigoplus_{i=0}^{\infty} \mathrm{Sym}^{i}\mathrm{Sym}^{d}(V), \,\,\,\, R = \bigoplus_{i=0}^{\infty}\mathrm{Sym}^{\mathrm{id}}(V) \] the coordinate ring of the Veronese ambient space and the homogeneous coordinate ring of the Veronese image, respectively. Consider the free graded \(S\)-resolution \[ 0 \leftarrow R \leftarrow F_{0} \leftarrow\dots \leftarrow F_{r} \leftarrow 0, \] where \[ F_{p} = \bigoplus_{j} S(-a_{p,j}) \] are free graded modules over \(S\). In order to keep track of the gradings one considers the finite dimensional vector spaces \(K_{p,q}(d)\) (i.e. the so-called Koszul cohomologies) and writes \[ F_{p} = K_{p,q}(d) \otimes S(-p-q). \] M. L. Green in [J. Differ. Geom. 20, 279–289 (1984; Zbl 0559.14009)] initiated geometrical studies of syzygies and he showed that for the \(d\)-th Veronese embedding one has \[ K_{p,q}(d) = 0 \,\,\, \text{if} \,\,\, q \geq 2 \,\,\, \text{and} \,\,\, d \geq p. \] In the case of Veronese embeddings the syzygies \(K_{p,q}(d)\) are representations of the general linear group \(\mathrm{GL}(V)\) and one may ask how the syzygies decompose into irreducible representations. It is difficult to expect that it is possible to determine the exact decomposition, thus looking at the asymptotic picture seems to be more accessible. Another result of Green stays that the whole \(p\)-th syzygy space \[ K_{p}(d) = \bigoplus_{q = - \infty}^{\infty}K_{p,q}(d) \] is captured by the weight \(q=1\) and for \(d \geq p\). The main result of the paper under review describes the asymptotics of \(K_{p,1}(d)\). More precisely the authors show the following.
Theorem.
Pick \(p \geq 1\) and assume that \(\dim V \geq p+1\). Then as \(d\) grows, \(K_{p,1}(d)\) contains
i) exactly on the order of \(d^{p}\) distinct irreducible representations,
ii) exactly on the order of \(d^{ {p+1 \choose 2} }\) irreducible representations counting multiplicities.