On generators and relations of the rational cohomology of Hilbert schemes (Q6100543)

The authors study the presentation of the algebra \(\mathcal A({d})\) (the rational cohomology algebra of the \(d\)th Hilbert scheme of \(\mathbb C^{2}\)) in terms of generators and relations. As shown by \textit{M. Lehn} and \textit{C. Sorger} [Duke Math. J. 110, No. 2, 345--357 (2001; Zbl 1093.14008)], the algebra \(\mathcal A(d)\) admits a combinatorial description as the associated graded algebra of a certain filtration on the rational group algebra \(\mathbb Q[\mathfrak S({d})]\) of the symmetric group \(\mathfrak S({d}) \). Moreover, \(\mathcal A(d)\) is the subalgebra of invariants of this action. \(\mathcal A(d)\) is a finite-dimensional, commutative graded \(\mathbb Q\)-algebra. It is concentrated in even degrees. Quoting from the authors' abstract: ``We determine two distinct, minimal sets of \(\lfloor d/2 \rfloor\) multiplicative generators of \(\mathcal A(d)\). Additionally, we prove when the lowest degree generating relations occur. For small values of \(d\), we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for \(\mathcal A(d)\).''