Kleinian singularities: some geometry, combinatorics and representation theory (Q6646758)

Kleinian singularities, or quotients of \(\mathbb{C}^2\) by discrete finite subgroups of \(SL(2; \mathbb{C})\), have provided a wealth of results in algebraic geometry and representation theory. A classic review is by Peter Slodowy from 1980. This current paper is a marvellous modern and rather pedagogical review of the subject, with injection of several new results and ideas.\N\NIt begins with elements of discrete groups of symmetries of Euclidean three-space and constructions in algebraic geometry around Kleinian singularities, before turning to the contemporary perspective of Hilbert and Quot Schemes, as well as relating the singularities to finite-dimensional and affine Lie algebras via the McKay ADE correspondence. Emphasis is paid to a combinatorial outlook where one considers generating functions of certain coloured partitions, which is in one-to-one correspondence with (fixed points on) the Hilbert scheme. The Abelian (type A) case is introduced in detail and recent results on types D and E are also presented.