Connections between discriminants of complex reflection groups and their representation theory

The following thesis explores an extension to the classical McKay correspondence, a theorem that touches on several areas of mathematics. Our extension comes by considering pseudo reflection groups which were not included in the original correspondence. The discriminant of a pseudo reflection group is a singular hypersurface expressed by a polynomial ∆ in the invariant ring of the group action. A main object of study is the matrix factorization (z, j) of ∆, and the corresponding Cohen-Macaulay module, arising from the arrangement of the hyperplanes fixed by the reflections. A key idea that we frequently use is that the matrix factorization (z, j) can be decomposed using the irreducible representations of the group. A McKay correspondence of refection groups generated by reflections of order 2 has been presented by Buchweitz–Faber–Ingalls. In Chapter 3 we follow the methods from Loc. cit and consider the complex reflection groups G(m, p,2), which appear in the Shephard–Todd classification and show that similar results hold. The matrix factorization is fully decomposed and the corresponding decomposition of the Cohen-Macaulay module is given. A collaboration with Eleonore Faber, Colin Ingalls and Marco Talarico has resulted in a description of the decomposition of the matrix factorization (z, j) of ∆ for the symmetric group Sn on n letters. In Chapter 4 a modification of higher Specht polynomials is used to present a computational way to explicitly calculate the decomposition of the corresponding Cohen-Macaulay module of (z, j). In Chapter 5, the Lusztig algebra for the pseudo reflection group G(m,1,2) which is Morita equivalent algebra to the skew group ring is calculated. This is achieved by using the McKay quivers for G(m,1,2) and calculating the required relations in terms of 2-paths. The Lusztig algebra can give us insights into how the McKay correspondence can generalise to pseudo reflection groups