PROJECTIVE REPRESENTATIONS AND SPIN CHARACTERS OF COMPLEX REFLECTION GROUPS

This volume consists of one expository paper and two research papers:T Hirai, A Hora and E Hirai, Introductory expositions on projective representations of groups (referred as [E])T Hirai, E Hirai and A Hora, Projective representations and spin characters of complex reflection groups G(m,p,n) and G(m,p, ), I;T Hirai, A Hora and E Hirai, Projective representations and spin characters of complex reflection groups G(m,p,n) and G(m,p, ), II, Case of generalized symmetric groups.Since Schur's trilogy on 1904 and so on, many mathematicians studied projective representations of groups and algebras, and also of their characters. Nevertheless, to invite mathematicians to this interesting and important areas, the paper [E] collects introductory expositions, with a historical plotting, for the theory of projective representations of groups and their characters. The paper [I] treats general theory for projective (or spin) representations and spin characters of complex reflection groups G(m,p,n) and G(m,p, )=lim n G(m,p,n), and clarifies the intimate relations between mother groups, G(m,1,n), G(m,1, )(p=1), called generalized symmetric groups, and their child groups, G(m,p,n), G(m,p, ),(p m, p> 1). Also we treat explicitly a case of spin type in connection with the case of non-spin type (i.e. of linear representations). A detailed and general account on the so-called Vershik-Kerov theory on limits of characters is added. The paper [II] treats spin irreducible representations and spin characters of generalized symmetric groups (mother groups) for other spin types.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets