Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces.
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We calculate the minimal faithful permutation degree for a class of finite complex reflection groups G(p, p, q), for p and q primes, and demonstrate that they form examples where the minimal degree of a direct product is strictly less than the sum of the minimal degrees of the direct factors.
The minimal faithful permutation degree of a finite group G, denoted by .G/, is the least non-negative integer n such that G embeds inside Sym.n/. In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups. We also exhibit new examples of finite groups that possess a quotient whose minimal degree is strictly greater than that of the group. Johnson [4] and Wright [15] first determined conditions under which (1.1) is an equality, however they were unaware of examples where the inequality was strict. Indeed, in the closing remarks of [15], due to the absence of any examples, Wright Brought to you by | University of St Andrews Scotland Authenticated Download Date | 5/28/15 11:41 AM
Background results and examplesWe give a series of theorems and examples that we will use implicitly, but frequently, throughout the sequel. First, we give a theorem due to Karpilovsky [5], which also serves as an introductory example; the proof can be found in [4] or [5]. Brought to you by | University of St Andrews Scotland Authenticated Download Date | 5/28/15 11:41 AM Brought to you by | University of St Andrews Scotland Authenticated Download Date | 5/28/15 11:41 AM
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