Enumerating Representations in Finite Wreath Products II: Explicit Formulae

✍ Scribed by Thomas W. Müller; John Shareshian

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 434 KB
• Volume: 171
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2002.2082

✦ Synopsis

We present several contributions to the enumerative theory of wreath product representations developed in a previous paper by the first named author (Adv. in Math. 153 (2000), 118-154). Theorem 3.1 of the present paper establishes an explicit formula for one of the key ingredients in the description of the corresponding generating functions given in M .

u uller (2000) (the exterior function F G ). Building on Theorem 1 in M .

u uller (2000) and the latter result, we derive explicit formulae for the exponential generating function of the series fjHomðG; R n Þjg in the case where G is dihedral or a finite abelian group, and the representation sequence fR n g is any of fHwS n g or fHwA n g with a fixed finite group H; or the sequence fW n g of Weyl groups of type D n : Moreover, we verify a conjecture concerning the asymptotic behaviour of the sequence fjHomðG; W n Þjg for finite groups G made in M .

u uller (2000) in the case when G is dihedral or abelian.