Asymptotics of McKay numbers for

✍ Scribed by Daniel M. Kane

• Book ID: 104024789
• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 257 KB
• Volume: 124
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.08.011

✦ Synopsis

For a partition Λ of n, let H (Λ) denote its hook product. If is prime and a 0 an integer, then define

These numbers are simply related to the McKay numbers in the representation theory of the symmetric group. Using a generating function of Nakamura and the "circle method," we determine asymptotic properties of p (a; n) and a (-1) a p (a; n), resolving questions of Ono. In particular we show that for fixed and n, p (a; n) roughly fits a given distribution that is dependent on , is centered near nc 1 √ n log n and has width c 2 √ n. We also give an asymptotic formula for a (-1) a p (a; n) that is valid whenever √ n is not, for any k, within a multiplicative factor of c log of k . This formula is of the form ±c(n)/n exp(κ(n)

√ n ) where c and κ are specific functions of n and the sign is determined by n.