The Isaacs–Navarro conjecture for the alternating groups

AbstractA recent refinement of the McKay conjecture is verified for the case of the alternating groups. The argument builds upon the verification of the conjecture for the symmetric groups [P. Fong, The Isaacs–Navarro conjecture for symmetric groups, J. Algebra 250 (1) (2003) 154–161]

The number of self-conjugate core partitions

A conjecture on the monotonicity of t-core partitions in an article of Stanton [Open positivity conjectures for integer partitions, Trends Math., 2:19-25, 1999] has been the catalyst for much recent research on t-core partitions. We conjecture Stanton-like monotonicity results comparing self-conjugate (t+2)- and t-core partitions of n. We obtain partial results toward these conjectures for values of t that are large with respect to n, and an application to the block theory of the symmetric and alternating groups. To this end we prove formulas for the number of self-conjugate t-core partitions of n as a function of the number of self-conjugate partitions of smaller n. Additionally, we discuss the positivity of self-conjugate 6-core partitions and introduce areas for future research in representation theory, asymptotic analysis, unimodality, and numerical identities and inequalities.Comment: 17 pages, 2 figures, to appear in Journal of Number Theory Updated to accepted version. Removed previously known results about self-conjugate 7-cores and corrected other historical informatio