Infinite Families of Congruences Modulo 5 for the Number of Irreducible Characters of the Alternating Groups

Abstract

In 1919, Ramanujan proved red three congruences for the partition function p(n) which denotes the number of partitions of n. The partition function p(n) can be understood as the number of irreducible characters of the symmetric group Sn . Recently, Nath and Sellers, and Xia established a number of congruences modulo 2, 3, and 5 for the numbers of spin characters of S ̂ n and A ̂ n , where S ̂ n and A ̂ n are the double covering group of Sn and the double covering group of the alternating group An , respectively. Motivated by their work, we establish infinite families of congruences modulo 5 for a(n), which denotes the number of irreducible characters of the alternating group An . In particular, we prove some strange congruences modulo 5 for a(n). For example, we prove that for k ≥ 0 , a ( 95 × 73 2 k + 1 24 ) ≡ − 3 k ( mod 5 ) .