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 S_n. Recently, Nath and Sellers, and Xia established a number of congruences modulo 2, 3, and 5 for the numbers of spin characters of ${\widehat{S}}_n$ and ${\widehat{A}}_n$, where ${\widehat{S}}_n$ and ${\widehat{A}}_n$ are the double covering group of S_n and the double covering group of the alternating group A_n, 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 A_n. In particular, we prove some strange congruences modulo 5 for a(n). For example, we prove that for $k\ge 0$, $$a\left(\frac{95\times {73}^{2k}+1}{24}\right)\equiv -3^k\hspace{1em}\left(\mathrm{\text{mod}}5\right).$$

KEYWORDS:

• partition
• congruence
• irreducible character
• alternating group

MATHEMATICS SUBJECT CLASSIFICATION:

• 11P83
• 05A17

Authors’ contributions

All authors read and approved the final manuscript. The first author wrote the original draft of the manuscript. The second and third authors review and edit the manuscript.

Data availability statement

Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.

Declaration of Interest

The authors declared that they have no conflicts of interest to this work.

Additional information

Funding

This work was supported by the Natural Science Foundation of Jiangsu Province of China (BK20221383 and BK20200267) and Qinglan Project.