Research
Preprints
Enumeration of Partitions modulo 4
We study the modular properties of the degrees of the irreducible representations of symmetric groups as well as alternating groups. The modulo 2 behavior of the irreducible representations of symmetric groups is well-known, due to Macdonald and McKay. We extend these results modulo 4 for symmetric/alternating groups on n letters for some natural numbers n.Â
MS Thesis (submitted May 2022)
Counting Cores and Bar cores: From Modular Forms to McKay numbers (You can request a copy here or email me!)
In this thesis, we delve into the theory of cores of partitions and tackle two main problems: the enumeration of t-cores and t-bar cores, and the computation of McKay numbers for the symmetric and alternating groups. For the enumeration problem, we discuss new and known explicit results for small values of t and bounds for general values of t which are obtained through the theory of modular forms. We also present new generating functions for t-bar cores in the case when t is even. For the computation of McKay numbers, we invoke the theory of p-core towers, for primes p, which serves as a direct application of the topic of enumeration of p-cores to other combinatorial problems. We also resolve the values for p=2 further and study them modulo 4. This thesis presents itself as a survey of existing results in literature that have paved the way towards solving the two problems and as a presentation of original contributions in the same direction.