CEU Electronic Theses and Dissertations, 2023
| Author | Madireddi, Sai Praveen |
|---|---|
| Title | An insight into the Foulkes conjecture and the Generalized Foulkes module |
| Summary | For integers, $a,b$, the Foulkes moudule $F_{(b)}^{(a)}$, is the permutation module of the symmetric group, $S_{ab}$ acting on partitions of the $ab$ elements into $b$ sets of size $a$ each. In 1950, Foulkes conjectured that if $b > a$ then $F_{(a)}^{(b)}$ is a submodule of $F_{(b)}^{(a)}$. The Foulkes conjecture and the properties of the Foulkes module have been of interest for researchers in both the fields of Algebraic Combinatorics and Representation theory of Symmetric groups. Another topic of interest is the Generalized Foulkes module where we generalize $F_{(b)}^{(a)}$ with the parameter being the partitions of $b$. In Chapter 3 we restrict the generalized Foulkes module to some large subgroups and in turn find a weak connection to Kronecker coefficients. We also prove a corollary that is based on this connection and obtain a result on the multiplicities of certain irreducible modules in the generalized Foulkes module. In Chapter 4 we compare the Foulkes character values on permutations of prime power order $p^k$. We prove that the character value decays exponentially as $k$ increases. We also prove that if $ab$ is a prime power then the restriction of Foulkes conjecture to the group generated by a long cycle is true. In Chapter 5 we compute the Foulkes character on involutions and examine if the conjecture is true when we restrict it to elementary abelian subgroups. |
| Supervisor | Hegedus, Pal |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2023/madireddi_sai.pdf |
Visit the CEU Library.
© 2007-2021, Central European University