Research Projects

This project aims to study word maps on groups and polynomial maps on Lie algebras using techniques from algebraic combinatorics, probability theory, computational algebra, and representation theory. In 2014, Lubotzky characterized word images for finite simple groups. In a recent work with Kaur and Kulshrestha, we showed that this characterization cannot be extended to non-simple groups. This posed the problem of characterizing word images for finite nilpotent groups, or in general for non-simple groups. As a part of this project, I aim to solve this problem. Each word map on a group induces a probability distribution on the group. Many mathematicians have studied these distributions in the context of the Amit-Ashurst conjecture. Kulshrestha and I have established several results concerning this conjecture for finite nilpotent groups of nilpotency class 2. I aim to extend our work to groups of higher nilpotency classes. I also wish to define explicitly the set of these probability distributions for extraspecial p-groups. This would help anticipate possible behaviour of these distributions for nilpotent groups in general, and thus will lead to a better understanding of these groups. It has been a problem of great interest to determine whether the set of commutators in a group is equal to its commutator subgroup. A similar well-studied problem for a Lie algebra is to check the equality of its derived subalgebra with the image set of the Lie bracket. We realised that most of the existing results for both problems pose several restrictions. In a series of two papers, Prof. Kulshrestha and I have developed a uniform and novel approach to determine the image of the commutator word map on nilpotent groups of class 2 and the Lie bracket map on Lie algebras. This approach relates these algebraic problems to a combinatorial problem of providing a consistent labeling on weighted graphs over local rings. The results thus obtained are free from existing restrictions. I believe that using our theory, more exciting results can be obtained. First, I aim to characterize commutators in finite simple Lie algebras. Additionally, I wish to characterize commutators for nilpotent groups of small order and Lie algebras of small dimension. Another problem that is also closely related to word maps is the description of products of conjugacy classes in a finite group. A lot of attempts have been made to determine the smallest integer k (depending on m) such that each element of an alternating group A_n is a product of cycles of length m. In collaboration with Kundu and Mishra, we have solved this problem completely. I aim to extend our results to more conjugacy classes of A_n. I also wish to describe explicitly the product of conjugacy classes of derangements in both S_n and A_n, as well as in simple transitive permutation groups in general. As a part of this project, I also plan to work on the problem of generating permutation groups by conjugating elements.