Researcher ORCID Identifier
Open Access Senior Thesis
Bachelor of Science
2023 Ian D Shors
My thesis examines a class of functions on the symmetric group called permutation statistics using tools from representation theory. In 2014, Axel Hultman gave formulas for computing expected values of permutation statistics sampled via random walks. I present analogous formulas for computing variances of these statistics involving Kronecker coefficients – certain numbers that arise in the representation theory of the symmetric group. I also explore deep connections between the study of moments of permutation statistics and the representation theory of the partition algebras, a family of algebras introduced by Paul Martin in 1991. By harnessing these partition algebras, I derive a new polynomial describing the mean statistic of the 2nd moment of the number of inversions of a permutation.
Shors, Ian, "Permutations, Representations, and Partition Algebras: A Random Walk through Algebraic Statistics" (2023). HMC Senior Theses. 277.