Alladi Ramakrishnan Hall

Odd dimensional representations in Young's lattice

Amritanshu Prasad

IMSc

Young's lattice is the set of integer partitions partially ordered by containment of Young diagrams. To each integer partition is associated an f-number, which can be though of as the number of standard Young tableaux whose shape is the given partition, or as the dimension of the irreducible representation of the symmetric group associated to the given partition. This number can be calculated using the stunning hook length formula of Frame, Robinson and Thrall.

Macdonald showed that the number of integer partitions of n with odd f-number is an easily calculated power of two. Digging a little deeper in Macdonald's result, we discovered that inside Young's lattice, the odd dimensional representations form an incomplete binary tree. Moreover, this tree exhibits a fractal structure and has a nice recursive description.

This talk is based on joint work with Arvind Ayyer and Steven Spallone.