3:30pm-3:55pm in PDL C-401
The number of standard tableaux of a given shape and major index \$r\$ mod \$n\$ give the irreducible multiplicities of certain induced or restricted representations. We give simple necessary and sufficient conditions classifying when this number is zero. This result generalizes the \$r=1\$ case due essentially to Klyachko (1974) and proves a recent conjecture due to Sundaram (2016) for the \$r=0\$ case. Indeed, we prove a stronger asymptotic uniform distribution result for "almost all'' shapes.
We'll discuss aspects of the proof, including a representation-theoretic formula due to Desarmenien, normalized symmetric group character estimates due to Fomin-Lulov, and new techniques involving "opposite hook lengths'' for classifying \$\lambda\$ where \$f^\lambda \leq n^d\$ for fixed \$d\$.