##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Josh Swanson

#### University of Washington

## On the Existence of Tableaux with Given Modular Major Index

##### Abstract:

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 $\lambda \vdash n$ where $f^\lambda \leq n^d$ for fixed $d$.

Host: Brendon Rhoades

### February 28, 2017

### 3:00 PM

### AP&M 7321

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