Flows on Honeycombs and Sums of Littlewood-Richardson Tableaux

Abstract

Suppose \mu and \mu' are two partitions. We will let \mu \oplus \mu' denote the "direct sum" of the partitions, defined as the sorted partition made of the parts of $\mu$ and $\mu'$. In this paper, we define a summation operation on two Littlewood-Richardson fillings of type (\mu, \nu;\lambda) and (\mu', \nu';\lambda'), which results in a Littlewood-Richardson filling of type (\mu\oplus \mu', \nu\oplus \nu' ;\lambda\oplus \lambda'). We give an algorithm to produce the sum, and show that it terminates in a Littlewood-Richardson filling by defining a bijection between a Littlewood-Richardson filling and a flow on a honeycomb, and then showing that the overlay of the two honeycombs of appropriate type corresponds to the sum of the two fillings.