Flows on honeycombs and sums of Littlewood–Richardson tableaux

Suppose $\mu$ and $\mu^\prime$ are two partitions. We will let $\mu \oplus \mu^\prime$ denote the direct sum of the partitions, defined as the sorted partition made of the parts of $\mu$ and $\mu^\prime$. In this paper, we define a summation operation on two Littlewood–Richardson fillings of type $(\mu , \nu ; \lambda)$ and $(\mu^\prime , \nu^\prime ; \lambda^\prime)$, which results in a Littlewood–Richardson filling of type $(\mu \oplus \mu^\prime , \nu \oplus \nu^\prime ; \lambda \oplus \lambda^\prime)$. 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.

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Published 4 June 2015