Journal of Combinatorics

Volume 6 (2015)

Number 3

Flows on honeycombs and sums of Littlewood–Richardson tableaux

Pages: 353 – 394

DOI: http://dx.doi.org/10.4310/JOC.2015.v6.n3.a6

Authors

Glenn Appleby (Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California, U.S.A.)

Tamsen Whitehead (Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California, U.S.A.)

Abstract

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.

Full Text (PDF format)

Published 4 June 2015