Flows on Honeycombs and Sums of Littlewood-Richardson Tableaux

TL;DR

This paper introduces a new summation operation on Littlewood-Richardson tableaux, connecting it to flows on honeycombs, with an algorithm and bijection demonstrating the operation's properties.

Contribution

It defines a novel sum for Littlewood-Richardson fillings and links it to honeycomb flows, providing an explicit algorithm and bijection for the operation.

Findings

The summation operation produces valid Littlewood-Richardson fillings.

A bijection between fillings and honeycomb flows is established.

Overlay of honeycombs corresponds to the sum of fillings.

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.