Puzzles, Tableaux and Mosaics

TL;DR

This paper introduces mosaics, a new combinatorial object, establishing bijections with puzzles and tableaux, and provides simplified proofs of key algebraic properties like commutativity and associativity in representation theory.

Contribution

It defines mosaics and their operations, linking puzzles and tableaux, and offers new, simpler proofs of fundamental algebraic rules such as the Littlewood-Richardson rule.

Findings

Mosaics are in bijection with Knutson-Tao puzzles.

Mosaics are also in bijection with Littlewood-Richardson skew-tableaux.

The paper provides bijective proofs of commutativity and associativity for related ring structures.

Abstract

We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is that we obtain bijective proofs of commutativity and associativity for the ring structures defined either of these objects. In particular, we obtain a new, easy proof of the Littlewood-Richardson rule. Finally we discuss how our operation is related to other known constructions, particularly jeu de taquin.