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From Notices of the AMS

The Symmetric Group through a Dual Perspective

A combinatorial view of the decomposition of $V^{\otimes 3}$ into $GL_9$ representations by primary colors and then the restriction of those into $S_9$ representations by the different shades of those regions.

by Rosa Orellana
Mike Zabrocki
Communicated by Emilie Purvine

Mathematicians began the study of representation theory over a hundred years ago. Since then it has become a centerpiece technique in fields such as algebra, topology, number theory, geometry, mathematical physics, quantum information theory, and complexity theory. A premise of representation theory is that we can study groups and algebras from how they act on vector spaces.

In this article we take this a step further; to study actions of a group or algebra we study what commutes with the action. The collection of all linear transformations that commute with the action is called the commutant or the centralizer. The centralizer is itself an algebra which is called the Schur-Weyl dual.

A reason why this has become such an important technique is that it can lead to beautiful connections between seemingly different areas of mathematics. One example of this is the discovery of the Jones polynomial. This polynomial is a one variable invariant for oriented knots or links [6][7].

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