An explicit bijection between semistandard tableaux and non-elliptic sl_{3} webs

An explicit bijection between semistandard tableaux and non-elliptic webs

Heather M. Russell Department of Mathematics
University of Southern California

The spider is a diagrammatic category used to study the representation theory of the quantum group . The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov-Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and joins of webs. We build on Tymoczko’s bijection to give a simple and explicit algorithm for constructing all non-elliptic webs.

1. Introduction

The spider, introduced by Kuperberg [10] and subsequently studied by many others [8, 9, 11, 12], is a diagrammatic, braided monoidal category encoding the representation theory of . The objects in this category, called sign strings, are finite words in the alphabet including the empty word. The morphisms are - linear combinations of certain graphs called webs. See Figure 1 for an example of a web.

The objects in the spider can be thought of as tensor products of the two dual 3-dimensional irreducible representations and of , and the morphisms can be thought of as intertwining maps between tensor products of these representations [10]. Spider categories for other Lie types have been defined. See for instance [2, 11, 12].

Figure 1. A web in .

Webs in the spider are oriented trivalent graphs drawn in a rectangular region with boundary points lying on the top and bottom edges of that region. Edges incident on the boundary points have orientations compatible with the source and target sign strings. We read webs from bottom to top. All vertices are either sources or sinks. Webs are also subject to Relations 1, 2, and 3 below which are often referred to as the circle, bigon, and square relations respectively. A web with no bigons, squares, or circles is called non-elliptic or irreducible. Every web is a linear combination of non-elliptic webs. We follow the normalization conventions found in Khovanov’s work on link homology [7].