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A relation for the Jones-Wenzl projector and tensor space representations of the Temperley-Lieb algebra. (English) Zbl 1455.16024

Summary: A relation for the Jones-Wenzl projector is proven. It has the following consequence for representations of the Temperley-Lieb algebra on tensor product spaces: if such a representation is built from a Hermitian \(n\times n\) matrix \(T\) of rank \(r\) such that \(T^2=QT\), then either \(n^2=Q^2r\) and \(Q^2=1,2,3\) or \(n^2\geq 4r\). For the latter class of representations, new examples are found. This includes explicit examples for \(r=2,3,4\) and any \(n\geq r\) (with one exception) and a solution for \(n=r+1\) with arbitrary \(r\).

MSC:

16S99 Associative rings and algebras arising under various constructions
15A24 Matrix equations and identities
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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