no. 3
Calibrated representations of two boundary Temperley–Lieb algebras
Daugherty, Zajj1; Ram, Arun2
1 Dept. of Mathematics and Statistics, Reed Colleg, Portland OR 97202 (USA)
2 School of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 Australia
Annals of Representation Theory, Volume 2 (2025) no. 3, pp. 405-438.

The two boundary Temperley–Lieb algebra $TL_k$ is a quotient of the type $C_k$ affine Hecke algebra $H_k$. The algebra $H_k$ has a diagrammatic presentation by braids with $k$ strands and two poles and $TL_k$ has a presentation via non-crossing diagrams with boundaries. The algebra $TL_k$ plays a role in the analysis of Heisenberg spin chains with boundaries. A calibrated representation of $TL_k$ is a $TL_k$-module for which all the Murphy elements (integrals) are simultaneously diagonalizable. In this paper we give a combinatorial classification and construction of all irreducible calibrated $TL_k$-modules and explain how these modules also arise from a Schur–Weyl duality with the quantum group $U_q\mathfrak{gl}_2$.

Received:
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Accepted:
Published online:
DOI: 10.5802/art.26
Classification: 20C08, 17B10, 17B37, 05E10
Keywords: Temperley–Lieb algebras, representations, Schur–Weyl duality

Daugherty, Zajj 1; Ram, Arun 2

1 Dept. of Mathematics and Statistics, Reed Colleg, Portland OR 97202 (USA)
2 School of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 Australia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Daugherty, Zajj; Ram, Arun. Calibrated representations of two boundary Temperley–Lieb algebras. Annals of Representation Theory, Volume 2 (2025) no. 3, pp. 405-438. doi : 10.5802/art.26. https://art.centre-mersenne.org/articles/10.5802/art.26/

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