Simple modules for Temperley-Lieb algebras and related algebras (Q1632461)

The main result of the paper under review gives algorithms which determine the dimensions of the simple modules of the Temperley-Lieb algebra \(TL_n(q + q^{-1})\) over any field \(k\) and for any value of \(q \in k\backslash \{0\}\). This extends previous results of [\textit{K. Iohara} et al., Math. Res. Lett. 26, No. 1, 121--158 (2019; Zbl 1448.17020)]. The algorithms presented in this paper are based on the fact that \(TL_n(q + q^{-1})\) is the endomorphism ring of the \(n\)th tensor power \(V_q^{\otimes n}\), where \(V_q\) is the natural 2-dimensional module of \(U_q({sl}_2)\) and, by previous results of the author et al. [Pac. J. Math. 292, No. 1, 21--59 (2018; Zbl 1425.17005)], it is a cellular algebra in the sense of \textit{J. J. Graham} and \textit{G. I. Lehrer} [Invent. Math. 123, No. 1, 1--34 (1996; Zbl 0853.20029)]. The algorithms are then obtained from knowledge of the characters of the (tilting) indecomposable summands of \(V_q^{\otimes n}\). When \(q\) is a root of unity, the dimensions of the simple modules for the Jones quotients \(Q_n(q+q^{-1})\) are recovered as part of the algorithm for \(TL_n(q + q^{-1})\), also extending previous results of the above mentioned paper of Iohara, Lehrer and Zhang. It is also shown that the methods developed in the paper allow to determine as well the dimensions of some simple modules for BMW-algebras with special parameters and other algebras related with endomorphism algebras of families of tilting modules for \(U_q({sl}_2)\).