The modular Temperley-Lieb algebra. (English) Zbl 1536.16011
The paper under review studies the composition multiplicities of the simple modules in standard and projective modules over the Temperley-Lieb algebra \(\operatorname{TL}_n(\delta)\) over a field of positive characteristic. In fact, closed formulas are obtained for these. Further, this is then used to derive a formula for the dimension of the simple modules.
The initial step is to use the truncation functors in order to reduce the problem to determining the multiplicity of the simple module \(L(n, n)\) in the standard modules \(S(n, r)\). A lower bound is first established using certain linear maps called candidate morphisms, extending ideas from J. J. Graham and G. I. Lehrer [Enseign. Math. (2) 44, No. 3–4, 173–218 (1998; Zbl 0964.20002)]. Subsequently, it is shown that this is also an upper bound, using restriction to \(\operatorname{TL}_{n-1}(\delta)\), as well as a certain central element \(F_n\) defined in [J. Belletête and Y. Saint-Aubin, J. Math. Phys. 55, No. 11, 111706, 41 p. (2014; Zbl 1348.82029)].
The paper concludes by describing connections and potential applications to the theory of Soergel bimodules in positive characteristic and Jones-Wenzl idempotents in mixed characteristic.
The initial step is to use the truncation functors in order to reduce the problem to determining the multiplicity of the simple module \(L(n, n)\) in the standard modules \(S(n, r)\). A lower bound is first established using certain linear maps called candidate morphisms, extending ideas from J. J. Graham and G. I. Lehrer [Enseign. Math. (2) 44, No. 3–4, 173–218 (1998; Zbl 0964.20002)]. Subsequently, it is shown that this is also an upper bound, using restriction to \(\operatorname{TL}_{n-1}(\delta)\), as well as a certain central element \(F_n\) defined in [J. Belletête and Y. Saint-Aubin, J. Math. Phys. 55, No. 11, 111706, 41 p. (2014; Zbl 1348.82029)].
The paper concludes by describing connections and potential applications to the theory of Soergel bimodules in positive characteristic and Jones-Wenzl idempotents in mixed characteristic.
Reviewer: Anna Lucia Rodriguez Rasmussen (Uppsala)
MSC:
| 16G99 | Representation theory of associative rings and algebras |
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