On the semi-simplicity of the cyclotomic Brauer algebras.
A fundamental result for Brauer algebras has been \textit{H. Wenzl}'s proof [Ann. Math. (2) 128, No. 1, 173-193 (1988; Zbl 0656.20040)] of the Hanlon-Wales conjecture: Over the complex numbers, a Brauer algebra is semisimple unless the parameter \(\delta\) takes one of finitely many integer values. Cyclotomic Brauer algebras (and their quantizations) were introduced by \textit{R. Häring-Oldenburg} [J. Pure Appl. Algebra 161, No. 1-2, 113-144 (2001; Zbl 1062.20004)]. These generalize Brauer algebras in the same way as cyclotomic (or Ariki-Koike) Hecke algebras generalize Hecke algebras of symmetric groups. The article under review generalizes Wenzl's result to the case of cyclotomic Brauer algebras. Here, there are finitely many parameters, and the condition for semisimplicity is in terms of certain polynomials in these parameters to avoid finitely many integer values. The proof first provides a number of results on cyclotomic Hecke algebras, in particular a Littlewood-Richardson rule. Then it is shown that cyclotomic Brauer algebras have a cellular structure similar to that of Brauer algebras (as in [\textit{S. König, C.-C. Xi}, Trans. Am. Math. Soc. 353, No. 4, 1489-1505 (2001; Zbl 0996.16009)]). A branching rule for cell (= Specht) modules is proved and induction and restriction is shown to have properties similar to the case of Brauer algebras (as in [J. Algebra 211, No. 2, 647-685 (1999; Zbl 0944.16002)]). With these preparations, the strategy of proof for Wenzl's theorem developed by \textit{W. F. Doran IV, D. B. Wales} and \textit{P. J. Hanlon} (Zbl 0944.16002) can be used.
scientific article; zbMATH DE number 2102475
scientific article