2025/02/01 by Gao, Mengmeng, Rui, Hebing
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2502.00420
Following Nazarov's suggestion~\citeNaz1, we refer to the cyclotomic Nazarov-Wenzl algebra as the cyclotomic Brauer algebra. When the cyclotomic Brauer algebra is isomorphic to the endomorphism algebra of MIi, r-- the tensor product of a simple scalar-type parabolic Verma module with the natural module in the parabolic BGG category \mathcal O of types Bn, Cn and Dn, its decomposition numbers can theoretically be computed, based on general results from \citeAST and \cite[Corollary~5.10]RS. This paper aims to establish explicit connections between the parabolic Verma modules that appear as subquotients of MIi, r and the right cell modules of the cyclotomic Brauer algebra under condition~\eqrefsimple111. It allows us to explicitly decompose MIi, r into a direct sum of indecomposable tilting modules by identifying their highest weights and multiplicities. Our result demonstrates that the decomposition numbers of such a cyclotomic Brauer algebra can be explicitly computed using the parabolic Kazhdan-Lusztig polynomials of types Bn, Cn, and Dn with suitable parabolic subgroups~\citeSo. Finally, condition~\eqrefsimple111 is well-supported by a result of Wei Xiao presented in Section~6.