Decomposition numbers of cyclotomic Brauer algebras over the complex field, I

TL;DR

This paper establishes explicit formulas for the decomposition numbers of cyclotomic Brauer algebras over the complex field, linking them to parabolic Kazhdan-Lusztig polynomials and module decompositions in type B, C, D categories.

Contribution

It connects the representation theory of cyclotomic Brauer algebras with parabolic Kazhdan-Lusztig polynomials, providing explicit decomposition formulas and module structure insights.

Findings

Decomposition numbers can be computed via parabolic Kazhdan-Lusztig polynomials.

Explicit module decompositions into tilting modules are established.

Condition~ ef{simple111} is validated by Wei Xiao's result.

Abstract

Following Nazarov's suggestion~\cite{Naz1}, 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 $M_{I_i, r}$-- the tensor product of a simple scalar-type parabolic Verma module with the natural module in the parabolic BGG category $\mathcal O$ of types $B_n$, $C_n$ and $D_n$, its decomposition numbers can theoretically be computed, based on general results from \cite{AST} and \cite[Corollary~5.10]{RS}. This paper aims to establish explicit connections between the parabolic Verma modules that appear as subquotients of $M_{I_i, r}$ and the right cell modules of the cyclotomic Brauer algebra under condition~\eqref{simple111}. It allows us to explicitly decompose $M_{I_i, r}$ 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 $B_n$, $C_n$, and $D_n$ with suitable parabolic subgroups~\cite{So}. Finally, condition~\eqref{simple111} is well-supported by a result of Wei Xiao presented in Section~6.