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

Mengmeng Gao, Hebing Rui

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.

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

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 -- the tensor product of a simple scalar-type parabolic Verma module with the natural module in the parabolic BGG category of types , and , 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 and the right cell modules of the cyclotomic Brauer algebra under condition~\eqref{simple111}. It allows us to explicitly decompose 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 , , and with suitable parabolic subgroups~\cite{So}. Finally, condition~\eqref{simple111} is well-supported by a result of Wei Xiao presented in Section~6.

Paper Structure

This paper contains 12 sections, 35 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. The cyclotomic Brauer algebra
  3. Cyclotomic Brauer algebras
  4. Degenerate cyclotomic Hecke algebras
  5. A weakly cellular basis of $\mathcal{B}_{a, r}(\mathbf u)$
  6. parabolic category $\mathcal{O}$ in types $B_n, C_n$ and $D_n$
  7. The symplectic and orthogonal Lie algebras
  8. Parabolic category $\mathcal{O}^\mathfrak p$
  9. Tensor modules in $\mathcal{O}^{\mathfrak p_{I_i}}$
  10. Classification of singular vectors in $M_{I_i, r}/M_{I_i, r} \langle E^f\rangle$
  11. Proof of Theorem \ref{['main4']} and Theorem \ref{['first']}
  12. Appendix: Proof of Theorem \ref{['saturated']} by Wei Xiao

Key Result

Theorem A

RS Suppose $\Phi\neq B_n$ if $i=1$, and $M^{\mathfrak p_{I_i}}({\lambda_{I_i, \mathbf c} })$ is simple (and hence tilting). If $p_t-p_{t-1}\ge 2r$ for all $1\le t\le k$, then $\text{End}_{\mathcal{O}^{\mathfrak p_{I_i}}}(M_{I_i, r})\cong \mathcal{B}_{a, r}^{\text{op}}(\mathbf u)$. Here $\mathcal{B}_

Theorems & Definitions (66)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • ...and 56 more