Decomposition numbers of cyclotomic Brauer algebras over the complex field, I
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2025
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| _version_ | 1866917908527972352 |
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| author | Gao, Mengmeng Rui, Hebing |
| author_facet | Gao, Mengmeng Rui, Hebing |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_00420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposition numbers of cyclotomic Brauer algebras over the complex field, I Gao, Mengmeng Rui, Hebing Representation Theory 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. |
| title | Decomposition numbers of cyclotomic Brauer algebras over the complex field, I |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2502.00420 |