Decomposition numbers of the cyclotomic Brauer algebra over the complex field, II

TL;DR

This work computes decomposition numbers for the cyclotomic Brauer algebra $B_{k,r}((-1)^k\mathbf u_k)$ over $\mathbb{C}$ with arbitrary parameters by embedding the level-$k$ algebra as an idempotent truncation of a higher-level algebra and realizing it as an endomorphism algebra in a parabolic category $\mathcal{O}$ of type $D_n$. The main bridge is that decomposition numbers are governed by parabolic Kazhdan–Lusztig polynomials for type $D_n$ with a parabolic $A$, under a key admissibility condition (Condition 1.2 / $\mathbf u_k$-admissibility) and a saturation hypothesis. By constructing a simple tilting parabolic Verma module $M^{\mathfrak p}(\lambda_{\mathbf c})$ and matching $B_{2k,r}(\mathbf u_{2k})$-endomorphisms to endomorphisms in $\mathcal{O}^{\mathfrak p}$, the authors reduce decomposition numbers to graded multiplicities $(T^{\mathfrak p}(\mu): M^{\mathfrak p}(\lambda))$, expressible via parabolic KL polynomials. Under a nonvanishing $\omega$-condition (when $r$ is even) the results yield explicit formulas for all decomposition numbers in terms of the parabolic KL data, providing a substantial Lie-theoretic interpretation of the cyclotomic Brauer algebra’s representation theory.

Abstract

Following Nazarov's suggestion, the cyclotomic Nazarov-Wenzl algebra is referred to as the cyclotomic Brauer algebra. This paper focuses on computing the decomposition numbers of the cyclotomic Brauer algebra over $\mathbb{C}$ with arbitrary parameters. We show that these decomposition numbers can be expressed in terms of the parabolic Kazhdan-Lusztig polynomials of type $D_n$, with a parabolic subgroup of type $A$, under Condition 1.2.

Theorems & Definitions (45)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Conjecture 1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4