The saturated condition for decomposition numbers of the cyclotomic Brauer algebra over the complex field: Conjecture 1 of Rui–Song

Over the complex field, the decomposition numbers of the cyclotomic Brauer algebra (equivalently, the cyclotomic Nazarov–Wenzl algebra) $B_{k,r}(((-1)^k\mathbf{u}_k))$ with arbitrary parameters were reduced by Rui and Song (2025), via a realization in parabolic BGG category $\mathcal{O}$ of type $D_n$ , to parabolic Kazhdan–Lusztig polynomials of type $D_n$ with a parabolic subgroup of type $A$ — but only under an unproven 'saturated condition' (Condition 1.2) and, when $r$ is even, under a nonvanishing hypothesis on the auxiliary parameters $\omega_i$ . This problem asks to prove or refute the paper's Conjecture 1, namely that the saturated condition holds, and that consequently the announced determination of all decomposition numbers follows whenever $r$ is odd or $r$ is even with some $\omega_i\neq 0$ . The conjecture is open.

Background

The Brauer algebra, introduced in 1937 as the centralizer algebra of orthogonal and symplectic group actions on tensor space, is spanned by diagrams of $r$ strands pairing $2r$ boundary nodes; its representation theory over $\mathbb{C}$ serves as the level-one prototype for the algebras studied here. The cyclotomic Brauer algebra — more precisely the cyclotomic Nazarov–Wenzl algebra $B_{k,r}(\mathbf{u}_k)$ , since the two names refer to the same object by convention — is its higher-level analogue: the quotient of the affine Wenzl algebra with generators $e_i, s_i, x_j$ (where $e_i$ cap strands, $s_i$ is a symmetric-group generator, and $x_j$ records a Jucys–Murphy-type eigenvalue) cut by the cyclotomic relation $(x_1-u_1)\cdots(x_1-u_k)=0$ for a parameter vector $\mathbf{u}_k\in\mathbb{C}^k$ . Ariki, Mathas and Rui (2006) proved that $B_{k,r}(\mathbf{u}_k)$ is a cellular algebra of dimension at most $k^r(2r-1)!!$ , with equality exactly when a family of scalar parameters $\omega = (\omega_i)_{i\ge 0}$ is determined by $\mathbf{u}_k$ through the so-called $\mathbf{u}_k$ -admissibility relation; for $k=1$ the algebra reduces to the classical Brauer algebra. In a cellular algebra, each cell module $S(\lambda)$ carries an invariant bilinear form whose radical quotient $D(\lambda)$ is either zero or simple, and the decomposition number $[S(\lambda):D(\mu)]$ — the multiplicity of $D(\mu)$ in a composition series of $S(\lambda)$ — is the basic invariant of the representation theory. For the classical Brauer algebra these numbers were computed over $\mathbb{C}$ by realizing the algebra inside parabolic BGG (Bernstein–Gelfand–Gelfand) category $\mathcal{O}$ : Ehrig and Stroppel (2016) identified it with an idempotent truncation of a level-two Brauer algebra arising as an endomorphism algebra in parabolic category $\mathcal{O}$ of an even orthogonal Lie algebra, obtaining Koszul gradings and a graded Kazhdan–Lusztig-theoretic interpretation, and Martin (2014) determined its Cartan decomposition matrices. In parabolic category $\mathcal{O}$ , parabolic Verma modules $M^p(\lambda)$ and indecomposable tilting modules $T^p(\lambda)$ (the modules with both a Verma and a dual-Verma flag) are the standard and tilting objects, and Soergel-type formulas express tilting multiplicities through parabolic Kazhdan–Lusztig polynomials — polynomials attached to a Weyl group with a parabolic subgroup that encode composition multiplicities of parabolic Verma modules. Rui and Song (2019) realized cyclotomic Nazarov–Wenzl algebras as endomorphism algebras of $M_{r,\mathbf{c}} := M^p(\lambda_{\mathbf{c}})\otimes V^{\otimes r}$ , the tensor product of a simple scalar-type parabolic Verma module with $r$ copies of the natural $\mathfrak{so}_{2n}$ -module $V$ , in parabolic category $\mathcal{O}$ of types $B_n, C_n, D_n$ , and showed that decomposition numbers of the resulting algebra equal multiplicities $(T^p(\mu):M^p(\lambda))$ . Their 2025 two-part work makes this effective for $B_{k,r}(((-1)^k\mathbf{u}_k))$ with arbitrary parameters: part I (Gao and Rui, with an appendix by Wei Xiao) connects parabolic Verma subquotients of $M_{I_i,r}$ to right cell modules of the algebra (the cell modules spanned by cellular basis elements indexed by Kazhdan–Lusztig right cells) under a simplicity condition on $M^p(\lambda_{\mathbf{c}})$ , and part II (Rui and Song) proves in type $D_n$ with parabolic of type $A$ that $B^{{op}}_{2k,r}(\mathbf{u}_{2k}) \cong \mathrm{End}_{\mathcal{O}_p}(M_{r,\mathbf{c}})$ and $B^{{op}}_{k,r}(((-1)^k\mathbf{u}_k)) \cong e\,\mathrm{End}_{\mathcal{O}_p}(M_{r,\mathbf{c}})\,e$ for an idempotent $e$ (Theorem C), where the highest weight is $\lambda_{\mathbf{c}} = \sum_{j=1}^k c_j(\varepsilon_{p_{j-1}+1}+\dots+\varepsilon_{p_j})$ with $c_j = u_j + p_{j-1} - n + \tfrac{1}{2}$ and suitable integers $0=p_0<p_1<\dots<p_k=n$ . Define $F_j = \{\mu \mid (M_{j,\mathbf{c}}:M^p(\mu))\neq 0\}$ for $0\le j\le r$ , where $(\cdot:\cdot)$ is parabolic Verma flag multiplicity, and let $\mu\prec\lambda$ mean that $\lambda$ is reached from $\mu$ by a finite chain in which each simple $\mathfrak{so}_{2n}$ -module $L(\gamma_{l-1})$ occurs as a composition factor of the parabolic Verma module $M^p(\gamma_l)$ . The load-bearing hypothesis of part II is the saturated condition: Condition 1.2 requires each $F_j$ to be an ideal for $\prec$ ( $\mu\prec\lambda\in F_j$ implies $\mu\in F_j$ ). Under Condition 1.2, Theorem D of part II identifies the nonzero cell modules, the nonzero simple modules, and the decomposition numbers $[S(\hat\lambda)e:D(\hat\mu)e] = (T^p(\hat\mu):M^p(\hat\lambda))$ of $B_{k,r}(((-1)^k\mathbf{u}_k))$ , i.e. parabolic Kazhdan–Lusztig polynomial values of type $D_n$ with parabolic subgroup of type $A$ ; parts (2)–(4) of Theorem D additionally require, when $r$ is even, that $\omega_i\neq 0$ for some $0\le i\le k-1$ . Theorem E proves Condition 1.2 in the complementary (generic) case where $\Phi_A\cap\Psi^+_{\lambda_{\mathbf{c}}} = \varnothing$ , with $\Phi_A = \{\varepsilon_i-\varepsilon_j \mid i<j\}\cap(\Phi^+\setminus\Phi_I)$ the roots outside the parabolic root set $\Phi_I$ of type $A$ that involve the first block coordinates, and $\Psi^+_{\lambda_{\mathbf{c}}} = \{\beta\in\Phi^+\setminus\Phi_I \mid \langle\lambda_{\mathbf{c}}+\rho,\beta^\vee\rangle\in\mathbb{Z}_{>0}\}$ the positive roots integral on $\lambda_{\mathbf{c}}+\rho$ ( $\rho$ the half-sum of positive roots, $\beta^\vee$ the coroot); for $k=1$ this holds unconditionally, recovering and strengthening the classical Brauer algebra results. What remains open — and is formulated by the authors as Conjecture 1 — is exactly the case $\Phi_A\cap\Psi^+_{\lambda_{\mathbf{c}}}\neq\varnothing$ ; the $\omega$ -nonvanishing restriction is carried by the conjecture itself. The source only records (Remark 2.2 of part II) that failure of that hypothesis is equivalent to an algebraic symmetry relation on $\mathbf{u}_k$ — equation (2.3) there; for $k=1$ , to $u_1=\tfrac{1}{2}$ — and Theorem D(2)–(4) is not established in that all-vanishing even- $r$ regime, but the source does not pose that regime as an open question. A caveat on names: the 'cyclotomic Brauer algebra' $B^m_n(\delta)$ of Bowman, Cox and De Visscher (2013), an enlargement of complex reflection group algebras of type $G(m,1,n)$ whose characteristic-zero decomposition numbers they determined by reduction to (walled) Brauer algebras, is a different algebra sharing the same name and does not bear on this problem.

Problem Statement

Resolve the open case left by Rui and Song's determination of the decomposition numbers of the cyclotomic Brauer algebra $B_{k,r}(((-1)^k\mathbf{u}_k))$ over $\mathbb{C}$ with arbitrary parameters, in the framework of parabolic BGG category $\mathcal{O}_p$ of type $D_n$ with parabolic subgroup of type $A$ (notation and definitions — $\lambda_{\mathbf{c}}$ , $F_j$ , $\prec$ , Condition 1.2, $\Phi_A$ , $\Psi^+_{\lambda_{\mathbf{c}}}$ , the parameters $\omega$ determined by $\mathbf{u}_k$ — as in the Background, following part II of the source). Suppose $\Phi_A\cap\Psi^+_{\lambda_{\mathbf{c}}}\neq\varnothing$ (Conjecture 1 of the source). Prove, or refute, that Condition 1.2 is nevertheless satisfied (i.e. that each $F_j$ is an ideal for the order $\prec$ : $\mu\prec\lambda\in F_j$ implies $\mu\in F_j$ ), and consequently that Theorem D(1)–(4) of the source hold — i.e. that the cell modules and simple modules of $B_{k,r}(((-1)^k\mathbf{u}_k))$ are characterized as stated and every decomposition number $[S(\hat\lambda)e:D(\hat\mu)e]$ equals the parabolic Verma multiplicity $(T^p(\hat\mu):M^p(\hat\lambda))$ , a parabolic Kazhdan–Lusztig polynomial value of type $D_n$ — whenever $r$ is odd, or $r$ is even and $\omega_i\neq 0$ for some $0\le i\le k-1$ . The scientific objective is to remove the last hypothesis under which the source's computation of all decomposition numbers of the cyclotomic Brauer algebra over $\mathbb{C}$ with arbitrary parameters stands, completing that determination within the range stated in Conjecture 1. The complementary regime where $r$ is even and $\omega_i = 0$ for all $i\ge 0$ (equivalently, where $\mathbf{u}_k$ satisfies relation (2.3) of the source; for $k=1$ , $u_1 = \tfrac{1}{2}$ ) lies outside the scope of Conjecture 1 — the source poses no open question for it — and is not part of this problem.

References

Hebing Rui and Linliang Song, 'Decomposition numbers of the cyclotomic Brauer algebra over the complex field, II', arXiv:2502.02081, doi:10.48550/arXiv.2502.02081, https://arxiv.org/abs/2502.02081 (2025)
Mengmeng Gao and Hebing Rui (with an appendix by Wei Xiao), 'Decomposition numbers of cyclotomic Brauer algebras over the complex field, I', arXiv:2502.00420, doi:10.48550/arXiv.2502.00420, https://arxiv.org/abs/2502.00420 (2025)
Susumu Ariki, Andrew Mathas and Hebing Rui, 'Cyclotomic Nazarov-Wenzl algebras', Nagoya Mathematical Journal 182 (2006) 47-134, arXiv:math/0506467, doi:10.1017/S0027763000026842, https://doi.org/10.1017/S0027763000026842
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Michael Ehrig and Catharina Stroppel, 'Koszul gradings on Brauer algebras', International Mathematics Research Notices 2016 (13) 3970-4011, arXiv:1504.03924, doi:10.1093/imrn/rnv267, https://doi.org/10.1093/imrn/rnv267
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C. Bowman, A. G. Cox and M. De Visscher, 'Decomposition numbers for the cyclotomic Brauer algebras in characteristic zero', Journal of Algebra 378 (2013) 104-117, arXiv:1205.3345, doi:10.1016/j.jalgebra.2012.12.020, https://doi.org/10.1016/j.jalgebra.2012.12.020
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