Blocks of the Brauer algebra in positive characteristic and decomposition numbers beyond the non-dividing regime

The Brauer algebra $B_r(\delta)$ is a diagram algebra governing Schur–Weyl duality for orthogonal and symplectic groups. Its blocks and decomposition numbers are fully described in characteristic zero, and Shalile (2015) determined all decomposition numbers over fields of characteristic $p>r$ (the non-dividing regime), explicitly postponing the block structure to future work. This problem asks for a structural, partition-level characterization of the blocks of $B_r(\delta)$ in positive characteristic — at least for $p>r$ , where decomposition numbers are known and the char-zero cap–curl description of Cox–De Visscher–Martin should degenerate in a controlled way — and for decomposition numbers and blocks beyond this regime. A rigorous closed-form block criterion in the non-dividing regime $p>r$ (part (1) of the Problem Statement) is an open structural target in its own right; the extension to dividing characteristics (part (2)) is a further open target. Later work of Li–Tange (Transformation Groups, published 2022) extended the decomposition numbers to all $p$ exceeding the largest hook length of the partitions involved, which strictly enlarges the $p>r$ regime but still leaves the genuinely dividing characteristics (small $p$ relative to $r$ ) untouched; no block classification for the fixed algebra $B_r(\delta)$ in any positive characteristic exists in the literature. The problem remains open.

Background

The Brauer algebra $B_r(\delta)$ was introduced by Richard Brauer in 1937 as the centralizer algebra of the orthogonal or symplectic group acting on tensor space $V^{\otimes r}$ , replacing the symmetric group that appears in classical Schur–Weyl duality for $GL(V)$ . It is a finite-dimensional algebra over a field $k$ , spanned by diagrams of $r$ upper and $r$ lower nodes joined pairwise by edges, with multiplication by stacking diagrams; the parameter $\delta\in k$ enters through the number of closed loops removed when stacking. $B_r(\delta)$ is a cellular algebra in the sense of Graham–Lehrer: it has a family of cell modules $\Delta(\lambda)$ indexed by partitions $\lambda$ of $r, r-2, r-4, \dots$ , and each cell module has (when nonzero) a simple head $L(\lambda)$ . The representation theory is governed by two kinds of invariants. First, the decomposition numbers $d_{\lambda\mu}=[\Delta(\lambda):L(\mu)]$ , the composition multiplicities of simples in cell modules. Second, the blocks: two simple modules lie in the same block if they are linked by a chain in which consecutive members occur as composition factors of a common cell module; equivalently, blocks correspond to the primitive central idempotents of the algebra. In characteristic zero the theory is complete: Cox–De Visscher–Martin (2009) determined the blocks of $B_r(\delta)$ explicitly, in terms of cap–curl diagrams — diagrammatic encodings of the pair $(\lambda,\mu)$ whose connected components (caps joining two addable positions, curls reflecting the half-integer shift $\delta/2$ ) decide block membership — and Martin and Cox–De Visscher computed the decomposition matrices over $\mathbb{C}$ . In positive characteristic $p$ , the picture is far less complete. A basic dichotomy separates the non-dividing regime $p>r$ or $p=0$ , where the group algebra of the symmetric group $S_r\subset B_r(\delta)$ is semisimple, from the dividing regime $p\le r$ , where it is not. Shalile (2015) studied the non-dividing regime with the Okounkov–Vershik–Jucys–Murphy approach: the Jucys–Murphy (JM) elements are a commuting family of distinguished algebra elements whose simultaneous eigenspaces on cell modules encode the partition combinatorics, and the arrow abacus is an infinite display of labelled runners on which partitions and the $p$ -shifted content of boxes are read off, playing the role that the classical abacus plays for symmetric groups. He proved that $L(\mu)$ is a composition factor of $\Delta(\lambda)$ exactly when $\lambda$ and $\mu$ share a common JM-weight, translated this into a diagrammatic criterion via up–down tableaux (sequences of partitions obtained by adding and removing one box at each step, which track induction and restriction along the tower of Brauer algebras), and thus determined all decomposition numbers for $p=0$ or $p>r$ . Since blocks are the transitive closure of nonzero decomposition numbers through cell modules, the blocks of $B_r(\delta)$ for $p>r$ are in principle computable from his theorem; but the paper does not extract a closed structural description of the blocks — the analogue, in positive characteristic, of the cap–curl block criterion of Cox–De Visscher–Martin in characteristic zero — and the author explicitly postponed the description of the blocks to future work. Two neighbouring results frame the gap. King (2014) determined the limiting blocks of $B_r(\delta)$ over any field of positive characteristic: the limiting blocks describe the stable behaviour as $r$ grows at fixed $\delta \bmod p$ and form a coarse shadow of, not a substitute for, the blocks of the fixed algebra $B_r(\delta)$ at given $r$ . Later, Li–Tange (published 2022) computed decomposition numbers of $B_r(\delta)$ under the hypothesis that $p$ exceeds the largest hook length $\lambda_1+\ell(\lambda)-1$ of the largest partition involved (with $p>2$ and $\delta$ in the prime field), using the symplectic Schur functor relating $B_r(\delta)$ -modules to modules for the symplectic group and a reduced Jantzen sum formula; this strictly extends the $p>r$ regime, since hook lengths of partitions of at most $r$ never exceed $r$ , yet it still excludes small primes. What remains is the structural block theory in positive characteristic and the dividing regime, where even the symmetric-group subalgebra is non-semisimple.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $p\ge 0$ and let $B_r(\delta)$ be the Brauer algebra over $k$ , with cell modules $\Delta(\lambda)$ labelled by the partitions of $r, r-2, r-4, \dots$ . Determine, in positive characteristic: (1) a structural, partition-level characterization of the blocks of $B_r(\delta)$ in the non-dividing regime $p>r$ — an explicit combinatorial condition on a pair of partitions $\lambda,\mu$ that holds exactly when the simple heads $L(\lambda)$ and $L(\mu)$ lie in the same block, generalizing the characteristic-zero cap–curl description of Cox–De Visscher–Martin and reducing to it at $p=0$ ; Shalile's determination of all decomposition numbers for $p>r$ makes the blocks algorithmically computable in this regime, so the target is the closed-form block criterion on the pair $(\lambda,\mu)$ itself, replacing the transitive-closure computation through up–down tableaux; and (2) beyond this regime, the decomposition numbers and blocks of $B_r(\delta)$ for dividing characteristics — the currently largest known region is $p$ exceeding the largest hook length of the partitions involved (Li–Tange), so the unresolved window is $p$ at or below those hook lengths, in particular $p\le r$ , where the group algebra of the symmetric group $S_r$ inside $B_r(\delta)$ is itself non-semisimple and neither Shalile's JM-weight criterion nor the symplectic Schur functor machinery of Li–Tange applies. Answers should state precisely for which pairs $(r,p,\delta\bmod p)$ they are valid, including the small-prime and integral-parameter cases.

References

Armin Shalile, "Decomposition numbers of Brauer algebras in non-dividing characteristic", Journal of Algebra 423 (2015) 963–1009. DOI: 10.1016/j.jalgebra.2014.10.040, https://doi.org/10.1016/j.jalgebra.2014.10.040
Anton G. Cox, Maud De Visscher, Paul P. Martin, "The blocks of the Brauer algebra in characteristic zero", Representation Theory 13 (2009) 272–308. DOI: 10.1090/s1088-4165-09-00305-7, https://doi.org/10.1090/S1088-4165-09-00305-7
Oliver King, "The limiting blocks of the Brauer algebra in characteristic p" (arXiv title: "The blocks of the Brauer algebra in characteristic p"), Journal of Algebra 397 (2014) 168–189. DOI: 10.1016/j.jalgebra.2013.08.040, https://doi.org/10.1016/j.jalgebra.2013.08.040; arXiv:1211.6303, https://arxiv.org/abs/1211.6303
O. H. King, P. P. Martin, A. E. Parker, "On central idempotents in the Brauer algebra", Journal of Algebra 512 (2018) 20–46. DOI: 10.1016/j.jalgebra.2018.06.024, https://doi.org/10.1016/j.jalgebra.2018.06.024; arXiv:1609.01183, https://arxiv.org/abs/1609.01183
Henry Li, Rudolf Tange, "A combinatorial translation principle and diagram combinatorics for the symplectic group", Transformation Groups 29 (2024) 231–252 (published online 2022). DOI: 10.1007/s00031-021-09685-6, https://doi.org/10.1007/s00031-021-09685-6; arXiv:2011.03606, https://arxiv.org/abs/2011.03606
Rudolf Tange, "A combinatorial translation principle and diagram combinatorics for the general linear group", Transformation Groups 28 (2023) 1687–1719 (published online 2022). DOI: 10.1007/s00031-022-09710-2, https://doi.org/10.1007/s00031-022-09710-2; arXiv:2011.05564, https://arxiv.org/abs/2011.05564
Rudolf Tange, "Injective and tilting resolutions and a Kazhdan–Lusztig theory for the general linear and symplectic group", Algebras and Representation Theory 26 (2023) 2819–2839 (published online 2022). DOI: 10.1007/s10468-022-10197-4, https://doi.org/10.1007/s10468-022-10197-4; arXiv:2109.10709, https://arxiv.org/abs/2109.10709
Armin Shalile, "On the modular representation theory of the partition algebra", Journal of Algebraic Combinatorics 42 (2015) 245–282. DOI: 10.1007/s10801-014-0580-z, https://doi.org/10.1007/s10801-014-0580-z; arXiv:1403.5132, https://arxiv.org/abs/1403.5132
Anton G. Cox, Maud De Visscher, "Diagrammatic Kazhdan–Lusztig theory for the (walled) Brauer algebra", Journal of Algebra 340 (2011) 151–181. DOI: 10.1016/j.jalgebra.2011.05.024, https://doi.org/10.1016/j.jalgebra.2011.05.024
Sigiswald Barbier, Anton Cox, Maud De Visscher, "The blocks of the periplectic Brauer algebra in positive characteristic", Journal of Algebra 534 (2019) 289–312. DOI: 10.1016/j.jalgebra.2019.06.016, https://doi.org/10.1016/j.jalgebra.2019.06.016; arXiv:1902.04382, https://arxiv.org/abs/1902.04382