The periplectic Brauer algebra AnA_n is the diagram algebra that, via Schur–Weyl duality, centralizes the periplectic Lie superalgebra p(n)\mathfrak p(n) acting on tensor powers of its natural module. It is never semisimple, and although it is standardly based it is not cellular, so its modular representation theory requires dedicated machinery. The blocks of AnA_n are now known over every field of odd characteristic, but the composition multiplicities of its standard (cell) modules in the simple modules — the decomposition numbers — are known only in characteristic zero (equivalently, for p>np>n). Barbier, Cox and De Visscher, after determining the blocks in all odd characteristic, single out the computation of these decomposition multiplicities in positive characteristic as an important open problem, related to the long-standing open problem of the decomposition numbers of the symmetric group, and out of reach by current methods. Determine the full decomposition matrix of AnA_n for every nn over every algebraically closed field of odd characteristic; the Cartan matrix then follows from it via the BGG-reciprocity factorization.

Background

A Brauer diagram on nn northern and nn southern nodes is a perfect matching of the 2n2n nodes, drawn as cups, caps and vertical propagating lines. The classical Brauer algebra is the span of such diagrams with multiplication given by stacking; it realizes the centralizer of the orthogonal and symplectic Lie algebras on tensor powers of their natural modules (Schur–Weyl duality). The periplectic Lie superalgebra p(n)\mathfrak p(n) (also called the strange or type-PP superalgebra) is the classical Lie superalgebra with even part gln\mathfrak{gl}_n and odd part the symmetric and exterior square representations; it is the one classical family whose invariant theory is controlled, on tensor powers of the natural module, by the periplectic Brauer algebra AnA_n: the same Brauer diagrams as the ordinary Brauer algebra at parameter δ=0\delta=0, with a multiplication deformed by signs. The algebra AnA_n is never semisimple, and unlike the Brauer algebra it is not cellular (a cellular algebra, in the sense of Graham–Wenzl, admits a basis with a compatible anti-involution from which cell modules are built); it is only standardly based, a weaker structure due to Du and Rui that still yields standard (cell) modules Wn(λ)W_n(\lambda) indexed by all partitions λ\lambda of the numbers i∈{n−2j∣0≤j≤n/2}i\in\{n-2j\mid 0\le j\le n/2\}, and simple modules Ln(λ)L_n(\lambda) labelled by the pp-restricted partitions of the numbers i∈{n−2j∣0≤j<n/2}i\in\{n-2j\mid 0\le j<n/2\}. The decomposition numbers are the composition multiplicities dμλ=[Wn(μ):Ln(λ)]d_{\mu\lambda}=[W_n(\mu):L_n(\lambda)], forming the decomposition matrix DD; the Cartan matrix CC records the multiplicities cλν=[Pn(λ):Ln(ν)]c_{\lambda\nu}=[P_n(\lambda):L_n(\nu)] of simples in the indecomposable projective covers. Two simples lie in the same block when they are linked through nonvanishing extensions, so the block decomposition is the coarse shadow of the decomposition matrix. A pp-modular system (K,R,k)(\mathbb K,R,\mathbb k) — a discrete valuation ring RR with fraction field K\mathbb K of characteristic 00 and residue field k\mathbb k of characteristic pp — is the standard device for comparing characteristic-zero and modular representation theory. Prior to Barbier–Cox–De Visscher, Coulembier had shown that AnA_n is quasi-hereditary in many cases, described its blocks in characteristic 00 and more generally whenever the characteristic pp satisfies p∉[2,n]p\notin[2,n] (two simples are then linked exactly when their labelling partitions have the same 22-core, i.e. the same staircase partition left after removing dominoes), and Coulembier–Ehrig had computed all decomposition multiplicities of cell and projective modules in characteristic zero through the combinatorics of skew Young diagrams, relating them to Kazhdan–Lusztig multiplicities of p(n)\mathfrak p(n). Barbier–Cox–De Visscher then determined the blocks over any field of odd characteristic: the blocks are a principal block together with exceptional blocks attached to staircase partitions ρr=(r,r−1,…,1)\rho_r=(r,r-1,\dots,1) satisfying 2r−1<p2r-1<p, r(r+1)/2+p−2r>nr(r+1)/2+p-2r>n and r(r+1)/2=n−2kr(r+1)/2=n-2k for some integer k≥0k\ge0, and there is a single block whenever n≥(p2+7)/8n\ge(p^2+7)/8. Along the way they proved a BGG-reciprocity with a Mullineux twist (the Mullineux map is the permutation of irreducible symmetric-group modules — equivalently of partitions — induced by tensoring with the sign representation): [Pn(λ):Wn(μ)]=[Wn(μT):Ln(λM)][P_n(\lambda):W_n(\mu)]=[W_n(\mu^T):L_n(\lambda^M)], whence the Cartan matrix factors as cλν=∑μdμTλM dμνc_{\lambda\nu}=\sum_{\mu}d_{\mu^T\lambda^M}\,d_{\mu\nu} in arbitrary characteristic (their Corollary 7.4). What remains unknown over odd characteristic p≤np\le n is the multiplicity matrix DD itself.

Problem Statement

Determine the decomposition numbers of the periplectic Brauer algebra in positive characteristic. Concretely: for every nn and every algebraically closed field of odd characteristic pp with p≤np\le n, compute the composition multiplicities dμλ=[Wn(μ):Ln(λ)]d_{\mu\lambda}=[W_n(\mu):L_n(\lambda)] of the standard (cell) modules Wn(μ)W_n(\mu) in the simple modules Ln(λ)L_n(\lambda); the Cartan numbers [Pn(λ):Ln(ν)][P_n(\lambda):L_n(\nu)] of the indecomposable projectives then follow from these multiplicities via the Mullineux-twisted BGG-reciprocity factorization cλν=∑μdμTλM dμνc_{\lambda\nu}=\sum_{\mu}d_{\mu^T\lambda^M}\,d_{\mu\nu}. Nonzero multiplicities occur only within the blocks of Barbier–Cox–De Visscher's Theorem 8.10, i.e. the principal and exceptional staircase blocks described in the background, and for p∉[2,n]p\notin[2,n] the multiplicities coincide with the characteristic-zero multiplicities of Coulembier–Ehrig.

References

  1. 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; arXiv:1902.04382; https://arxiv.org/abs/1902.04382
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