Diagrammatic identification of the radical of the Brauer algebra at orthogonal and symplectic parameters (Gavarini's Conjectures 5.10 and 5.14)

For the Brauer algebra $B_f^{(x)}$ at the non-semisimple integral parameters $x=n$ (orthogonal type) and $x=-2n$ (symplectic type), Gavarini (2008) proved that an explicit invariant-theoretic span $R_f^{(x)}$ of diagrammatic minors/Pfaffians lies in the Jacobson radical of the top layer $B_f^{(x)}([(f-n+1)/2])$ of the standard arc filtration, and conjectured equality there (Conjecture 5.10), with the parallel module-level conjecture $\operatorname{Rad}(H^\mu_{f,k})=R_f^{(x)}\cdot H^\mu_{f,k}$ for the standard modules in the top range (Conjecture 5.14), in characteristic 0 and extended to characteristic $p>f$. Only special cases ($x=0$; $n=1$ with $f$ even) are proved. The problem is to prove or refute these identities in general. Later work resolved the adjacent kernel (second-fundamental-theorem) side and described cell-module (standard-module) radicals at permissible points by other spanning sets, but the radical identification itself remained open: Hu and Xiao in 2012 still listed describing the radical of $B_n(m)$ (the Brauer algebra on $2n$ vertices with loop parameter $m$, in their notation) in the non-semisimple case as a desirable target, and it remains open.

Background

The Brauer algebra $B_f^{(x)}$, introduced by Richard Brauer in 1937, is the algebra over a field $\mathbb{k}$ with basis the set $\mathcal{D}_f$ of Brauer $f$-diagrams: perfect matchings on $2f$ vertices arranged in a top and a bottom row of $f$ vertices each. Edges within a row are called arcs (or horizontal edges) and edges joining the two rows are called vertical or propagating edges; multiplication is by vertical concatenation with each closed loop erased at the cost of a factor $x$, the loop parameter. Let $\mathcal{D}_{f,h}$ denote the set of $f$-diagrams with exactly $h$ arcs, and $B_f^{(x)}(h):=\mathbb{k}\text{-span}\big(\bigcup_{j\ge h}\mathcal{D}_{f,j}\big)$; the $B_f^{(x)}(h)$ form a chain of two-sided ideals, the standard series, from $B_f^{(x)}(0)=B_f^{(x)}$ down to $B_f^{(x)}([f/2])$. Brauer introduced this algebra to centralize the action of the orthogonal group $O(V)$, $\dim V=n$, or the symplectic group $Sp(W)$, $\dim W=2n$, on tensor powers of the natural representation: there are epimorphisms $\pi_V:B_f^{(n)}\twoheadrightarrow\operatorname{End}_{O(V)}(V^{\otimes f})$ and $\pi_W:B_f^{(-2n)}\twoheadrightarrow\operatorname{End}_{Sp(W)}(W^{\otimes f})$. By a theorem of Wenzl, in characteristic 0 the algebra $B_f^{(x)}$ is semisimple for all non-integral $x$ and for integral $x$ outside a finite range, so precisely at parameters such as $x=n$ and $x=-2n$ (when $f$ is large relative to $n$) the algebra becomes non-semisimple and acquires a nonzero Jacobson radical $\operatorname{Rad}(B_f^{(x)})$, its largest nilpotent two-sided ideal. Since $O(V)$ and $Sp(W)$ are linearly reductive (all finite-dimensional representations completely reducible) in characteristic 0, the centralizer algebra $\operatorname{End}_G(U^{\otimes f})$ is semisimple, forcing $\operatorname{Rad}(B_f^{(x)})\subseteq\ker(\pi_U)$. By the second fundamental theorem of classical invariant theory, in Gavarini's formulation, $\ker(\pi_V)$ is the span of all diagrammatic minors of order $n+1$ and $\ker(\pi_W)$ is the span of all diagrammatic Pfaffians of order $2(n+1)$: explicit signed alternating sums of $f$-diagrams, the diagram avatars of the determinants of $(n+1)\times(n+1)$ submatrices of the coordinate matrix and of Pfaffians (square roots of determinants) of skew-symmetric $(2n+2)\times(2n+2)$ submatrices, which vanish identically on a space of dimension at most $n$ (respectively $2n$). Gavarini then proved (Theorems 5.3 and 5.5) exactly which of these minors and Pfaffians belong to the radical of each layer and quotient of the standard series, and in particular that the span $R_f^{(x)}$ of the order-$(n+1)$ minors (for $x=n$), respectively order-$2(n+1)$ Pfaffians (for $x=-2n$), that individually lie in the top layer $B_f^{(x)}([(f-n+1)/2])$, is contained in $\operatorname{Rad}\big(B_f^{(x)}([(f-n+1)/2])\big)$. The subtlety is that the opposite inclusion is not formal: the radical of the layer is contained in $\ker(\pi_U)\cap B_f^{(x)}([(f-n+1)/2])$, i.e. in the intersection of the top layer with the span of all minors/Pfaffians, and such an intersection can a priori contain linear combinations of lower-layer minors/Pfaffians that no individual generator captures. Whether such accidental radical elements exist is the content of the conjectures. On the module side, the standard (cell) modules of $B_f^{(x)}$ are Kerov's modules $H^\mu_{f,k}=M^\mu\otimes H_{f,k}$, where $M^\mu$ is a Specht module (the standard module of the symmetric group labeled by the partition $\mu\vdash f-2k$) and $H_{f,k}$ is a space of diagram junctions with $k$ arcs; every simple module is a quotient of some $H^\mu_{f,k}$, and the conjectural radical of these modules is the submodule generated by the same invariant-theoretic spans. Gavarini proved the conjectures in the case $x=0$ (Proposition 5.9) and, for $x=1$ with $f$ even, the algebra-level case and the top module case (via the Temperley–Lieb algebra — the subalgebra of $B_f^{(1)}$ spanned by the planar $f$-diagrams, i.e. those whose edges can be drawn without crossings — and pointed chord diagrams); he also extended the whole framework, and the conjectures, to positive characteristic $p>f$, while explicitly leaving smaller characteristics unresolved because the linear reductivity of $O(V)$ and $Sp(W)$ fails there. After 2008 the kernel side advanced: Bowman–Enyang–Goodman (published online 2018, print 2020) constructed integral cellular (Murphy-type) bases of the centralizer algebras — a cellular basis being a basis layered by a chain of one-sided ideals so that the successive quotient layers realize the standard (cell) modules — with explicit bases of the kernels, and Hu–Xiao (2012) proved the Lehrer–Zhang conjecture that the annihilator of tensor space in the orthogonal case is a single explicitly generated ideal over any field of characteristic not 2; Bowman–Enyang–Goodman also identified the radical of each cell module at a permissible point (a point of the parametrizing spectrum at which the cell module's canonical bilinear form is nonzero) with the span of its non-permissible Murphy basis elements. None of these works states or claims the identification of the radical with Gavarini's invariant-theoretic spans, and the algebra-level question remains untouched by them.

Problem Statement

Over a field $\mathbb{k}$ of characteristic 0 (equivalently, by Gavarini's §5.15 extension, of characteristic $p>f$), prove or refute Gavarini's Conjecture 5.10: for all $f\in\mathbb{N}_+$ and $n\in\mathbb{N}_+$, writing $[(f-n+1)/2]$ for the integer part, the Jacobson radical of the top layer of the standard series of the Brauer algebra equals the invariant-theoretic generator span lying in that layer, namely $\operatorname{Rad}\big(\mathcal{B}_f^{(n)}([(f-n+1)/2])\big)=R_f^{(n)}$ in the orthogonal case and $\operatorname{Rad}\big(\mathcal{B}_f^{(-2n)}([(f-n+1)/2])\big)=R_f^{(-2n)}$ in the symplectic case, where $R_f^{(n)}$ (resp. $R_f^{(-2n)}$) is the $\mathbb{k}$-span of the diagrammatic minors of order $n+1$ (resp. diagrammatic Pfaffians of order $2(n+1)$) that individually belong to $\mathcal{B}_f^{(x)}([(f-n+1)/2])$; and prove or refute the companion Conjecture 5.14: for all $f,n\in\mathbb{N}_+$, all $k$ with $[(f-n+1)/2]\le k\le[f/2]$ and all partitions $\mu\vdash(f-2k)$, the radical of the standard (Kerov) module equals the generated submodule, $\operatorname{Rad}\big(H^\mu_{f,k}\big)=R_f^{(n)}\cdot H^\mu_{f,k}$, with the analogous statement for $\mathcal{B}_f^{(-2n)}$-modules and $R_f^{(-2n)}$. Gavarini's proof of Conjecture 5.14 is conditional on Conjecture 5.10 (it invokes the standard identity $\operatorname{Rad}(M)=\operatorname{Rad}(A)\cdot M$), so an affirmative resolution of Conjecture 5.10 settles Conjecture 5.14 as well, while a refutation of Conjecture 5.10 leaves the weaker Conjecture 5.14 open; 5.14 therefore remains a separately attackable deliverable. The known inclusions $R_f^{(x)}\subseteq\operatorname{Rad}(\cdot)$ are proved, so the entire content is the absence of additional radical elements; since each conjecture is a universal claim over all $(f,n)$, a single explicit instance with an element of $\operatorname{Rad}\big(\mathcal{B}_f^{(x)}([(f-n+1)/2])\big)$ (or of $\operatorname{Rad}(H^\mu_{f,k})$) lying outside $R_f^{(x)}$ (resp. outside $R_f^{(x)}\cdot H^\mu_{f,k}$) refutes it. A secondary, source-flagged open direction with the same objective of a complete diagrammatic radical description: determine whether, and under which precise conditions on the characteristic, these identities persist when $\operatorname{char}(\mathbb{k})\le f$, where the linear reductivity arguments fail.

References

1. Fabio Gavarini, On the radical of Brauer algebras, Mathematische Zeitschrift 260 (2008), no. 3, 673–697. DOI: 10.1007/s00209-007-0296-z, https://doi.org/10.1007/s00209-007-0296-z (arXiv:math/0606501, https://arxiv.org/abs/math/0606501)
2. Christopher Bowman, John Enyang, Frederick M. Goodman, The cellular second fundamental theorem of invariant theory for classical groups, International Mathematics Research Notices 2020 (2020), no. 9, 2626–2683 (published online 2018). DOI: 10.1093/imrn/rny079, https://doi.org/10.1093/imrn/rny079 (arXiv:1610.09009, https://arxiv.org/abs/1610.09009)
3. Jun Hu, Zhankui Xiao, On a theorem of Lehrer and Zhang, Documenta Mathematica 17 (2012), 245–270. DOI: 10.4171/dm/367, https://doi.org/10.4171/dm/367 (arXiv:1105.5287, https://arxiv.org/abs/1105.5287)
4. Jun Hu, Zhankui Xiao, Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality, Transactions of the American Mathematical Society, Series B 8 (2021), 823–848. DOI: 10.1090/btran/84, https://doi.org/10.1090/btran/84
5. Stephen Doty, Jun Hu, Schur–Weyl duality for orthogonal groups, Proceedings of the London Mathematical Society (3) 98 (2009), 679–713 (published online 2008). DOI: 10.1112/plms/pdn044, https://doi.org/10.1112/plms/pdn044 (content described in Gavarini §4.9 and in Hu–Xiao)