Tensor-ideal classification for the oriented Brauer–Clifford category

The oriented Brauer–Clifford category is the strict monoidal supercategory OBC\mathcal{OBC} equipped with full monoidal superfunctors

OBCRepkQ(n)\mathcal{OBC}\to \operatorname{Rep}_{\Bbbk}\mathrm{Q}(n)

for nNn\in\mathbb{N}, and let RepQ=OBC\underline{\operatorname{Rep}}\,Q=\mathcal{OBC}^{\oplus\sharp}. A tensor ideal is a collection of morphism spaces closed under tensoring and composition, and a thick tensor Ob\mathrm{Ob}-ideal is the corresponding ideal notion in RepQ\underline{\operatorname{Rep}}\,Q. Tensor-ideal classification conjecture. The tensor ideals in OBC\mathcal{OBC} are precisely the kernels of the superfunctors

OBCRepkQ(n).\mathcal{OBC}\to \operatorname{Rep}_{\Bbbk}\mathrm{Q}(n).

These are in natural bijection with the thick tensor Ob\mathrm{Ob}-ideals in RepQ\underline{\operatorname{Rep}}\,Q. The claim proposes a classification of tensor ideals for the oriented Brauer–Clifford category and their relationship with ideals in the associated Deligne-type category; the supplied text does not state whether it has been proved or disproved.

Sources & referencesView supporting material

Primary source

Kevin Coulembier, “Tensor ideals, Deligne categories and invariant theory”, arXiv:1712.06248 (2018).

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