Block equivalence conjecture for orthosymplectic supergroups

Let m,nm,n be natural numbers and fix a block B{\mathcal B} of atypicality kk in F(SOSP(m2n)){\mathcal F}(\operatorname{SOSP}(m|2n)). Let Hk\mathbb{H}_k^\infty be the limiting graded diagram algebra for atypicality kk, and let I\mathbb{I} be the subspace spanned by diagrams containing a non-propagating line. Block equivalence conjecture. The block B{\mathcal B} is equivalent to the category of finite-dimensional Hk/I\mathbb{H}_k^\infty/\mathbb{I}-modules. In the fuller formulation, the equivalence sends each projective P(λ)P(\lambda) to the indecomposable projective module P(λ)P(\lambda^\circ) and each irreducible highest-weight module L(λ)L(\lambda) to L(λ)L(\lambda^\circ). This conjectural equivalence would give a diagrammatic description of every block of finite-dimensional representations of the orthosymplectic supergroup. The source provides no evidence of resolution.

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Primary source

Michael Ehrig and Catharina Stroppel, “Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)”, arXiv:1306.4043 (2013).

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