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5. Parity Sheaves and Torsion in IC Sheaves

    1. Problem 5.1.

      [Achar–Williamson]
      1. For which primes p is it true that all indecomposable B-constructible parity \F_p sheaves on G/B are perverse?
      2. Same question for I-constructible parity \F_p-sheaves on Gr.


      (Expect this to be true for large p.)
        • Problem 5.2.

          [Mautner] For which primes p do the IC sheaves on \operatorname{Perv}_I(Gr, \Z_p) have torsion-free stalks?

          (Related to Problem 10.1 part 2, and to Ext-vanishing between reduced standard and costandard modules.)
            • Problem 5.3.

              [Williamson] Give nontrivial sufficient conditions for indecomposable parity sheaves on G/B or Gr or Fl to be simple perverse sheaves.
                • Problem 5.4.

                  [Achar] Is there a faster algorithm to compute indecomposable parity sheaves on \operatorname{Perv}_{\mathrm{sph}}(Gr, \F_p) than by computing the p-canonical basis?

                  (Motivated by A. Broer’s algorithm in characteristic 0.)

                      Cite this as: AimPL: Sheaves and modular representations of reductive groups, available at http://aimpl.org/sheavemodular.