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Categorifying Hecke algebras at prime roots of unity, part I
We equip the type diagrammatic Hecke category with a special derivation,
so that after specialization to characteristic it becomes a -dg
category. We prove that the defining relations of the Hecke algebra are
satisfied in the -dg Grothendieck group. We conjecture that the -dg
Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it
with a basis which may differ from both the Kazhdan-Lusztig basis and the
-canonical basis. More precise conjectures will be found in the sequel.
Here are some other results contained in this paper. We provide an incomplete
proof of the classification of all degree derivations on the diagrammatic
Hecke category, and a complete proof of the classification of those derivations
for which the defining relations of the Hecke algebra are satisfied in the
-dg Grothendieck group. In particular, our special derivation is unique up
to duality and equivalence. We prove that no such derivation exists in
simply-laced types outside of finite and affine type . We also examine a
particular Bott-Samelson bimodule in type , which is indecomposable in
characteristic but decomposable in all other characteristics. We prove that
this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any
characteristic, which is the analogue in the -dg setting of being
indecomposable.Comment: 44 pages, many figures, color viewing essential. V2 contains
corrections from referee reports. To appear in Transactions of the AM
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