Abstract
We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a ‘super-strong linkage principle’ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are ‘generically’ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.
Highlights
Cyclotomic quiver Hecke algebras are of central interest in Khovanov homology, knot theory, group theory, and higher representation theory
This paper seeks to generalize their work to the modular representation theory of these algebras, where almost nothing is known or even conjectured
In the main result of this section, we prove that Ah(n, θ, κ) has a basis indexed by pairs of paths in a certain alcove geometry
Summary
Cyclotomic quiver Hecke algebras (and their quasiherediary covers, the diagrammatic Cherednik algebras) are of central interest in Khovanov homology, knot theory, group theory, and higher representation theory. Dλ,μ(t ) = ddeth(λ),deth(μ)(t ) for all λ, μ ∈ Pn (h) The results of this paper were very much inspired by ideas and conjectures of Paul Martin His programme of studying non-Lie theoretic algebras via alcove geometries and his use of path theoretic bases for encoding representation-theoretic structures
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