CYCLOTOMIC QUIVER HECKE ALGEBRAS OF TYPE A

Abstract

Contents n 1.6. Semisimple cyclotomic Hecke algebras of type A 1.7. Gram determinants and the Jantzen sum formula 1.8. The blocks of H F n 2. Cyclotomic quiver Hecke algebras of type A 2.1. Graded algebras 2.2. Cyclotomic quiver Hecke algebras 2.3. Nilpotence and small representations 2.4. Semisimple KLR algebras 2.5. The nil-Hecke algebra 3. Isomorphisms, Specht modules and categorification 3.1. The Graded Isomorphism Theorem 3.2. Graded Specht modules 3.3. Blocks and dual Specht modules 3.4. Induction and restriction 3.5. Grading Ariki's Categorification Theorem 3.6. Homogeneous Garnir relations 3.7. Graded adjustment matrices 4. Seminormal bases and the KLR grading 4.1. Gram determinants and graded dimensions 4.2. A deformation of the KLR grading 4.3. A distinguished homogeneous basis 4.4. A simple conjecture References

Restricting Specht modules of cyclotomic Hecke algebras

“…The general cyclotomic case was established by Ariki and Koike [3]. Here we follow the exposition of [17,25].…”

On the semisimplicity of the cyclotomic quiver Hecke algebra of type 𝐶

“…(ii) For all 1 j l, n−1 2 κ j − n−1 2 . Our proof that R Λ n is semisimple when the above two conditions hold is inspired by an argument from Mathas's survey [Mat15] in type A. In the other direction, when the conditions fail, we explicitly construct modules that have one-dimensional submodules, which we show have no complement, thus concluding that R Λ n is not semisimple.…”

On Bases of Some Simple Modules of Symmetric Groups and Hecke Algebras

“…for all admissible r and s. The algebra H n (ξ) is cellular, with cell modules S λ H parameterised by the partitions λ of n; see [22].…”

“…Recall that for p > 0, there is a unique square matrix A p -known as an adjustment matrix-such that D p = D 0 A p . Similarly, there is a unique graded adjustment matrix A p (q) such that D p (q) = D 0 (q)A p (q), see [22,Theorem 6.35], [4, Theorem 5.17]. Write A p (q) = (a λµ (q)) λ,µ∈RPar e, 2 (n) .…”