On the structure of cyclotomic nilHecke algebras

Hu, Jun; Liang, Xinfeng

doi:10.2140/pjm.2018.296.105

Cited by 9 publications

(10 citation statements)

References 28 publications

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“…, and for each [14,Theorem 2.34], Theorem 4.8 and Corollary 3.21, one sees that γ is a K-linear isomorphism.…”

Section: 1mentioning

confidence: 95%

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Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

Hu¹,

Shi²

2021

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In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra R Λ (β) associated to an arbitrary symmetrizable Cartan matrix A = (a ij ) i,j ∈ I, where Λ ∈ P + and β ∈ Q + n . As applications, we obtain some necessary and sufficient conditions for the KLR idempotent e(ν) (for any ν ∈ I β ) to be nonzero in the cyclotomic quiver Hecke algebra R Λ (β). We decompose dim R Λ (β) into a sum of some products of dim R Λ i (β i ) with Λ = i Λ i and β = i β i . We construct some explicit monomial bases for the subspaces e( ν)R Λ (β)e(µ) and e( ν)R Λ (β)e(µ) of R Λ (β), where µ ∈ I β is arbitrary and ν ∈ I β is a certain specific n-tuple (see Section 4). Finally, we show that R Λ (β) is indecomposable if β = n i=1 α i with {α i |1 ≤ i ≤ n} being distinct.

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“…, and for each [14,Theorem 2.34], Theorem 4.8 and Corollary 3.21, one sees that γ is a K-linear isomorphism.…”

Section: 1mentioning

confidence: 95%

“…As is well known, constructing monomial bases for the cyclotomic quiver Hecke algebra R Λ (β) is a challenging problem. The first author of this paper and Liang have constructed a monomial basis for the cyclotomic nilHecke algebra in [14]. In general, even in the special case of type A, none of such monomial bases is known at the moment.…”

mentioning

confidence: 99%

Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

Hu¹,

Shi²

2021

Preprint

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“…The readers can find that the above corollary is an analogue of the corresponding result [4,Theorem 2.34] for the usual cyclotomic nilHecke algebra of type A. However, the original argument does not transfer to the cyclotomic odd nilHecke case.…”

Section: Introductionmentioning

confidence: 90%

Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

Hu¹,

Shi²

2021

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In this paper we derive a closed formula for the (Z × Z 2 )-graded dimension of the cyclotomic quiver Hecke superalgebra R Λ (β) associated to an arbitrary Cartan superdatum (A, P, Π, Π ∨ ), polynomialsn and Λ ∈ P + . As applications, we obtain a necessary and sufficient condition for which e(ν) = 0 in R Λ (β). We construct an explicit monomial basis for the bi-weight space e( ν)R Λ (β)e( ν), where ν is a certain specific ntuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β = α 1 +α 2 +• • •+αn with α 1 , • • • , αn distinct. We construct an explicit monomial basis of R Λ (β) and show that it is indecomposable in this case.

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“…The starting point of this work is to answer the above question. Using Borel's picture for the cohomology of the Grassmannian G n,ℓ , we construct an explicit Z-algebra isomorphism between a natural Z-form B of the basic algebra of H (0) ℓ,n and the cohomology ring H * (G n,ℓ , Z) of the Grassmannian G n,ℓ such that the purely algebraic defined element b(λ)(denoted by b λ in [7]) is sent to the geometrically defined Schubert class basis element (a 1 , • • • , a n ), where λ ∈ P(0) is the ℓ-multipartition associated to the n-tuple…”

Section: Z(h (0)mentioning

confidence: 99%

“…In a recent work [7], the second author and Xinfeng Liang constructed a monomial basis of the cyclotomic nilHecke algebra H…”

Section: Introductionmentioning

confidence: 99%