Quiver Hecke algebras for alternating groups

Boys, Clinton; Mathas, Andrew

doi:10.1007/s00209-016-1732-8

Cited by 2 publications

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“…Before going further, we give the following lemma. Let us mention that we can find a noncyclotomic statement in [Rou,Corollary 3.8]; see also [Ma2,Proposition 2.4.6] and [BoMa,Lemma 1.16]. Lemma 6.32.…”

Section: A Distinguished Subalgebramentioning

confidence: 96%

Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras

Rostam¹

2017

Advances in Mathematics

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Section: A Distinguished Subalgebramentioning

confidence: 96%

Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras

Rostam¹

2017

Advances in Mathematics

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Cyclotomic quiver Hecke algebras and Hecke algebra of $G(r,p,n)$

Rostam

2018

Trans. Amer. Math. Soc.

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Given a quiver automorphism with nice properties, we give a presentation of the fixed point subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r, 1, n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r, p, n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In addition, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras.Thibon [LLT]. The theorem has the following consequence in characteristic 0. If all u k for 1 ≤ k ≤ r are powers of q, then determining the decomposition matrix of H n (q, u) or the canonical basis of a certain integrable highest weight sl e -module L(Λ) are equivalent problems, where sl e denotes the Kac-Moody algebra of type A (1) e−1 . Together with the work of Uglov [Ug] which computes this canonical basis, we are thus able to explicitly describe the decomposition matrix of H n (q, u).Once again in the semisimple case, Ariki [Ar95] used Clifford theory to determine all irreducible modules for H Λ p,n (q). In the modular case, Genet and Jacon [GeJa] and Chlouveraki and Jacon [ChJa] gave a parametrisation of the simple modules of H Λ p,n (q) over C, and Hu [Hu04, Hu07] classified them over a field containing a primitive pth root of unity. Furthermore, Hu and Mathas [HuMa09, HuMa12] gave a procedure to compute the decomposition matrix of H Λ p,n (q) in characteristic 0, under a separation condition (where the Hecke algebra is not semisimple in general). Let us also mention the work of Geck [Ge], who deals with the case of type D (corresponding to r = p = 2).Partially motivated by Ariki's theorem, Khovanov and Lauda [KhLau09, KhLau11] and Rouquier [Rou] independently introduced the algebra R n (Γ), known as a quiver Hecke algebra or KLR algebra. This led to a categorification result:We will first need to generalise the main result of [BrKl]. Surprisingly, combined with a theorem of [Ro17], this leads to the isomorphism realising the above Morita equivalence. We then exploit the fact that H Λ p,n (q) is the fixed point subalgebra of H n (q, u) for a particular automorphism σ, the shift automorphism. After some technical work, we obtain the cyclotomic quiver Hecke-like presentation for H Λ p,n (q). We deduce that H Λ p,n (q) depends only on the quantum characteristic and is a graded subalgebra of H n (q, u). We note that Boys and Mathas [Bo, BoMa] already studied a restriction of the isomorphism of [BrKl] to the fixed point subalgebra of an automorphism, in order to study alternating (cyclotomic) Hecke algebras. The approach taken here is globally similar, however, we are here able to use a trick of Stroppel and Webster [StWe] to simplify the final proof (see §4.1).We now give a brief overview of this article. Let r, p, d, n ∈ N * be some integers with r = dp, an element q = 0, 1 of a field F , a primitive pth root...

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Quiver Hecke algebras for alternating groups

Cited by 2 publications

References 20 publications

Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras

Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras

Cyclotomic quiver Hecke algebras and Hecke algebra of $G(r,p,n)$

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