Brundan, Jonathan; Kleshchev, Alexander Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. (English) Zbl 1201.20004 Invent. Math. 178, No. 3, 451-484 (2009). This is an important paper that has opened up a new line of investigation within modular representation theory. Several papers that relate directly to it have already appeared. It is concerned with cyclotomic quotients of the affine Hecke algebra and of its rational degeneration. A main outcome of the paper is that these algebras are \(\mathbb{Z}\)-graded algebras in a non-trivial way, thus allowing for the study of graded representation theory of these algebras. The group algebra of the symmetric group in positive characteristic and the Iwahori-Hecke algebra of type \(A\), for any value of the parameter \(q\), are covered as special cases. Let \(F\) be a ground field containing \(q\) and let \(e\) be the smallest positive integer satisfying \[ 1+q+\cdots+q^{e-1}=0 \] (where \(e:=0\) if no such integer exists). Let \(I=\mathbb{Z}/e\mathbb{Z}\) and let \(\Gamma\) be a cyclic quiver of type \(A^{(1)}_{e-1}\) (\(=A_\infty\) if \(e=0\)) with vertex set \(I\). The authors define the algebra \(H_d\) to be either the affine Hecke algebra on generators \(T_1,\dots,T_{d-1}\) and \(X_1^{\pm 1},\dots,X_{d}^{\pm 1}\) or its rational degeneration on generators \(s_1,\dots,s_{d-1}\) and \(x_1,\dots,x_d\) depending on \(q\neq 1\) or \(q=1\). They next define the algebra \(H_d^\Lambda\) as a cyclotomic quotient of \(H_d\), depending on an additional parameter \(\Lambda\). There is a system of idempotents \(e(\mathbf i)\) for \(\mathbf i\in I^d\) which gives rise to blocks algebras \(H_\alpha^\Lambda:=e_\alpha H_d^\Lambda\) where \(e_\alpha:=\sum_{\mathbf i\in I^\alpha}e(\mathbf i)\) for \(I^\alpha\) an orbit of the symmetric group \(S_d\) in \(I^d\). These algebras are the first main object of the paper. The second main object of the paper is the algebra \(R_\alpha^\Lambda\) introduced by Khovanov and Lauda, and independently by Rouquier in the affine case. The authors denote it the cyclotomic Khovanov-Lauda algebra and present it via generators \[ \{e(\mathbf i)\mid\mathbf i\in I^\alpha\}\cup\{y_1,\dots,y_d\}\cup\{\psi_1,\dots,\psi_{d-1}\} \] and a list of 10 kinds of relations between them that depend on \(\Gamma\). This varies slightly from the presentation used by Khovanov-Lauda that does not include the \(e=2\) case. The presentation makes \(R_\alpha^\Lambda\) a \(\mathbb{Z}\)-graded algebra in a concrete way. The authors now obtain their results by proving in their main Theorem that \(H_\alpha^\Lambda\) and \(R_\alpha^\Lambda\) are isomorphic algebras. The main part of the proof consists in finding appropriate elements \(e(\mathbf i)\), \(y_r\) and \(\psi_r\) in \(H_\alpha^\Lambda\) and verifying that these satisfy the above relations. This verification is the most difficult part of the proof and involves long calculations. The calculations are done first in the degenerate case, which is the easiest, then in the non-degenerate case. In the degenerate case, for example, the \(e(\mathbf i)\) are the elements mentioned above, \(y_r:=\sum_{\mathbf i\in I^\alpha}(x_r-i_r)e(\mathbf i)\) whereas \(\psi_r\) are suitably adjusted versions of the following intertwining elements \[ \phi_r:=s_r+\sum_{i_r\neq i_{r+1}}\frac{1}{x_r-x_{r+1}}\,e(\mathbf i)+\sum_{i_r=i_{r+1}}e(\mathbf i). \] Throughout, expressions involving power series in \(y_r\) are manipulated. They make sense because \(y_r\) are nilpotent. Apart from the results on \(\mathbb{Z}\)-gradings, the authors draw another couple of consequences from their main Theorem. One very immediate Corollary is that the cylotomic Hecke algebra is isomorphic to its degenerate analogue if \(q\) is generic and the characteristic of \(F\) is zero. Another consequence is a proof of a Conjecture due to A. Mathas, stating that the decomposition numbers of the cyclotomic Hecke algbra depend only on \(e\) and the characteristic of \(F\), not on \(F\) itself. Finally, they also explain how the case \(e=0\) may be seen as Young’s seminormal form. We point out that for Frobenius kernels of algebraic groups and their quantum analogues, \(\mathbb{Z}\)-gradings have been constructed by H. H. Andersen, J. C. Jantzen and W. Soergel [Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque. 220. Paris: Société Mathématique de France (1994; Zbl 0802.17009)]. The methods used there are completely different. Reviewer: Steen Ryom-Hansen (Talca) Cited in 14 ReviewsCited in 178 Documents MSC: 20C08 Hecke algebras and their representations 20C20 Modular representations and characters 16G20 Representations of quivers and partially ordered sets 16W50 Graded rings and modules (associative rings and algebras) 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory Keywords:affine Hecke algebras; cyclotomic Hecke algebras; rational degenerations; Iwahori-Hecke algebras; group algebras; symmetric groups; quiver algebras; graded algebras; generators; relations; decomposition numbers Citations:Zbl 0802.17009 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Ariki, S.: On the decomposition numbers of the Hecke algebra of G(m,1,n). J. Math. Kyoto Univ. 36, 789–808 (1996) · Zbl 0888.20011 [2] Ariki, S., Koike, K.: A Hecke algebra of (\(\mathbb{Z}\)/r\(\mathbb{Z}\))S n and construction of its irreducible representations. Adv. 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