On graded presentations of Hecke algebras and their generalizations
Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 1-38.

In this paper, we define a number of closely related isomorphisms. On one side of these isomorphisms sit a number of of algebras generalizing the Hecke and affine Hecke algebras, which we call the “Hecke family”; on the other, we find generalizations of KLR algebras in finite and affine type A, the “KLR family.”

We show that these algebras have compatible isomorphisms generalizing those between Hecke and KLR algebras given by Brundan and Kleshchev. This allows us to organize a long list of algebras and categories into a single system, including (affine/cyclotomic) Hecke algebras, (affine/cyclotomic) q-Schur algebras, (weighted) KLR algebras, category 𝒪 for 𝔤𝔩 N and for the Cherednik algebras for the groups /eS n , and give graded presentations of all of these objects.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.84
Classification: 20C08
Keywords: Hecke algebra, Khovanov-Lauda-Rouquier algebra, Cherednik algebra

Webster, Ben 1

1 Department of Pure Mathematics University of Waterloo & Perimeter Institute for Mathematical Physics Waterloo, ON, Canada
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Webster, Ben. On graded presentations of Hecke algebras and their generalizations. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 1-38. doi : 10.5802/alco.84. https://alco.centre-mersenne.org/articles/10.5802/alco.84/

[1] Ariki, Susumu Graded q-Schur algebras, 2009 (https://arxiv.org/abs/0903.3453)

[2] Ariki, Susumu; Koike, Kazuhiko A Hecke algebra of (/r)𝔖 n and construction of its irreducible representations, Adv. Math., Volume 106 (1994) no. 2, pp. 216-243 | DOI | MR | Zbl

[3] Brundan, Jonathan; Kleshchev, Alexander Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math., Volume 178 (2009) no. 3, pp. 451-484 | DOI | MR | Zbl

[4] Dipper, Richard; James, Gordon; Mathas, Andrew Cyclotomic q-Schur algebras, Math. Z., Volume 229 (1998) no. 3, pp. 385-416 | DOI | MR | Zbl

[5] Dipper, Richard; Mathas, Andrew Morita equivalences of Ariki–Koike algebras, Math. Z., Volume 240 (2002) no. 3, pp. 579-610 | DOI | MR | Zbl

[6] Gautam, Sachin; Toledano Laredo, Valerio Yangians and quantum loop algebras, Selecta Math. (N.S.), Volume 19 (2013) no. 2, pp. 271-336 | DOI | MR | Zbl

[7] Ginzburg, Victor; Guay, Nicolas; Opdam, Eric; Rouquier, Raphaël On the category 𝒪 for rational Cherednik algebras, Invent. Math., Volume 154 (2003) no. 3, pp. 617-651 | DOI | MR | Zbl

[8] Green, Richard M. The affine q-Schur algebra, J. Algebra, Volume 215 (1999) no. 2, pp. 379-411 | DOI | MR | Zbl

[9] Hu, Jun; Mathas, Andrew Seminormal forms and cyclotomic quiver Hecke algebras of type A, Math. Ann., Volume 364 (2016) no. 3-4, pp. 1189-1254 | DOI | MR | Zbl

[10] Jimbo, Michio A q-analogue of U(𝔤𝔩(N+1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys., Volume 11 (1986) no. 3, pp. 247-252 | DOI | MR | Zbl

[11] Khovanov, Mikhail; Lauda, Aaron D. A diagrammatic approach to categorification of quantum groups. I, Represent. Theory, Volume 13 (2009), pp. 309-347 | DOI | MR | Zbl

[12] Khovanov, Mikhail; Lauda, Aaron D. A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc., Volume 363 (2011) no. 5, pp. 2685-2700 | DOI | MR | Zbl

[13] Lusztig, George Affine Hecke algebras and their graded version, J. Amer. Math. Soc., Volume 2 (1989) no. 3, pp. 599-635 | DOI | MR | Zbl

[14] Macdonald, Ian G. Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki : volume 1994/95, exposés 790-804 (Astérisque), Société mathématique de France, 1996 no. 237, pp. 189-207 (Exp. No. 797) | MR | Zbl

[15] Macdonald, Ian G. Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge University Press, Cambridge, 2003, x+175 pages | DOI | MR | Zbl

[16] Maksimau, Ruslan; Stroppel, Catharina Higher level affine Schur and Hecke algebras (https://arxiv.org/abs/1805.02425)

[17] Miemietz, Vanessa; Stroppel, Catharina Affine quiver Schur algebras and p-adic GL n , Selecta Math. (N.S.), Volume 25 (2019) no. 32, 66 pages | DOI | MR | Zbl

[18] Rouquier, Raphaël 2-Kac-Moody algebras (https://arxiv.org/abs/0812.5023)

[19] Stroppel, Catharina; Webster, Ben Quiver Schur algebras and q-Fock space (https://arxiv.org/abs/1110.1115)

[20] Uglov, Denis Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials, Physical combinatorics (Kyoto, 1999) (Progr. Math.), Volume 191, Birkhäuser, 2000, pp. 249-299 | DOI | MR | Zbl

[21] Webster, Ben Representation theory of the cyclotomic Cherednik algebra via the Dunkl-Opdam subalgebra (https://arxiv.org/abs/1609.05494) | MR

[22] Webster, Ben Weighted Khovanov-Lauda-Rouquier algebras (to appear in Documenta Mathematica, https://arxiv.org/abs/1209.2463)

[23] Webster, Ben Knot invariants and higher representation theory, Mem. Amer. Math. Soc., 250, American Mathematical Society, 2017 no. 1191, 141 pages | DOI | MR | Zbl

[24] Webster, Ben Rouquier’s conjecture and diagrammatic algebra, Forum Math. Sigma, Volume 5 (2017), Paper no. e27, 71 pages | DOI | MR | Zbl

[25] Weibel, Charles A. The K-book. An introduction to algebraic K-theory, Graduate Studies in Mathematics, 145, American Mathematical Society, Providence, RI, 2013, xii+618 pages | MR | Zbl

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