Hostname: page-component-cb9f654ff-mnl9s Total loading time: 0 Render date: 2025-08-03T11:05:57.588Z Has data issue: false hasContentIssue false
  • English
  • Français

TAME BLOCK ALGEBRAS OF HECKE ALGEBRAS OF CLASSICAL TYPE

Part of: Representation theory of rings and algebras Representation theory of groups

Published online by Cambridge University Press:  20 September 2019

SUSUMU ARIKI
SUSUMU ARIKI*
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan e-mail: ariki@ist.osaka-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We classify tame block algebras of Hecke algebras of classical type over an algebraically closed field of characteristic not equal to two.

Keywords

Hecke algebratame representation typetilting mutation

MSC classification

Primary: 20C08: Hecke algebras and their representations
Secondary: 20C33: Representations of finite groups of Lie type 16G20: Representations of quivers and partially ordered sets 16G60: Representation type (finite, tame, wild, etc.)

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 2 , October 2021 , pp. 179 - 201
Check for updates
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Communicated by A. Henderson

On the occasion of Professor Richard Dipper’s retirement

This research is supported in part by JSPS Grants-in-Aid for Scientific Research Grant Number JP18K03212.

References

Adachi, T., Aihara, T. and Chan, A., ‘Classification of two-term tilting complexes over Brauer graph algebras’, Math. Z. 290 (2018), 1–36.CrossRefGoogle Scholar
Aihara, T., ‘Tilting connected symmetric algebras’, Algebr. Represent. Theory 16 (2013), 873–894.CrossRefGoogle Scholar
Aihara, T., ‘Derived equivalences between symmetric special biserial algebras’, J. Pure Appl. Algebra 219 (2015), 1800–1825.CrossRefGoogle Scholar
Aihara, T. and Iyama, O., ‘Silting mutation in triangulated categories’, J. Lond. Math. Soc. (2) 85 (2012), 633–668.CrossRefGoogle Scholar
Aihara, T. and Mizuno, Y., ‘Classifying tilting complexes over preprojective algebras of Dynkin type’, Algebra Number Theory 11 (2017), 1287–1315.CrossRefGoogle Scholar
Antipov, M. and Zvonareva, A., ‘On stably biserial algebras and the Auslander–Reiten conjecture for special biserial algebras’, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Stekelov. (POMI) 460 (2017), 5–34.Google Scholar
Ariki, S., ‘On the decomposition numbers of the Hecke algebra of G (m, 1, n)’, J. Math. Kyoto Univ. 36 (1996), 789–808.Google Scholar
Ariki, S., ‘Lectures on cyclotomic Hecke algebras’, in: Quantum Groups and Lie Theory (Durham 1999), London Mathematical Society Lecture Note Series, 290 (Cambridge University Press, Cambridge, 2001), 1–22.Google Scholar
Ariki, S., ‘On the classification of simple modules for cyclotomic Hecke algebras of type G (m, 1, n) and Kleshchev multipartitions’, Osaka J. Math. 38 (2001), 827–837.Google Scholar
Ariki, S., ‘Finite dimensional Hecke algebras’, in: Trends in Representation Theory of Algebras and Related Topics, EMS Series of Congress Reports (European Mathematical Society, Zurich, 2008), 1–48.Google Scholar
Ariki, S., ‘Representation type for block algebras of Hecke algebras of classical type’, Adv. Math. 317 (2017), 823–845.CrossRefGoogle Scholar
Ariki, S., Iijima, K. and Park, E., ‘Representation type of finite quiver Hecke algebras of type A ℓ (1) for arbitrary parameters’, Int. Math. Res. Not. IMRN 15 (2015), 6070–6135.CrossRefGoogle Scholar
Ariki, S. and Mathas, A., ‘The number of simple modules of the Hecke algebras of type G (r, 1, n)’, Math. Z. 233 (2000), 601–623.CrossRefGoogle Scholar
Brundan, J., Dipper, R. and Kleshchev, A., Quantum Linear Groups and Representations of GLn(𝔽q), Memoirs of the American Mathematical Society, 149 (American Mathematical Society, Providence, RI, 2001).Google Scholar
Brundan, J. and Kleshchev, A., ‘Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras’, Invent. Math. 178 (2009), 451–484.CrossRefGoogle Scholar
Brundan, J. and Kleshchev, A., ‘Graded decomposition numbers for cyclotomic Hecke algebras’, Adv. Math. 222 (2009), 1883–1942.CrossRefGoogle Scholar
Chan, A., König, S. and Liu, Y., ‘Simple-minded systems, configurations and mutations for representation-finite self-injective algebras’, J. Pure Appl. Algebra 219 (2015), 1940–1961.CrossRefGoogle Scholar
Chuang, J. and Rouquier, R., ‘Derived equivalences for symmetric groups and sl 2 -categorification’, Ann. of Math. (2) 167(1) (2008), 245–298.CrossRefGoogle Scholar
Dipper, R., ‘On the decomposition numbers of the finite general linear groups’, Trans. Amer. Math. Soc. 290 (1985), 315–344.CrossRefGoogle Scholar
Dipper, R., ‘On the decomposition numbers of the finite general linear groups II’, Trans. Amer. Math. Soc. 292 (1985), 123–133.CrossRefGoogle Scholar
Dipper, R. and James, G., ‘Identification of the irreducible modular representations of GLn(q)’, J. Algebra 104 (1986), 266–288.CrossRefGoogle Scholar
Dipper, R. and James, G., ‘Representations of Hecke algebras of general linear groups’, Proc. Lond. Math. Soc. (3) 52 (1986), 20–52.CrossRefGoogle Scholar
Dipper, R. and James, G., ‘Blocks and idempotents of Hecke algebras of general linear groups’, Proc. Lond. Math. Soc. (3) 54 (1987), 57–82.CrossRefGoogle Scholar
Dipper, R. and James, G., ‘The q-Schur algebra’, Proc. Lond. Math. Soc. (3) 59 (1989), 23–50.CrossRefGoogle Scholar
Dipper, R. and James, G., ‘Representations of Hecke algebras of type B n ’, J. Algebra 146 (1992), 454–481.CrossRefGoogle Scholar
Dipper, R., James, G. and Murphy, E., ‘Hecke algebras of type B n at roots of unity’, Proc. Lond. Math. Soc. (3) 70 (1995), 505–528.CrossRefGoogle Scholar
Erdmann, K. and Skowroński, A., ‘From Brauer graph algebras to biserial weighted surface algebras’, J. Algbr. Comb. (to appear) arXiv:1706.07693.Google Scholar
Geck, M., ‘Hecke algebras of finite type are cellular’, Invent. Math. 169 (2007), 501–517.CrossRefGoogle Scholar
Hu, J., ‘Crystal bases and simple modules for Hecke algebra of type D n ’, J. Algebra 267 (2003), 7–20.CrossRefGoogle Scholar
James, G., ‘The irreducible representations of the finite general linear groups’, Proc. Lond. Math. Soc. (3) 52 (1986), 236–268.CrossRefGoogle Scholar
Kang, S.-J. and Kashiwara, M., ‘Categorification of highest weight modules via Khovanov–Lauda–Rouquier algebras’, Invent. Math. 190(3) (2012), 699–742.CrossRefGoogle Scholar
Kashiwara, M., ‘Biadjointness in cyclotomic Khovanov–Lauda–Rouquier algebras’, Publ. Res. Inst. Math. Sci. 48 (2012), 501–524.CrossRefGoogle Scholar
Krause, H., ‘Stable equivalence preserves representation type’, Comment. Math. Helv. 72 (1997), 266–284.CrossRefGoogle Scholar
Lyle, S. and Mathas, A., ‘Blocks of cyclotomic Hecke algebras’, Adv. Math. 216 (2007), 854–878.CrossRefGoogle Scholar
Rickard, J., ‘Morita theory for derived categories’, J. Lond. Math. Soc. (2) 39 (1989), 436–456.CrossRefGoogle Scholar
Rickard, J., ‘Derived categories and stable equivalence’, J. Pure Appl. Algebra 61 (1989), 303–317.CrossRefGoogle Scholar
Rickard, J., ‘Derived equivalences as derived functors’, J. Lond. Math. Soc. (2) 43 (1991), 37–48.CrossRefGoogle Scholar