Skip to main content
SpringerLink
Log in

Cellular Structure of q-Brauer Algebras

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

Let R be a commutative noetherian domain. The q-Brauer algebras over R are shown to be cellular algebras in the sense of Graham and Lehrer. In particular, they are iterated inflations of Hecke algebras of type A. When R is a field of arbitrary characteristic, we determine for which parameters the q-Brauer algebras are quasi-hereditary. Then, using the general theory of cellular algebras we parametrize all irreducible representations of q-Brauer algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Birman, J.S., Wenzl, H.: Braids, link polynomials and a new algebra. Trans. Am. Math. Soc. 313(1), 249–273 (1989). MR0992598

    Article  MathSciNet  MATH  Google Scholar 

  2. Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. 63, 854–872 (1937)

    MathSciNet  Google Scholar 

  3. Brown, Wm. P.: An algebra related to the orthogonal group. Mich. Math. J. 3, 1–22 (1955). MR0072122

    Article  Google Scholar 

  4. Brown, W.P.: The semisimplicity of \(\omega\sb f\sp n\). Ann. Math. (2) 63, 324–335 (1956). MR0075931

    Article  MATH  Google Scholar 

  5. Cline, E., Parshall, B., Scott, L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988). MR0961165

    MathSciNet  MATH  Google Scholar 

  6. Dipper, R., James, G.: Representations of Hecke algebras of general linear groups. Proc. Lond. Math. Soc. (3) 52(1), 20–52 (1986). MR0812444

    Article  MathSciNet  MATH  Google Scholar 

  7. Dipper, R., James, G.: Blocks and idempotents of Hecke algebras of general linear groups. Proc. Lond. Math. Soc. (3) 54(1), 57–82 (1987). MR0872250

    Article  MathSciNet  MATH  Google Scholar 

  8. Doran, W.F., IV, Wales, D.B., Hanlon, P.J.: On the semisimplicity of the Brauer centralizer algebras. J. Algebra 211(2), 647–685 (1999). MR1666664

    Article  MathSciNet  MATH  Google Scholar 

  9. Dung, N.T.: A cellular basis of the q-Brauer algebra related with Murphy bases of the Hecke algebras of the symmetric groups. arXiv:1302.4272

  10. Hanlon, P., Wales, D.: On the decomposition of Brauer’s centralizer algebras. J. Algebra 121(2), 409–445 (1989). MR0992775

    Article  MathSciNet  MATH  Google Scholar 

  11. Hanlon, P., Wales, D.: Computing the discriminants of Brauer’s centralizer algebras. Math. Comput. 54(190), 771–796 (1990). MR1010599

    MathSciNet  MATH  Google Scholar 

  12. Geck, M.: Hecke algebras of finite type are cellular. Invent. Math. 169(3), 501–517 (2007). MR2336039

    Article  MathSciNet  MATH  Google Scholar 

  13. Graham, J.J., Lehrer, G.I.: Cellular algebras. Invent. Math. 123(1), 1–34 (1996). MR1376244

    Article  MathSciNet  MATH  Google Scholar 

  14. König, S., Xi, C.: On the structure of cellular algebras. In: Algebras and modules, II (Geiranger, 1996), pp. 365–386. CMS Conf. Proc., vol. 24. Amer. Math. Soc., Providence, RI (1996). MR1648638

  15. König, S., Xi, C.: Cellular algebras: inflations and Morita equivalences. J. Lond. Math. Soc. (2) 60(3), 700–722 (1999). MR1753809

    Article  MATH  Google Scholar 

  16. König, S., Xi, C.: When is a cellular algebra quasi-hereditary? Math. Ann. 315(2), 281–293 (1999). MR1721800

    Article  MathSciNet  MATH  Google Scholar 

  17. König, S., Xi, C.: A characteristic free approach to Brauer algebras. Trans. Am. Math. Soc. 353(4), 1489–1505 (2001). MR1806731

    Article  MATH  Google Scholar 

  18. Mathas, A.: Iwahori–Hecke algebras and Schur algebras of the symmetric group. University Lecture Series, vol. 15. Amer. Math. Soc., Providence, RI (1999). MR1711316

  19. Molev, A.I.: A new quantum analog of the Brauer algebra. Czechoslov. J. Phys. 53(11), 1073–1078 (2003). MR2074086

    Article  MathSciNet  Google Scholar 

  20. Murphy, G.E.: The representations of Hecke algebras of type \(A\sb n\). J. Algebra 173(1), 97–121 (1995). MR1327362

    Article  MathSciNet  MATH  Google Scholar 

  21. Murakami, J.: The Kauffman polynomial of links and representation theory. Osaka J. Math. 24(4), 745–758 (1987). MR0927059

    MathSciNet  MATH  Google Scholar 

  22. Parshall, B., Scott, L.: Derived categories, quasi-hereditary algebras and algebraic groups. In: Proceeding of the Ottawa-Moosonee Workshop in Algebra 1987. Math. Lecture Note Series, Carleton University (1988)

  23. Wenzl, H.: On the structure of Brauer’s centralizer algebras. Ann. Math. (2) 128(1), 173–193 (1988). MR0951511

    Article  MathSciNet  MATH  Google Scholar 

  24. Wenzl, H.: A q-Brauer algebra. J. Algebra 358, 102–127 (2012). MR2905021

    Article  MathSciNet  MATH  Google Scholar 

  25. Wenzl, H.: Quotients of representation rings. Represent. Theory 15, 385–406 (2011). MR2801174

    Article  MathSciNet  MATH  Google Scholar 

  26. Wenzl, H.: Fusion symmetric spaces and subfactors. Pac. J. Math. 259(2), 483–510 (2012). MR2988502

    Article  MathSciNet  MATH  Google Scholar 

  27. Xi, C.: Partition algebras are cellular. Compos. Math. 119(1), 99–109 (1999). MR1711582

    Article  MATH  Google Scholar 

  28. Xi, C.: On the quasi-heredity of Birman–Wenzl algebras. Adv. Math. 154(2), 280–298 (2000). MR1784677

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Education, Vinh University, Leduan 182, Vinh City, Vietnam

    Dung Nguyen Tien

Authors
  1. Dung Nguyen Tien
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dung Nguyen Tien.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nguyen Tien, D. Cellular Structure of q-Brauer Algebras. Algebr Represent Theor 17, 1359–1400 (2014). https://doi.org/10.1007/s10468-013-9452-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-013-9452-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation