Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Formulas for primitive idempotents in Frobenius algebras and an application to decomposition maps

Authors: Max Neunhöffer and Sarah Scherotzke
Journal: Represent. Theory 12 (2008), 170-185
MSC (2000): Primary 16G30; Secondary :, 16G99, 20C08, 20F55
Published electronically: March 19, 2008
MathSciNet review: 2390671
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the first part of this paper we present explicit formulas for primitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the well-known Frobenius-Schur relations for semisimple algebras.

The second part of this paper considers $ \mathcal{O}$-free $ \mathcal{O}$-algebras of finite $ \mathcal{O}$-rank over a discrete valuation ring $ \mathcal{O}$ and their decomposition maps under modular reduction modulo the maximal ideal of $ \mathcal{O}$, thereby studying the modular representation theory of such algebras.

Using the formulas from the first part we derive general criteria for such a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective indecomposable modules.

Finally, we show how this approach could eventually be used to attack a conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori-Hecke algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly.

References [Enhancements On Off] (What's this?)

  • 1. Richard Brauer, On hypercomplex arithmetic and a theorem of Speiser, Festschrift zum 60. Geburtstag von Prof. Dr. Andreas Speiser, Füssli, Zürich, 1945, pp. 233–245. MR 0014082
  • 2. Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979
  • 3. Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548
  • 4. Joseph Chuang and Kai Meng Tan, Filtrations in Rouquier blocks of symmetric groups and Schur algebras, Proc. London Math. Soc. (3) 86 (2003), no. 3, 685–706. MR 1974395, 10.1112/S0024611502013953
  • 5. Meinolf Geck, Brauer trees of Hecke algebras, Comm. Algebra 20 (1992), no. 10, 2937–2973. MR 1179271, 10.1080/00927879208824499
  • 6. Meinolf Geck, Representations of Hecke algebras at roots of unity, Astérisque 252 (1998), Exp. No. 836, 3, 33–55 (English, with French summary). Séminaire Bourbaki. Vol. 1997/98. MR 1685620
  • 7. Matthew Fayers and Kai Meng Tan, Adjustment matrices for weight three blocks of Iwahori-Hecke algebras, J. Algebra 306 (2006), no. 1, 76–103. MR 2271573, 10.1016/j.jalgebra.2006.01.054
  • 8. Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
  • 9. Meinolf Geck and Raphaël Rouquier, Centers and simple modules for Iwahori-Hecke algebras, Finite reductive groups (Luminy, 1994) Progr. Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, pp. 251–272. MR 1429875
  • 10. Gordon James, The decomposition matrices of 𝐺𝐿_{𝑛}(𝑞) for 𝑛\le10, Proc. London Math. Soc. (3) 60 (1990), no. 2, 225–265. MR 1031453, 10.1112/plms/s3-60.2.225
  • 11. G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442
  • 12. Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • 13. Jürgen Müller.
    Zerlegungszahlen für generische Iwahori-Hecke-Algebren von exzeptionellem Typ.
    Ph.D. thesis, RWTH Aachen, 1995.
    See http://www.math. rwth-aachen. de/~Juergen.Mueller/preprints/jm3.pdf
  • 14. Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 0000016
  • 15. Max Neunhöffer. Untersuchungen zu James' Vermutung über Iwahori-Hecke-Algebren vom Typ A.
    Ph.D. thesis, RWTH Aachen, 2003.
    See http://www.math.rwth-aachen. de/~Max.Neunhoeffer/Publications/phd.html
  • 16. Max Neunhöffer, Kazhdan-Lusztig basis, Wedderburn decomposition, and Lusztig’s homomorphism for Iwahori-Hecke algebras, J. Algebra 303 (2006), no. 1, 430–446. MR 2253671, 10.1016/j.jalgebra.2006.04.005

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 16G30, :, 16G99, 20C08, 20F55

Retrieve articles in all journals with MSC (2000): 16G30, :, 16G99, 20C08, 20F55

Additional Information

Max Neunhöffer
Affiliation: School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland, United Kingdom

Sarah Scherotzke
Affiliation: Mathematical Institute, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom

Keywords: Frobenius algebra, symmetric algebra, idempotent, explicit formula, Frobenius-Schur relations, projective indecomposable module, simple module, Grothendieck group, decomposition map, Coxeter group, Iwahori-Hecke algebra, James' conjecture
Received by editor(s): May 8, 2007
Received by editor(s) in revised form: February 9, 2008
Published electronically: March 19, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.