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

Neunhöffer, Max; Scherotzke, Sarah

doi:10.1090/S1088-4165-08-00326-9

Formulas for primitive idempotents in Frobenius algebras and an application to decomposition maps
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by Max Neunhöffer and Sarah Scherotzke

Represent. Theory 12 (2008), 170-185

DOI: https://doi.org/10.1090/S1088-4165-08-00326-9

Published electronically: March 19, 2008

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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

Richard Brauer, On hypercomplex arithmetic and a theorem of Speiser, Festschrift zum 60. Geburtstag von Prof. Dr. Andreas Speiser, Orell Füssli, Zürich, 1945, pp. 233–245. MR 0014082
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, Inc.), New York-London, 1962. MR 0144979
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
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, DOI 10.1112/S0024611502013953
Meinolf Geck, Brauer trees of Hecke algebras, Comm. Algebra 20 (1992), no. 10, 2937–2973. MR 1179271, DOI 10.1080/00927879208824499
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
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, DOI 10.1016/j.jalgebra.2006.01.054
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
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
Gordon James, The decomposition matrices of $\textrm {GL}_n(q)$ for $n\le 10$, Proc. London Math. Soc. (3) 60 (1990), no. 2, 225–265. MR 1031453, DOI 10.1112/plms/s3-60.2.225
G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442, DOI 10.1090/crmm/018
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
Jürgen Müller. Zerlegungszahlen für generische Iwahori-Hecke-Algebren von exzeptionellem Typ. Ph.D. thesis, RWTH Aachen, 1995. See http://www.math. rwth-aachen. de/\verb! !Juergen.Mueller/preprints/jm3.pdf
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
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/\verb! !Max.Neunhoeffer/Publications/phd.html
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, DOI 10.1016/j.jalgebra.2006.04.005