On simple modules over twisted finite category algebras

Linckelmann, Markus; Stolorz, Michał

doi:10.1090/S0002-9939-2012-11381-4

On simple modules over twisted finite category algebras
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by Markus Linckelmann and Michał Stolorz PDF

Proc. Amer. Math. Soc. 140 (2012), 3725-3737 Request permission

Abstract:

The purpose of this paper is to show that the recent proof by Ganyushkin, Mazorchuk and Steinberg of the parametrisation of simple modules over finite semigroup algebras due to Clifford, Munn and Ponizovskiĭ carries over to twisted finite category algebras. We observe that the parametrisations of simple modules over Brauer algebras, Temperley-Lieb algebras, and Jones algebras due to Graham and Lehrer can be obtained as special cases of our main result. We further note that the notion of weights in the context of Alperin’s weight conjecture extends to twisted finite category algebras.

References

Hans Joachim Baues and Günther Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), no. 2-3, 187–211. MR 814176, DOI 10.1016/0022-4049(85)90008-8
D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
A. H. Clifford, Matrix representations of completely simple semigroups, Amer. J. Math. 64 (1942), 327–342. MR 6551, DOI 10.2307/2371687
A. H. Clifford, Basic representations of completely simple semigroups, Amer. J. Math. 82 (1960), 430–434. MR 116062, DOI 10.2307/2372967
Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 889050, DOI 10.1515/9783110858372.312
Olexandr Ganyushkin, Volodymyr Mazorchuk, and Benjamin Steinberg, On the irreducible representations of a finite semigroup, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3585–3592. MR 2529864, DOI 10.1090/S0002-9939-09-09857-8
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
J. A. Green, On the structure of semigroups, Ann. of Math. (2) 54 (1951), 163–172. MR 42380, DOI 10.2307/1969317
J. A. Green, On the indecomposable representations of a finite group, Math. Z. 70 (1958/59), 430–445. MR 131454, DOI 10.1007/BF01558601
James A. Green, Polynomial representations of $\textrm {GL}_{n}$, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR 606556
Reinhard Häring-Oldenburg, Cyclotomic Birman-Murakami-Wenzl algebras, J. Pure Appl. Algebra 161 (2001), no. 1-2, 113–144. MR 1834081, DOI 10.1016/S0022-4049(00)00100-6
Markus Linckelmann, Fusion category algebras, J. Algebra 277 (2004), no. 1, 222–235. MR 2059628, DOI 10.1016/j.jalgebra.2003.12.010
Markus Linckelmann, Alperin’s weight conjecture in terms of equivariant Bredon cohomology, Math. Z. 250 (2005), no. 3, 495–513. MR 2179608, DOI 10.1007/s00209-004-0753-x
M. Linckelmann, On inverse categories and transfer in cohomology, Skye conference proceedings (2009), to appear.
Wolfgang Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989. Mathematica Gottingensis. MR 1027600, DOI 10.1007/BFb0083681
W. D. Munn, Matrix representations of semigroups, Proc. Cambridge Philos. Soc. 53 (1957), 5–12. MR 82050, DOI 10.1017/s0305004100031935
I. S. Ponizovskiĭ, On matrix representations of associative systems, Mat. Sb. N.S. 38(80) (1956), 241–260 (Russian). MR 0081292
John Rhodes and Benjamin Steinberg, The $q$-theory of finite semigroups, Springer Monographs in Mathematics, Springer, New York, 2009. MR 2472427, DOI 10.1007/b104443
Hebing Rui and Changchang Xi, The representation theory of cyclotomic Temperley-Lieb algebras, Comment. Math. Helv. 79 (2004), no. 2, 427–450. MR 2059440, DOI 10.1007/s00014-004-0800-6
Hebing Rui and Weihua Yu, On the semi-simplicity of the cyclotomic Brauer algebras, J. Algebra 277 (2004), no. 1, 187–221. MR 2059627, DOI 10.1016/j.jalgebra.2004.02.017
Benjamin Steinberg, Möbius functions and semigroup representation theory. II. Character formulas and multiplicities, Adv. Math. 217 (2008), no. 4, 1521–1557. MR 2382734, DOI 10.1016/j.aim.2007.12.001
Jacques Thévenaz, $G$-algebras and modular representation theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1365077
Peter Webb, An introduction to the representations and cohomology of categories, Group representation theory, EPFL Press, Lausanne, 2007, pp. 149–173. MR 2336640
Stewart Wilcox, Cellularity of diagram algebras as twisted semigroup algebras, J. Algebra 309 (2007), no. 1, 10–31. MR 2301230, DOI 10.1016/j.jalgebra.2006.10.016
Changchang Xi, Partition algebras are cellular, Compositio Math. 119 (1999), no. 1, 99–109. MR 1711582, DOI 10.1023/A:1001776125173
Fei Xu, Representations of categories and their applications, J. Algebra 317 (2007), no. 1, 153–183. MR 2360144, DOI 10.1016/j.jalgebra.2007.07.021