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Cell algebras. (English) Zbl 1334.16014

Summary: A family of associative algebras called cell algebras is defined and studied. These algebras generalize the cellular algebras of Graham and Lehrer. Standard results for cellular algebras carry over nicely to the more general cell algebras, including the characterization of their irreducible modules in terms of a bilinear form, a description of their decomposition and Cartan matrices, and a description of their hereditary ideals and possible quasi-hereditary algebra structures.
As examples of cell algebras which are not cellular, the semigroup algebras \(R[\mathcal T_r]\) and \(R[\mathcal{PT}_r]\) corresponding to the full transformation semigroup \(\mathcal T_r\) and the partial transformation semigroup \(\mathcal{PT}_r\) are shown to be cell algebras and cell algebra bases are obtained for these (and related) algebras. The general cell algebra theory is then applied to classify the irreducible representations of these algebras (when \(R\) is any field of characteristic 0 or \(p\)). In certain cases the algebras are found to be quasi-hereditary.

MSC:

16G30 Representations of orders, lattices, algebras over commutative rings
20C08 Hecke algebras and their representations
20M25 Semigroup rings, multiplicative semigroups of rings
16S36 Ordinary and skew polynomial rings and semigroup rings
20M20 Semigroups of transformations, relations, partitions, etc.
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[1] Clifford, A. H.; Preston, G. B., The Algebraic Theory of Semigroups, vol. 1, Math. Surveys, vol. 7 (1961), American Mathematical Society: American Mathematical Society Providence · Zbl 0111.03403
[2] Cline, E.; Parshall, B.; Scott, L., Finite dimensional algebras and highest weight categories, Math. Ann., 259, 153-199 (1982)
[3] Curtis, Charles W.; Reiner, Irving, Representation Theory of Finite Groups and Associative Algebras (2006), AMS Chelsea Publishing: AMS Chelsea Publishing Providence, RI, reprint of the 1962 original · Zbl 1093.20003
[4] East, James, Cellular algebras and inverse semigroups, J. Algebra, 296, 2, 505-519 (2006) · Zbl 1127.16021
[5] Ganyushkin, O.; Mazorchuk, V., Classical Finite Transformation Semigroups. An Introduction, Algebra and Applications, vol. 9 (2009), Springer-Verlag London, Ltd.: Springer-Verlag London, Ltd. London · Zbl 1166.20056
[6] Graham, J. J.; Lehrer, G. I., Cellular algebras, Invent. Math., 123, 1-34 (1996) · Zbl 0853.20029
[7] Guo, Xiaojiang; Xi, Changchang, Cellularity of twisted semigroup algebras, J. Pure Appl. Algebra, 213, 1, 71-86 (2009) · Zbl 1158.16010
[8] Howie, John M., Fundamentals of Semigroup Theory, London Mathematical Society Monographs, New Series, vol. 12 (1995), Oxford Science Publications, The Clarendon Press, Oxford University Press: Oxford Science Publications, The Clarendon Press, Oxford University Press New York · Zbl 0835.20077
[9] May, Robert, Representations of certain generalized Schur algebras, J. Algebra, 333, 180-201 (2011) · Zbl 1242.20072
[10] May, Robert, Double coset algebras, J. Pure Appl. Algebra, 218, 2081-2095 (2014) · Zbl 1327.20001
[11] May, Robert; Abrams, William, A generalization of the Schur algebra to \(k [\tau_r]\), J. Algebra, 295, 524-542 (2006) · Zbl 1102.20010
[12] Mathas, Andrew, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group (1999), American Mathematical Society: American Mathematical Society Providence · Zbl 0940.20018
[13] Putcha, Mohan, Complex representations of finite monoids II. Highest weight categories and quivers, J. Algebra, 205, 53-76 (1998) · Zbl 0913.20041
[14] Wilcox, Stewart, Cellularity of diagram algebras as twisted semigroup algebras, J. Algebra, 309, 1, 10-31 (2007) · Zbl 1154.16021
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