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Chuang, J.; Turner, W. (2007)
Publisher: Elsevier BV
Journal: Advances in Mathematics
Languages: English
Types: Article
Subjects: QA, Mathematics(all), Mathematics - Combinatorics, Mathematics, 20G05, Mathematics - Representation Theory

We construct algebras from rhombohedral tilings of Euclidean space obtained as projections of certain cubical complexes. We show that these ‘Cubist algebras’ satisfy strong homological properties, such as Koszulity and quasi-heredity, reflecting the combinatorics of the tilings. We construct derived equivalences between Cubist algebras associated to local mutations in tilings. We recover as a special case the Rhombal algebras of Michael Peach and make a precise connection to weight 2 blocks of symmetric groups. © 2007 Elsevier Inc. All rights reserved.

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    • 1. Istv´an A´goston, Vlastimil Dlab, and Erzs´ebet Luk´acs, Standardly stratified extension algebras, Comm. Algebra 33 (2005), no. 5, 1357-1368.
    • 2. Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR MR1322847 (96k:17010)
    • 3. Joseph Chuang, The derived categories of some blocks of symmetric groups and a conjecture of Brou´e, J. Algebra 217 (1999), no. 1, 114-155.
    • 4. Joseph Chuang and Radha Kessar, Symmetric groups, wreath products, Morita equivalences, and Brou´e's abelian defect group conjecture, Bull. London Math. Soc. 34 (2002), no. 2, 174-184.
    • 5. Joseph Chuang and Rapha¨el Rouquier, Derived equivalences for symmetric groups and sl2-categorification, Ann. of Math. (2) (2005), to appear, math.RT/0407205.
    • 6. Joseph Chuang and Kai Meng Tan, Representations of wreath products of algebras, Preprint, 2001.
    • 7. E. Cline, B. Parshall, and L. Scott, Algebraic stratification in representation categories, J. Algebra 117 (1988), no. 2, 504-521. MR MR957457 (90d:18004)
    • 8. , Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85-99.
    • 9. Vlastimil Dlab, Quasi-hereditary algebras revisited, An. S¸tiin¸t. Univ. Ovidius Constan¸ta Ser. Mat. 4 (1996), no. 2, 43-54, Representation theory of groups, algebras, and orders (Constan¸ta, 1995).
    • 10. Y. Drozd and V. Mazorchuk, Koszul duality for extension algebras of standard modules, Tech. Report 2004:48, Uppsala University Department of Mathematics, 2004.
    • 11. K. Erdmann and S. Martin, Quiver and relations for the principal p-block of Σ2p, J. London Math. Soc. (2) 49 (1994), no. 3, 442-462.
    • 12. R. Fr¨oberg, Koszul algebras, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., vol. 205, Dekker, New York, 1999, pp. 337-350.
    • 13. Daniel Huybrechts and Richard Thomas, P-objects and autoequivalences of derived categories, Math. Res. Lett. 13 (2006), no. 1, 87-98.
    • 14. G. D. James, The representation theory of the symmetric groups, Springer, Berlin, 1978.
    • 15. , Some combinatorial results involving Young diagrams, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 1-10.
    • 16. Steffen K¨onig and Changchang Xi, On the structure of cellular algebras, Algebras and modules, II (Geiranger, 1996), CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 365-386.
    • 17. Joakim Linde, Cristopher Moore, and Mats G. Nordahl, An n-dimensional generalization of the rhombus tiling, Discrete models: combinatorics, computation, and geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discr`et. (MIMD), Paris, 2001, pp. 023-042 (electronic).
    • 18. Roberto Mart´ınez-Villa, Graded, selfinjective, and Koszul algebras, J. Algebra 215 (1999), no. 1, 34-72.
    • 19. Roberto Mart´ınez-Villa and Alex Martsinkovsky, Cohomology of tails, Tate-Vogel cohomology, and noncommutative Serre duality over Koszul quiver algebras, J. Algebra 280 (2004), no. 1, 58-83.
    • 20. Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999.
    • 21. Gabriele Nebe, The principal block of ZpS2p, J. Group Theory 5 (2002), no. 2, 163-176. MR MR1888074 (2002j:20027)
    • 22. M. Peach, Rhombal algebras and derived equivalences, Ph.D. thesis, University of Bristol, 2004, available at http://www.maths.bris.ac.uk/~majc/.
    • 23. M. J. Richards, Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 383-402.
    • 24. J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436-456.
    • 25. Jeremy Rickard, talk at MSRI, Berkeley, November 1990.
    • 26. J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), no. 2, 441-455.
    • 27. , Symmetric group blocks of defect two, Quart. J. Math. Oxford Ser. (2) 46 (1995), no. 182, 201-234.
    • 28. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
    • 29. K.M. Tan, Small defect blocks of symmetric group algebras, Ph.D. thesis, University of Cambridge, 1998.
    • 30. Will Turner, On seven families of algebras, 2005.
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