Representation theory of q -rook monoid algebras
Rowena Paget
Mathematical Institute 24-29 St. Giles Oxford OX1 3LB England
DOI: 10.1007/s10801-006-0010-y
Abstract
We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.
Pages: 239–252
Keywords: keywords rook monoid; cellular algebra; iwahori-Hecke algebra; representation
Full Text: PDF
References
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16. C.C. Xi, “Partition algebras are cellular,” Compos. Math. 119 (1999), 99-109.
2. R. Dipper, and G. James, “Representations of Hecke algebras of general linear groups,” Proc. London Math. Soc. 52(3) (1986), 20-52. Springer J Algebr Comb (2006) 24:239-252
3. R. Dipper, G. James, “Blocks and idempotents of Hecke algebras of general linear groups,” Proc. London Math. Soc. 54(3) (1987), 57-82.
4. J. East, “Cellular algebras and inverse semigroups,” preprint.
5. J. Graham and G. Lehrer, “Cellular algebras,” Inv. Math. 123 (1996), 1-34.
6. C. Grood, “A Specht module analog for the rook monoid,” Electron. J. Combin. 9 (2002), 10 pp.
7. T. Halverson, “Representations of the q-rook monoid,” J. Algebra 273 (2004), 227-251.
8. T. Halverson and A. Ram, “q-rook monoid algebras, Hecke algebras and Schur-Weyl duality,” Zap. Nauchn. Sem. St. Petersburg, Otdel. Mt. Inst. Steklov (POMI) 283 (2001), 224-250.
9. G. James and A. Kerber, “The representation theory of the symmetric group,” vol. 16 of Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, Mass., 1981.
10. S. K\ddot onig and C. Xi, “On the structure of cellular algebras,” In Algebras and Modules II (Geiranger 1996) CMS Conf. Proc. 24, Amer. Math. Soc., Providence 1998, pp. 365-386.
11. S. K\ddot onig and C. Xi, “Cellular algebras: Inflations and Morita equivalences,” J. London Math. Soc. 60(2) (1999), 700-722.
12. S. K\ddot onig, and C.C. Xi, “A characteristic free approach to Brauer algebras,” Trans. Amer. Math. Soc. 353(4) (2001), 1489-1505.
13. W.D. Munn, “The characters of the symmetric inverse semigroup,” Proc. Cambridge Philos. Soc. 53 (1957), 13-18.
14. L. Solomon, “Representations of the rook monoid,” J. Algebra 256 (2002), 309-342.
15. L. Solomon, “The Iwahori algebra of Mn(Fq ). A presentation and a representation on tensor space,” J. Algebra 273 (2004), 206-226.
16. C.C. Xi, “Partition algebras are cellular,” Compos. Math. 119 (1999), 99-109.