Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-16T17:12:06.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Cellular bases of generalised q-Schur algebras

Published online by Cambridge University Press:  10 August 2016

STEPHEN DOTY  and
ANTHONY GIAQUINTO
STEPHEN DOTY
Affiliation:
Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660, U.S.A. e-mail: [email protected], [email protected]
ANTHONY GIAQUINTO
Affiliation:
Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60660, U.S.A. e-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Starting from their defining presentation by generators and relations, we develop the basic structure and representation theory of generalised q-Schur algebras of finite type.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 3 , May 2017 , pp. 533 - 560
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Beilinson, A. A., Lusztig, G. and MacPherson, R.. A geometric setting for the quantum deformation of GL n . Duke Math. J. 61 (1990), 655–677.Google Scholar
[2] Bourbaki, N.. Algebra I, Chapters 1–3 (Translated from the French). Reprint of the 1989 English translation “Elements of Mathematics” (Berlin) (Springer–Verlag, Berlin, 1998).Google Scholar
[3] Chriss, N. and Ginzburg, V.. Representation Theory and Complex Geometry (Birkhäuser Boston 1997).Google Scholar
[4] Dipper, R. and James, G. D.. The q-Schur algebra. Proc. London Math. Soc. 59 (1989), 23–50.CrossRefGoogle Scholar
[5] Dipper, R. and James, G. D.. q-tensor space and q-Weyl modules. Trans. Amer. Math. Soc. 327 (1991), 251–282.Google Scholar
[6] Donkin, S.. On Schur algebras and related algebras I. J. Algebra 104 (1986), 310–328.Google Scholar
[7] Donkin, S.. On Schur algebras and related algebras II. J. Algebra 111 (1987), 354–364.Google Scholar
[8] Donkin, S.. On Schur algebras and related algebras III. integral representations. Math. Proc. Camb. Phil. Soc. 116 (1994), 37–55.Google Scholar
[9] Doty, S.. Presenting generalised q-Schur algebras. Represent. Theory 7 (2003), 196–213 (electronic).CrossRefGoogle Scholar
[10] Doty, S.. Constructing quantised enveloping algebras via inverse limits of finite dimensional algebras. J. Algebra 321 (2009), 1225–1238.Google Scholar
[11] Doty, S. and Giaquinto, A.. Generators and relations for Schur algebras. Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 54–62 (electronic).Google Scholar
[12] Doty, S. and Giaquinto, A.. Presenting Schur algebras. Internat. Math. Res. Not. 36 (2002), 1907–1944.Google Scholar
[13] Goodman, F. and Graber, J.. On cellular algebras with Jucys Murphy elements. J. Algebra 330 (2011), 147–176.Google Scholar
[14] Graham, J. J. and Lehrer, G. I.. Cellular algebras. Invent. Math. 123 (1996), 1–34.Google Scholar
[15] Green, J. A.. Polynomial representations of GL n , Lecture Notes in Math. 830. (Springer–Verlag, Berlin–New York, 1980), (Second edition 2007.)Google Scholar
[16] Humphreys, J. E.. Introduction to Lie Algebras and Representation Theory (Springer 1972).Google Scholar
[17] Jantzen, J. C.. Lectures on Quantum Groups. American Math. Soc. (1996).Google Scholar
[18] Jimbo, M.. A q-analogue of $U(\mathfrak{gl}(N+1))$ , Hecke algebras and the Yang–Baxter equation. Lett. Math. Phys. 11 (1986), 247–252.Google Scholar
[19] Kashiwara, M., Miwa, T., Petersen, J.–U. H. and Yung, C. M.. Perfect crystals and q-deformed Fock spaces. Selecta Math. (N.S.) 2 (1996), 415–499.Google Scholar
[20] König, S. and Xi, C.. On the structure of cellular algebras. Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., 24 (Amer. Math. Soc., Providence, RI, 1998), 365–386.Google Scholar
[21] König, S. and Xi, C.. Cellular algebras and quasi-hereditary algebras: a comparison. Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 71–75 (electronic).Google Scholar
[22] Lusztig, G.. Canonical bases in tensor products. Proc. Nat. Acad. Sci. U.S.A. 89 (1992), 8177–8179.CrossRefGoogle ScholarPubMed
[23] Lusztig, G.. Introduction to Quantum Groups (Birkhäuser, Boston 1993).Google Scholar