Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T12:04:26.450Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical groups for Hopf algebra modules

Published online by Cambridge University Press:  07 November 2018

DARIJ GRINBERG ,
JIA HUANG  and
VICTOR REINER
DARIJ GRINBERG
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. e-mail: [email protected]
JIA HUANG
Affiliation:
Department of Mathematics and Statistics, University of Nebraska at Kearney, Kearney, NE 68849, U.S.A. e-mail: [email protected]
VICTOR REINER
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 473 - 503
Copyright
Copyright © Cambridge Philosophical Society 2018

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.)

Footnotes

†

Supported by NSF grants DMS-1148634, 1601961.

References

REFERENCES

Benkart, G., Klivans, C. and Reiner, V.Chip firing on Dynkin diagrams and McKay quivers. Math. Z. 290 (2018) 615–648. A preprint appears on arXiv: 1601.06849v2.CrossRefGoogle Scholar
Brauer, R.A note on theorems of Burnside and Blichfeldt. Proc. Amer. Math. Soc. 15 (1964), 31–34.CrossRefGoogle Scholar
Brauer, R.On the Cartan invariants of groups of finite order. Ann. of Math. 42 (1941), 53–61.CrossRefGoogle Scholar
Burciu, S.Kernels of representations and coideal subalgebras of Hopf algebras. Glasgow Math. J. 54 (2012), 107–119. A preprint appears on arXiv: 1012.3096v1.CrossRefGoogle Scholar
Cibils, C.A quiver quantum group. Comm. Math. Phys. 157 (1993), 459–477.CrossRefGoogle Scholar
Curtis, C. W. and Reiner, I.Representation Theory of Finite Groups and Associative Algebras and (John Wiley and Sons, 1962).Google Scholar
Curtis, C. W. and Reiner, I.Methods of Representation Theory, Volume I (Wiley, 1981).Google Scholar
Dăscălescu, S., Năstăsescu, C. and Raianu, S.Hopf Algebras. An Introduction (Marcel Dekker, Inc., New York, 2001).Google Scholar
Dummit, D. and Foote, R. M.Abstract Algebra (3rd edition) (John Wiley and Sons, Inc., 2004.)Google Scholar
Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V.Tensor Categories (Amer. Math. Soc., Providence, RI, 2015).CrossRefGoogle Scholar
Gaetz, C.Critical groups of group representations. Lin. Alg. Appl. 508 (2016), 91–99.CrossRefGoogle Scholar
Grinberg, D.math.stackexchange answer # 2147742 (“Nonsingular M-matrices are nonsingular”). http://math.stackexchange.com/q/2147742Google Scholar
Humphreys, J. E.Modular Representations of Finite Groups of Lie Type (Cambridge University Press, Cambridge, 2006).Google Scholar
James, G. and Kerber, A.The Representation Theory of the Symmetric Group (Addison–Wesley Publishing Co., Reading, Mass., 1981).Google Scholar
James, G. and Liebeck, M.Representations and Characters of Groups (Cambridge University Press, New York, 2001).CrossRefGoogle Scholar
Lorenzini, D.Smith normal form and Laplacians. J. Combin. Theory Ser. B 98 (2008), 1271–1300.CrossRefGoogle Scholar
Holroyd, A. E., Levine, L., Mészáros, K., Peres, Y., Propp, J. and Wilson, D.B.Chip-firing and rotor-routing on directed graphs. In and Out of Equilibrium 2, Progress in Probability 60 (2008) 331–364. Updated version available at arXiv: 0801.3306v4.Google Scholar
Li, L. and Zhang, Y.The Green rings of the generalized Taft Hopf algebras. Hopf algebras and tensor categories, Contemp. Math. 585 (Amer. Math. Soc., Providence, RI, 2013), 275–288.CrossRefGoogle Scholar
Lorenz, M.Representations of finite-dimensional Hopf algebras. J. Algebra 188 (1997), 476–505.CrossRefGoogle Scholar
Montgomery, S.Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics 82, (1993).CrossRefGoogle Scholar
Nichols, W. D.Bialgebras of type one. Comm. Algebra 6 (1978), 1521–1552.CrossRefGoogle Scholar
Pareigis, B.When Hopf algebras are Frobenius algebras. J. Algebra 18 (1971), 588–596.CrossRefGoogle Scholar
Passman, D. S. and Quinn, D.Burnside's theorem for Hopf algebras. Proc. Amer. Math. Soc. 123 (1995), 327–333.Google Scholar
Perkinson, D., Perlman, J. and Wilmes, J.Primer for the algebraic geometry of sandpiles. Contemp. Math. 605 (Amer. Math. Soc., Providence, RI). arXiv preprint arXiv:1112.6163v2 (2011) arXiv: 1112.6163v2.Google Scholar
Plemmons, R. J.M-matrix characterisations. I. Nonsingular M-matrices. Linear Algebra Appl. 18 (1977), 175–188.CrossRefGoogle Scholar
Radford, D. E.Hopf Algebras (World Scientific, 2012).Google Scholar
Rieffel, M. A.Burnside's theorem for representations of Hopf algebras. J. Algebra 6 (1967), 123–130.CrossRefGoogle Scholar
Schneider, H.-J. Lectures on Hopf algebras. Lecture notes, 31 March 2006. http://www.famaf.unc.edu.ar/series/pdf/pdfBMat/BMat31.pdfGoogle Scholar
Serre, J.-P.Linear Representations of Finite Groups (Springer, 1977).CrossRefGoogle Scholar
Steinberg, R.Complete sets of representations of algebras. Proc. Amer. Math. Soc. 13 (1962), 746–747.CrossRefGoogle Scholar
Webb, P.A Course in Finite Group Representation Theory (Cambridge University Press, 2016).CrossRefGoogle Scholar
Witherspoon, S. J.Cohomology of Hopf algebras. Lecture notes, 10 January 2017. http://www.math.tamu.edu/~sjw/pub/hopf-cohom.pdfGoogle Scholar