Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T04:16:15.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalised Cogrowth series, random walks, and the group determinant

Published online by Cambridge University Press:  14 August 2017

STEPHEN P. HUMPHRIES
STEPHEN P. HUMPHRIES*
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. e-mail: steve@math.byu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We associate to a group G a series that generalises the cogrowth series of G and is related to a random walk on G. We show that the series is rational if and only if G is finite, generalizing a result of Kouksov [Kou]. We show that when G is finite, the series determines G. There are naturally occurring ideals and varieties that are acted on by Aut(G). We study these and generalize this to the context of S-rings over finite groups. There is an associated representation of Aut(G) and we characterize when this is irreducible.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 445 - 465
Copyright
Copyright © Cambridge Philosophical Society 2017 

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

[AL] Adams, William W. and Loustaunau, Philippe. An introduction to Gröbner bases. Graduate Studies in Mathematics 3 (American Mathematical Society, Providence 1994).Google Scholar
[AM] Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra (Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969), ix+128 pp.Google Scholar
[Ber] Bergeron, F. Algebraic combinatorics and coinvariant spaces. CMS Treatises in Mathematics. Canadian Mathematical Society, Ottawa, ON (A K Peters, Ltd., Wellesley, MA, 2009), viii+221.Google Scholar
[Bou] Bourbaki, N. Elements of Mathematics: Algebra I (Addison-Wesley, 1974).Google Scholar
[Br] Brauer, R. Representations of finite groups. Lectures in Modern Mathematics, vol. I, Editor Saaty, T. L. (Wiley, New York, 1963), 133–175.Google Scholar
[Cur] Curtis, C. W. Pioneers of representation theory: Frobenius, Burnside, Schur and Brauer. History of Mathematics 15 (American Mathematical Society, Providence, RI; London Mathematical Society, London, 1999), 287 pages.Google Scholar
[FS] Formanek, E. and Sibley, D. The group determinant determines the group. Proc. Amer. Math. Soc. 112 (1991), 649–656.Google Scholar
[Fr] Frobenius, F.G. Über vertauschbare Matrizen. S'ber. Akad. Wiss. Berlin (1896), 601–614.Google Scholar
[Ge] Gerstenhaber, M. On nilalgebras and linear varieties of nilpotent matrices. III. Ann. of Math. (2) 70 (1959) 167–205.Google Scholar
[G] Göbel, M. Computing bases for rings of permutation-invariant polynomials. J. Symbolic Comput. 19 (1995), no. 4, 285–291.Google Scholar
[Gr] Grigorchuk, R. I. Symmetrical random walks on discrete groups. Multicomponent random systems, pp. 285-325. Adv. Probab. Related Topics, 6 (Dekker, New York, 1980).Google Scholar
[GB] Grove, L. C. and Benson, C. T. Finite reflection groups. Second edition. Graduate Texts in Mathematics, 99 (Springer-Verlag, New York, 1985). x+133 pp.Google Scholar
[HoJ] Hoehnke, H.-J. and Johnson, K. W.. 3-characters are sufficient for the group determinant. Second International Conference on Algebra (Barnaul, 1991)), 193–206. Contemp. Math. 184 (Amer. Math. Soc., Providence, RI, 1995).Google Scholar
[Hul] Hulek, K. Elementary algebraic geometry. Translated from the 2000 German original by Helena Verrill. Student Mathematical Library, 20 (American Mathematical Society, Providence, RI, 2003). viii+213 pp.Google Scholar
[Hu] Humphries, S. P. Cogrowth of groups and the Dedekind–Frobenius group determinant. Math. Proc. Camb. Phil. Soc. 121 (1997), 193–217.Google Scholar
[HR] Humphries, S. P. and Rode, E. L. Weak Cayley tables and generalised centraliser rings of finite groups. To appear in Math Proc. Camb. Phil. Soc. (2012).Google Scholar
[HJM] Humphries, S. P., Johnson, K. W. and Misseldine, A. Commutative S-rings of maximal dimension, preprint (2013).Google Scholar
[Isa] Isaacs, I. M. Finite group theory. Graduate Studies in Mathematics, 92 (American Mathematical Society, Providence, RI, 2008). xii+350 pp.Google Scholar
[Ja] Jantzen, J. C. Nilpotent orbits in representation theory. Lie theory, 1-211. Progr. Math. 228 (Birkhuser Boston, Boston, MA, 2004).Google Scholar
[J] Johnson, K. W. On the group determinant. Math. Proc. Camb. Phil. Soc. 109 (1991), 299–311.Google Scholar
[Ke] Keller, J. Representations associated to the group matrix. MS thesis. Brigham Young University (2014), 45 pages.Google Scholar
[Kou] Kouksov, D. On rationality of the cogrowth series. Proc. Amer. Math. Soc. 126 (1998), 2845–2847.Google Scholar
[L] Lam, T. Y. Representations of finite groups: a hundred years. I. Not. Amer. Math. Soc. 45 (1998), no. 3, 361–372.Google Scholar
[Mac] Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. (The Clarendon Press, Oxford University Press, New York, 1995).Google Scholar
[MA] Bosma, W. and Cannon, J. MAGMA (University of Sydney, 1994).Google Scholar
[Man] Mansfield, R. A group determinant determines its group. Proc. Amer. Math. Soc. 116 (1992), 939–941.Google Scholar
[Mil] Milnor, J. Singular points of complex hypersurfaces. Annals of Math. Stud., No. 61 (Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1968), iii+122 pp.Google Scholar
[OV] Okounkov, A. and Vershik, A. A new approach to representation theory of symmetric groups. Selecta Math. (N.S.) 2 (1996), no. 4, 581–605.Google Scholar
[Rod] Rode, E. The 3-S-ring determines a finite group. Preprint (2012).Google Scholar
[R] Roggenkamp, K. W. From Dedekind's group determinant to the isomorphism problem. C. R. Math. Acad. Sci. Soc. R. Can. 21 (1999), 97–126.Google Scholar
[Sch] Schur, I. Zur Theorie der einfach transitiven Permutationsgruppen. Sitz. Preuss. Akad. Wiss. Berlin, Phys-math Klasse (1933), 598–623.Google Scholar
[Sc] Scott, W. R. Group theory (Dover, 1987).Google Scholar
[Sk] Skrzyński, M. On basic geometric properties of the cones of nilpotent matrices. Univ. Iagel. Acta Math. No. 33 (1996), 219–228.Google Scholar
[St1] Stanley, R. P. Hilbert functions of graded algebras. Advances in Math. 28 (1978), no. 1, 57–83.Google Scholar
[St2] Stanley, R. P. Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 475–511.Google Scholar
[St] Sturmfels, B. Algorithms in invariant theory. Second edition. Texts and Monographs in Symbolic Computation. (Springer Wien New York, Vienna, 2008), vi+197 p.Google Scholar
[VZ] Vyshnevetskiy, A. L. and Zhmud, E. M. Random walks on finite groups converging after finite number of steps. Algebra Discrete Math. (2008), no. 2, 123–129.Google Scholar
[Wie] Helmut, W. Zur theorie der einfach transitiven permutationsgruppen II. Math. Z. 52 (1949), 384–393.Google Scholar
[Wo] Woess, W. Cogrowth of groups and simple random walks. Arch. Math. (Basel) 41 (1983), no. 4, 363–370.Google Scholar