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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: [email protected]
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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 

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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