Some local literature resources

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Table of contents

1 Abstracts for Talks 2 Some Links 3 Partial orders 4 Homology

Abstracts for Talks

Nathan Fox (University of Minnesota):

• Cayley Graphs and Spectra of Semidirect Products

• Abstract: The spectrum of a graph is the set of eigenvalues of its adjacency matrix. A group, together with a set of generators, gives a Cayley graph, and a semidirect product is an interesting way of producing new groups. Beginning with the simplest examples, dihedral groups, I compared the spectra of cyclic groups to those of their semidirect products. It was found that many of the interesting identities that result can be described through graph coverings, number theory, representation theory, and even field theory.

Casey Green (University of Rochester),

Richard Gustavson (Cornell University):

• Laplacians of Covering Complexes

• Abstract: The Laplacian operator on a simplicial complex encodes information about the adjacencies between the simplices of common dimensions. Since the Laplacian is not a topological invariant, we look for relationships between simplicial complexes that might translate to relationships between their Laplacians. We found that one such relationship is that of a covering of a simplicial complex, which is a larger complex that contains the same adjacency relationships between its simplices. We show that the Laplacian spectrum of a complex is contained inside the Laplacian spectrum of any of its covering complexes.

Tejas Shah (Tufts University),

Meryl Spencer (Grinnell University):

• Cyclic Cayley Graphs and their Characteristic Polynomials

• Abstract:

Joseph Thurman (Vanderbilt University),

Kelsey Watson (Valparaiso University):

• Generators of the Symmetric Group

• Abstract: A permutation is an arrangement of n objects such that order matters. The collection of all permutations of length n forms the symmetric group. Composition of permutations generates a subgroup of , which can be represented graphically using Cayley graphs. This research deals with cases when two permutations generate .

Elizabeth Wicks (University of Washington-Seattle)

Some Links

Here is a link to the catalog for the library at Canisius [1] (http://cando.canisius.edu/search/X)

Here is a link to the journal databases at Canisius: [2] (http://library.canisius.edu/databases)

look for the icons called:

• JSTOR - Scanned journal articles up from more than 3-to 5-years ago.
• MathSciNet - Reviews and bibliographic data from Mathematical Reviews and Current Mathematical Publications.
• ScienceDirect - Full text of Elsevier Science journals in the physical sciences.

Here is a link to the catalog for the library at SUNY-Buffalo [3] (http://ublib.buffalo.edu/libraries/e-resources/bison/index.html)

• (hit the "Connect to BISON catalog" icon)

A. M. Nikitin, The Ihara-Selberg zeta function of a finite graph and symbolic dynamics, Algebra i Analiz 13 (2001), no. 5, 134–149; translation in St. Petersburg Math. J. 13 (2002), no. 5, 809–820.

A. M. Nikitin and A. B. Venkov, The Selberg trace formula, Ramanujan graphs and some problems in mathematical physics. (Russian), Algebra i Analiz 5 (1993), no. 3, 1–76; translation in St. Petersburg Math. J. 5 (1994), no. 3, 419–484.

[[4] (http://people.missouristate.edu/lesreid/reu/2009/)] has projects from the REU at Missouri State in the summer of 2009, including a powerpoint by Christopher Tomaszewski, "Spectral Analysis of Cyclic and Elementary Abelian Subgroup Lattices"

• Coxeter groups are almost convex, by Michael W. Davis and Michael Shapiro.

Abstract:  Cannon introduced the notion of almost convexity for the Cayley graph of a finitely generated group. In this paper, we observe that standard facts about Coxeter groups imply that the Cayley graph associated to any Coxeter system is almost convex.

• On the eigenvalues of the Coxeter Laplacian, by D.N. Akhiezer, Journal of Algebra. Volume 313, Issue 1, 1 July 2007, Pages 4-7, Special Issue in Honor of Ernest Vinberg.

Abstract: We find some eigenvalues of the Laplacian on the Cayley graph of a Coxeter group with respect to its Coxeter generators and give an upper bound for the minimal positive eigenvalue.

• Semi-edges, reflections and Coxeter groups, by Ralf Gramlich; Georg W. Hofmann; Karl-Hermann Neeb; 
Trans. Amer. Math. Soc. 359 (2007), 3647-3668.

• Characterization of Coxeter Systems by Reflections on the Cayley Graph
by Georg Hofmann, Geometriae Dedicata, Volume 39, Number 1 / July, 1991


Abstract: Coxeter systems are defined in terms of a presentation of a group by generators and relations. Many characterizations of Coxeter systems are known, the most prominent is the Exchange Condition. I propose to speak about a new characterization of Coxeter systems: A system (W, S) consisting of a group W and a generating set S is a Coxeter system if and only if the elements of S act on the Cayley graph of the group W (with respect to S) by reflections. To make this precise a new notion of a reflection on a graph is provided. This new characterization has proven to be an effective tool to identify as Coxeter groups reflection groups acting properly discontinuously on certain topological spaces, for example manifolds. These results are part of my PhD thesis `the Geometry of Reflection Groups' at the Technical University Darmstadt, 2004.

• Constance Davis, A bibliographical survey of groups with two generators and their relations, Courant Institute of Mathematical Sciences, New York University, 1972. at UB SCI/ENGR, Z6654 .D2

• Book: Topics in Geometric Group Theory (2000 Edition), by Pierre De La Harpe.

• Book: Concrete Abstract Algebra: From Numbers to Groebner Bases, by Niels Lauritzen.

• Michael J. Wester, A Critique of the Mathematical Abilities of CA Systems, 1999, [5] (http://www.math.unm.edu/~wester/cas/book/Wester.pdf)

• Book: Groebner Bases, Coding, and Cryptography, by Massimiliano Sala.

• Groebner bases for codes, by Mario de Boer and Ruud Pellikaan

• Discrete Fourier Transform and Gröbner Bases, Lecture Notes in Computer Science (Springer), Volume 1719/1999,

• Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, by A. Poli, M. C. Gennero and D. Xin.

Abstract: Using multivariate polynomials, Gröbner bases have a great theoretical interest in decoding cyclic codes beyond their BCH capability, but unfortunately have a high complexity. From engineers point of view, the complexity comes in particular from the number of needed indeterminates, from the maximal number of needed polynomials during Buchberger’s algorithm (this number is unknown), and from the maximal number of attempts before recovering the error polynomial e(X). In this paper we propose a new algorithm, using Gröbner bases and Discrete Fourier Transform. In most of the cases this algorithm needs fewer indeterminates than Chen et al. algorithm, and at most as many as for XP algorithm (sometimes less). In some cases the maximal number of needed polynomials for calculations is reduced to 1. Finally, it is shown that only one attempt is needed for recovering e(X).

Here are some notes from Wikipedia on Coxeter groups [6] (http://en.wikipedia.org/wiki/Coxeter_group)

Partial orders

A choice of reflection generators gives rise to a length function l on a Coxeter group, namely the minimum number of uses of generators required to express a group element; this is precisely the length in the word metric in the Cayley graph. An expression for v using l(v) generators is a reduced word. For example, the permutation (13) in S₃ has two reduced words, (12)(23)(12) and (23)(12)(23). The function defines a map generalizing the sign map for the symmetric group.

Using reduced words one may define three partial orders on the Coxeter group, the weak order, the absolute order and the Bruhat order (named for François Bruhat). An element v exceeds an element u in the Bruhat order if some (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some letters (in any position) are dropped. In the weak order, v ≥ u if some reduced word for v contains a reduced word for u as an initial segment. Indeed, the word length makes this into a graded poset. The Hasse diagrams corresponding to these orders are objects of study, and are related to the Cayley graph determined by the generators. The absolute order is defined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxeter generators.

For example, the permutation (1 2 3) in S₃ has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.

Homology

Since a Coxeter group W is generated by finitely many elements of order 2, its abelianization is an elementary abelian group|elementary abelian 2-group, i.e. it is isomorphic to the direct sum of several copies of the cyclic group Z₂. This may be restated in terms of the first group homology|homology group of W.

The Schur multiplier M(W) (related to the second homology) was computed in a paper by Ihara and Yokonuma (1965) for finite reflection groups and in a paper by Yokonuma (1965) for affine reflection groups, with a more unified account given in a paper by Howlett (1988). In all cases, the Schur multiplier is also an elementary abelian 2-group. For each infinite family {W_n} of finite or affine Weyl groups, the rank of M(W) stabilizes as n goes to infinity.