1,698 research outputs found
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Orthogonal Abelian Cartan Subalgebra Decompositions of Classical Lie Algebras Over Finite Commutative Rings
Orthogonal decompositions of classical Lie algebras over the complex numbers of types A, B, C and D were studied in the early 1980s and attracted further attention in the past decade, especially in the type A case, due to its application in quantum information theory. In this dissertation, we consider the orthogonal decomposition problem of Lie algebras of type A, B, C and D over a finite commutative ring with identity. We first establish the appropriate definition of orthogonal decomposition under our setting, and then derive some general properties that rely on the finite commutative rings theory. Our goal is to construct interesting orthogonal decompositions of these Lie algebras. We begin with Lie algebras of type A by searching for sufficient conditions for the existence of such an orthogonal decomposition. Our study in the special case when the ring is a finite field provides us important information that leads to the approach we develop in this dissertation.
We then apply our results on the orthogonal decomposition of type A Lie algebras to obtain a construction of the orthogonal decomposition of Lie algebras of type C. We also provide methods of constructing orthogonal decompositions for Lie algebras of types B and D
Waring's Problem in Finite Rings
In this paper we obtain sharp results for Waring's problem over general
finite rings, by using a combination of Artin-Wedderburn theory and Hensel's
lemma and building on new proofs of analogous results over finite fields that
are achieved using spectral graph theory. We also prove an analogue of
S\'ark\"ozy's theorem for finite fields.Comment: 34 page
On the Weight Distribution of Codes over Finite Rings
Let R > S be finite Frobenius rings for which there exists a trace map T from
R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is
an S-linear subring-subcode of a left linear code over R. We consider functions
f for which the homogeneous weight distribution of C can be computed. In
particular, we give constructions of codes over integer modular rings and
commutative local Frobenius that have small spectra.Comment: 18 p
Nonexistence of Certain Skew-symmetric Amorphous Association Schemes
An association scheme is amorphous if it has as many fusion schemes as
possible. Symmetric amorphous schemes were classified by A. V. Ivanov [A. V.
Ivanov, Amorphous cellular rings II, in Investigations in algebraic theory of
combinatorial objects, pages 39--49. VNIISI, Moscow, Institute for System
Studies, 1985] and commutative amorphous schemes were classified by T. Ito, A.
Munemasa and M. Yamada [T. Ito, A. Munemasa and M. Yamada, Amorphous
association schemes over the Galois rings of characteristic 4, European J.
Combin., 12(1991), 513--526]. A scheme is called skew-symmetric if the diagonal
relation is the only symmetric relation. We prove the nonexistence of
skew-symmetric amorphous schemes with at least 4 classes. We also prove that
non-symmetric amorphous schemes are commutative.Comment: 10 page
Matrix factorizations and link homology
For each positive integer n the HOMFLY polynomial of links specializes to a
one-variable polynomial that can be recovered from the representation theory of
quantum sl(n). For each such n we build a doubly-graded homology theory of
links with this polynomial as the Euler characteristic. The core of our
construction utilizes the theory of matrix factorizations, which provide a
linear algebra description of maximal Cohen-Macaulay modules on isolated
hypersurface singularities.Comment: 108 pages, 61 figures, latex, ep
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