Modular representation theory of BIB designs

Available online 8 November 2016Our aim is to study the modular representation theory of coherent configurations. Let p be a prime. We consider structures of modular adjacency algebras of coherent configurations obtained from combinatorial designs. The structures of standard modules of modular adjacency algebras provide more information than p-ranks of incidence matrices of combinatorial designs. (C) 2016 Elsevier Inc. All rights reserved.ArticleLINEAR ALGEBRA AND ITS APPLICATIONS. 514:174-197 (2017)journal articl

Simplicity of p-blocks of modular adjacency algebras of association schemes

Available online 16 November 2016A criterion is given for blocks of modular adjacency algebras of association schemes to be simple. (C) 2016 Elsevier Inc. All rights reserved.ArticleJOURNAL OF ALGEBRA. 474:126-133 (2017)journal articl

Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory

The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlinear $\sigma$-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these correspondences of the ADE type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possible guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions.Comment: 35 pages, 4 figures; minor corrections, comments on toric geometry and references adde

Extensibility of Association Schemes and GRH-Based Deterministic Polynomial Factoring

The subject of the present work is the application of the theory of combinatorial schemes to problems in computational algebra. The principal notions of combinatorial schemes which are studied in this work are association schemes (Bannai & Ito (1984), Zieschang (1996, 2005)), m-schemes (Ivanyos, Karpinski & Saxena (2009), Arora et al. (2012)), and presuperschemes (Smith (1994, 2007), Wojdylo (1998, 2001)). The main computational problems considered in this work are polynomial factoring over finite fields, the Schurity problem of association schemes (and its relaxation in the notion of extensibility), and matrix multiplication. We show that each of the latter problems admits a deep connection to the theory of combinatorial schemes, and describe natural algebraic-combinatorial frameworks which capture the essence of their algebraic complexity. As a logical application, we delineate how structural results for combinatorial schemes can translate to fundamental improvements in the realm of computational algebra

Families of Association Schemes on Triples from Two-Transitive Groups

Association schemes on triples (ASTs) are ternary analogues of classical association schemes. Analogous to Schurian association schemes, ASTs arise from the actions of two-transitive groups. In this paper, we obtain the sizes and third valencies of the ASTs obtained from the two-transitive permutation groups by determining the orbits of the groups' two-point stabilizers. Specifically, we obtain these parameters for the ASTs obtained from the actions of $S_n$ and $A_n$, $PGU(3,q)$, $PSU(3,q)$, and $Sp(2k,2)$, $Sz(2^{2k+1})$ and $Ree(3^{2k+1})$, some subgroups of $A\Gamma L(k,n)$, some subgroups of $P\Gamma L(k,n)$, and the sporadic two-transitive groups. Further, we obtain the intersection numbers for the ASTs obtained from these subgroups of $P\Gamma L(k,n)$ and $A \Gamma L(k,n)$, and the sporadic two-transitive groups. In particular, the ASTs from these projective and sporadic groups are commutative.Comment: 20 pages, 5 table

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective

In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student.Comment: 513 pages, 71 figs, Edited Excerpts from the first 3 volumes of the author's PhD Thesi

On algebraic singularities, finite graphs and D-brane gauge theories: A String theoretic perspective

In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student