We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple \Bbb {C}-algebra T e T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y.

In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter.

We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”.

We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur.

We close with some conjectures and open problems.

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The Subconstituent Algebra of an Association Scheme, (Part I)

A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

Fusion Systems in Algebra and Topology

Part 1 Linear algebra over division rings: matrices over division rings matrix representations of subspaces systems of linear equations. Part 2 Affine geometry and projective geometry: affine spaces and affine groups projective spaces and projective groups one-dimensional projective geometry. Part 3 Geometry of rectangular matrices: the space of rectangular matrices proof of the fundamental theorem application to graph theory. Part 4 Geometry of alternate matrices: the space of alternate matrices maximal sets. Part 5 Geometry of symmetric matrices: the space of symmetric matrices proof of the fundamental theorem I-III. Part 6 Geometry of hermitian matrices: maximal sets of rank 1 proof of the fundamental theorem (the case n is greater than or equal to 3) the maximal set of rank 2 (n=2) proof of the fundamental theorem (the case n=2) and others.

Geometry of Matrices

An n-point metric space (X, D) can be represented by an n
 ×
 n
 
$ n \times n $

 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/c4259681-9cc7-449e-9db1-d407160aea9b/content/inline-math8_1.tif"/>
 table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their dissimilarity (computed, say, by comparing their DNA). It is difficult to see any structure in a large table of numbers, and so we would like to represent a given metric space in a more comprehensible way.

Low-Distortion Embeddings of Finite Metric Spaces

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/62438/1/1063-1.pdf

Groups and Generalized Polygons (Algebraic Combinatorics)

Let G be a finite group, and $g \geq 2$. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g=3 (including equations for the corresponding curves), and for $g \leq 10$ we classify those loci corresponding to "large" G.

The locus of curves with prescribed automorphism group

Primitive axial algebras of Jordan type

https://spiral.imperial.ac.uk/bitstream/10044/1/18379/2/Mathematische%20Zeitschrift_272_1-2_2012.PDF

Majorana representations of the symmetric group of degree 4

Universal axial algebras and a theorem of Sakuma

We investigate graphs that are isometrically embeddable into the metric space l 1 .

Sergey Shpectorov

Papers

The locus of curves with prescribed automorphism group

Primitive axial algebras of Jordan type

Majorana representations of the symmetric group of degree 4

Universal axial algebras and a theorem of Sakuma

On scale embeddings of graphs into hypercubes