Harvey I. Blau

Table algebras

This article surveys table algebras as algebraic abstractions of association schemes and other structures. Various definitions are compared; some basic properties are developed; and some theorems and applications are presented.

On Brauer stars

This paper contains some results on how representations in the principal p-block of a finite group with a cyclic Sylow p-subgroup yield information on the structure of the group. Throughout the paper, G denotes a finite group, p a fixed odd prime dividing 1 GI, P a Sylow p-subgroup of G with ]P] = p”, and B,(p) the principal p-block of G. If P is cyclic, then the ordinary and modular irreducible characters in any p-block of G can be described in terms of a graph, called the Bruuer tree belonging to the block (as in [5, VII.61). A tree is called a star if there are no paths with more than two edges, i.e.,

Reality-Based Algebras, Generalized Camina-Frobenius Pairs, and the Nonexistence of Degree Maps

The notion of a generalized Camina-Frobenius pair is extended to reality-based algebras, and a construction that characterizes such pairs is given. Zero-product sets are defined, and a best-possible upper bound on their size is proved and related to Camina-Frobenius pairs. It is shown that there exist commutative reality-based algebras with zero-product sets and, hence, no degree map, of every dimension at least 4. All such 4-dimensional algebras are constructed explicitly.

On block induction

This paper generalizes a result of Okuyama [4, Theorem 31 on the situation wherein a block of a subgroup of a finite group induces to a block of the group (in the classical sense of Brauer; see [ 1, p. 1361). Our generalization, which comprises Lemma A and Theorem B, is proved mainly by observing that Okuyama’s arguments hold under less stringent hypotheses. Our results are applied in Theorem C to establish a necessary and sufficient condition for when a block of defect zero of a subgroup induces to a block of the group. Another application is Theorem D, which concerns block induction from a normal subgroup. Throughout the paper, H denotes a subgroup of an arbitrary finite group G and p is a fixed prime rational integer. Let R be the ring of integers in a p-adic number field K of characteristic zero, let rt be a generator of J(R), and set R = R/rcR. If M is an RG-module, i@ denotes M/nM. Assume that K and R are splitting fields for all subgroups of G. Let v denote the p-adic valuation on K, scaled so that v(p) = 1. Let v( / G I) = a and r( 1 H 1) = m, so that IGJP=p”, /I1l,,=p”‘. Let B, h denote fixed p-blocks of G (resp. H), where B has defect d Let Irr (B) (resp. Irr(b)), denote the set of ordinary irreducible characters in B (resp. h). If 0 is a rational integral combination of Brauer characters of G, let 8 denote the generalized character of G defined by [ 1, IV.1.21,

A Class of P-polynomial Table Algebras with and without Integer Multiplicities

The homogeneous, monotonic, P-polynomial table algebras with valency k ≥ 2 are classified. It is also determined which of these algebras, when integral, have integer multiplicities. In particular, it is shown that all multiplicities are integers only if k = 2 or the diameter d = 2. Some of these algebras come from distance-regular graphs, and some do not.

On the Calculation of Multiplicities for P-Polynomial C-Algebras

An alternative rational function, with polynomial components of smaller degree, is constructed to compute multiplicities for a P-polynomial C-algebra whose generating tri-diagonal matrix has a set of repeated column entries. As a consequence, some upper bounds are derived for the diameter of the algebra. The bound in the case of an integral table algebra generalizes a well known result of Bannai and Ito for distance-regular graphs.

Normal series and character values in p-standard table algebras

ABSTRACT We investigate the character values and structures of p-standard table algebras (A,B) with o(B) = pN. If N≤3, then B has a complete normal series. If for every χ∈Irr(B), χ has at most p distinct classes of character values, and if either B has a complete normal series or p = 2, then B is an elementary abelian p-group.

Nonvanishing Elements in Table Algebras

It is proved that nonvanishing elements of a commutative nilpotent table algebra must be linear, which generalizes the known result on the nonvanishing elements of a finite nilpotent group. Other results on nonvanishing elements of finite groups are generalized to the context of a commutative table algebra whose dual is also a commutative table algebra.

Submodules of indecomposable modules in blocks with cyclic defect group

This paper investigates the submodule lattice of an arbitrary indecomposable module in a block with cyclic defect group of a group algebra over a field of prime characteristic. We set our notation, recall some important known facts about such modules, and then state our results. G denotes a finite group, p a fixed prime divisor of ICI, F a field of characteristic p, and B a block of the group algebra FG with cyclic defect group D, JDI > 1. Here, “module” will always mean a finitely generated FG-module, each of whose indecomposable summands belongs to (is “in”) B. If U is a submodule of M, this is denoted by U< M. The following description of the indecomposable modules in B may be found in [ 1, VII.121. There is a connected tree T (the “Brauer tree”) associated with B. The edges of r correspond to the simple FG-modules in B. We write edge E, ++ module L,, and sometimes say “edge L,.” There are r edges, where elp1. If t :=(I01 1)/e> I, then there is one node of r designated the