A Remark on Orthogonality Relations in Finite Groups
On the basis of Prof. R. Brauer’s fundamental work, certain orthogonality relations for characters of finite groups have recently been studied by Brauer himself, M. Osima, and one of the present writers; see Iizuka [7] and the references there. In the present short note some general remarks on orthogonality relations, dealing with “blocks” and “sections” of general type, are given first. They are of elementary, and often formal, nature and their proofs are merely combinations of known arguments. So, no deep significance is claimed on them, in comparison with the above alluded results based on deeper arithmetico-group-theoretical considerations. However, applied to blocks and sections of such deeper nature, our remarks give some rather useful informations on them. Thus, for instance, the “maximality” feature of 77-blocks is given a formulation (Prop. 5 below) finer than the one given in [71 Further, some new types of blocks and sections can be constructed, again in application of our remarks to such classical ones. These new blocks and sections give thus new orthogonality relations and we hope that some of them may turn to have some significance. There arize also several problems, which are stated at the end of the present note and to some of which we wish to come back elsewhere.
A generalization of Brauer characters
1. Statement of results. The Brauer characters, or modular characters, of a finite group G with respect to a prime p are defined only on the p-regular elements of G. Brauer [1], [3] has overcome this limitation to some extent by using the Brauer characters of certain subgroups of G, in terms of which he has defined the generalized decomposition matrix D of G. In this paper we study some complex-valued class-functions on G which are closely related to these Brauer characters of subgroups. These functions, which we call 0-functions, were defined by Brauer in [3, (7D)]. They behave in many ways like Brauer characters; for example, they have orthogonality relations (?4) and they can be distributed among the p-blocks of G (?9). In fact, our central motivating idea has been that the 0-functions may be regarded as group-characters of a sort. Explicitly, if x is any p-element of G and VI any irreducible Brauer character of the centralizer C(x) of x in G, there is a unique function 0 defined on G such that: (a) 0 is a class-function on G; (b) 4(xy) = +f(y) if y is p-regular in C(x); (c) O(x1y) = 0 if x1 is a p-element of G which is not conjugate to x and if y is p-regular in C(x1). We call these functions 0 the 0-functions of G for the prime p. (Actually the definition of Brauer characters depends on the choice of a prime ideal divisor p of p in a suitable algebraic number field [6, p. 589]; accordingly, the +-functions also depend on p.) For x = 1, b is simply the Brauer character il of G, if we extend the latter by giving it the value 0 on all p-singular elements (see the redefinition of Brauer characters in ?2). Dually we define the 1D-functions of G, starting from principal indecomposable characters P of C(x); for x = 1, these are the principal indecomposable characters of G. G has exactly k distinct +-functions, where k is the number of conjugate classes of G. If XI, ", Xk are the (ordinary) irreducible characters of G, there are unique complex numbers dij such that
Avocado Flowering 1
The avocado (Persea americana Mill.) has increased in importance over the years in many countries around the world. The United States, Mexico and Brazil lead in world commercial production, while Israel, South Africa, Australia, New Zealand, the Philippines, Spain, Cyprus, Canary Islands, Chile, Ecuador, Venezuela, Greece, and the Caribbean area are actively increasing commercial production (B.O. Bergh, personal communication; Gustafson 1976). The history of the avocado has been detailed by Gustafson (1973, 1975,1976) and others (Bergh 1975b, Knight 1980; Popenoe 1924; Ruehle 1958; Schroeder 1958; Williams 1976). Several reviews have been published on various aspects of the avocado, including its botany (Knight 1980; Purseglove 1968; Williams 1976), breeding (Bergh 1975b), horticulture (Hodgson 1930b; Ruehle 1958; Ryerson et al. 1924), cultivars (Campbell and Malo 1976), and pollination (Gustafson and Bergh 1966). A recent collection of papers discussing the horticultural aspects of avocado production is found in Sauls et al. (1976).