By Gabor Halasz, Laszlo Lovasz, Miklos Simonovits, Vera T. Sós

Considering that his dying in 1996, many clinical conferences were devoted to the reminiscence of Paul Erd?s. From July four to eleven, 1999, the convention "Paul Erd?s and his arithmetic" used to be held in Budapest, with the formidable target of unveiling the complete variety of Erd?s' paintings - a tough job in view of Erd?s' versatility and his vast scope of curiosity in arithmetic. in line with this target, the themes of lectures, given by means of the top experts of the topics, incorporated quantity thought, combinatorics, research, set concept, likelihood, geometry and parts connecting them, like ergodic concept. The convention has contributed to altering the typical view that Erd?s labored purely in combinatorics and combinatorial quantity concept. within the current volumes, the editors have gathered, in addition to a few own memories via Paul's previous buddies, customarily survey articles on his paintings, and on components he initiated or labored in.

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**Example text**

N}, let cµν ∈ K Γ be the function which associates to each g ∈ Γ its (µ, ν)-coeﬃcient gµν . Denote by A or AK (n) the K-subalgebra of K Γ generated by the functions cµν (µ, ν ∈ n); the elements of A are, by deﬁnition, the polynomial functions on Γ. Since K is inﬁnite, the cµν are algebraically independent over K, so that A can be regarded as the algebra of all polynomials over K in n2 “indeterminates” cµν (µ, ν ∈ n). For each r ≥ 0 we denote by AK (n, r) the subspace of A consisting of the elements expressible as polynomials which are homogeneous of degree r 2 as K-space; in parin the cµν .

Young [58, 1902]. The observation that the Dλ,K can be constructed over an arbitrary ﬁeld—or equivalently that the (Tl : Ti ) generate a Z-form Dλ,Z in Dλ,Q —was made by G. Higman [23, 1965]. The Vλ,K (and the Z-form Vλ,Z ) were constructed, independently of all this, by R. Carter and G. Lusztig [6, 1974]. They called these “Weyl modules”, and their construction was based on methods used in the theory of semisimple algebraic groups. Towber [52] showed that Dλ,K and Vλ,K are dual to each other—his framework is “functorial” and more general than ours.

Deruyts [13], in 1892. Although Schur refers to two later papers of Deruyts, there is no sign in [47] that he appreciated that Deruyts had really given a complete set of irreducible modules in MC (n, r). The discovery of the basis of the “standard” (Tl : Ti ), seems to go back to A. Young [58, 1902]. The observation that the Dλ,K can be constructed over an arbitrary ﬁeld—or equivalently that the (Tl : Ti ) generate a Z-form Dλ,Z in Dλ,Q —was made by G. Higman [23, 1965]. The Vλ,K (and the Z-form Vλ,Z ) were constructed, independently of all this, by R.