Representation theory of finite groups 1.1 Basic facts 1.2 Schur's lemma and the commutant 1.3 Characters and the projection formula 1.4 Permutation representations 1.5 The group algebra and the Fourier transform 1.6 Induced representations 2 The theory of Gelfand-Tsetlin bases 2.1 Algebras of conjugacy invariant functions 2.2 Gelfand-Tsetlin bases 3 The Okounkov-Vershik approach 3.1 The Young poset 3.2 The Young-Jucys-Murphy elements and a Gelfand-Tsetlin basis for S n 3.3 The spectrum of the Young-Jucys-Murphy elements and the branching graph of S n 3.4 The irreducible representation of S n 3.5 Skew representations and the Murnhagan-Nakayama rule 3.6 The Frobenius-Young correspondence 3.7 The Young rule

In the present paper we consider a q-analog of t − (v, k, λ)-designs. It is canonic since it arises by replacing sets by vector spaces over GF (q), and their orders by dimensions. These generalizations were introduced by Thomas [Geom. Dedicata vol. 63, pp. 247-253 (1996)] they are called t − (v, k, λ; q)-designs. A few of such q-analogs are known today, they were constructed using sophisticated geometric arguments and case-by-case methods. It is our aim now to present a general method that allows systematically to construct such designs, and to give complete catalogs (for small parameters, of course) using an implemented software package.In order to attack the (highly complex) construction, we prepare them for an enormous data reduction by embedding their definition into the theory of group actions on posets, so that we can derive and use a generalization of the Kramer-Mesner matrix for their definition, together with an improved version of the LLL-algorithm. By doing so we generalize the methods developed in a research project on t − (v, k, λ)-designs on sets, obtaining this way new results on the existence of t − (v, k, λ; q)-designs on spaces for further quintuples (t, v, k, λ; q) of parameters. We present several 2 − (6, 3, λ; 2)-designs, 2 − (7, 3, λ; 2)-designs and, as far as we know, the very first 3-designs over GF (q).

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This volume deals with the Cauchy or initial value problem for linear differential equations. It treats in detail some of the applications of linear space methods to partial differential equations, especially the equations of mathematical physics such as the Maxwell, Schrödinger and Dirac equations. Background material presented in the first chapter makes the book accessible to mathematicians and physicists who are not specialists in this area as well as to graduate students.

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