Symmetry And Group

## Group representation theory by Larry L Dornhoff

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By Larry L Dornhoff

Pt. A. usual illustration theory.--pt. B. Modular illustration thought

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The table algebras Tm(A) for m ~ 2 never arise as (Z(CG), Cla(G)), (Ch(G), Irr(G)), or any substructure thereof, for any finite group G. The equation XQXm-1 = CUm-1 + f3iQ holds in T m (A), whereas there existnoelements b, c in Cla(G) or Irr(G) with c =I- b, b =I- b, and SUPPB(bc) = b} [AI, Corollary E']. rc, We recently have determined all integral table algebras (A, B) which are homogeneous of degree 3, and for which B contains a faithful, real element [BX2]. The full result is too detailed to give here, so we restrict to the case where B contains no nontriviallinear elements.

26)] and [Sy]). Here we will only use the canonical induction formula Qc. 2. Without going into details we should try to make clear that the group Rf(G) is a very natural construction. Note that the groups R(H), H ::: G, are tied together by homomorphisms Cg,H: R(H) ---+ R( gH), res~: R(H) ---+ R(K), ind~: R(K) ---+ R(H), for K ::: H ::: G, g E G, called conjugation, restriction and induction. These homomorphisms satisfy certain compatibility conditions which are the axioms of a Mackey functor on G.

Austral. Math. Soc. 20 (1979), 35-55. Degrees and Diagrams of Integral Table Algebras 27 [Sc] I. Schur, Zur Theorie der einfach transitiven Permutations-gruppen, Sitzungsber. Preuss. Akad. -Math. K1, 1933,598-623. [SI] p. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Math. 815, Springer-Verlag, Berlin, 1980. edu Canonical Induction Formulae and the Defect of a Character R. Bo/tje Abstract. We explain the idea and the machinery of canonical induction formulae with going as little into details as possible and show that a certain version keeps track of the i-defect d(X) of an irreducible character X of a finite group G by mapping X precisely to the d (X)-th layer in the filtration A ~ i-I A ~ i- 2 A ...