We provide an alternative constructive proof of the classical Brauer theorem for finite groups based on the well-known description of the complex irreducible representations of the symmetric groups *Sn *. The theorem is first proved for *Sn * and then for general *G* by embedding in *Sn * and applying the Mackey subgroup theorem.

Let *τ* be the involution changing the sign of two coordinates in ℙ4. We prove that *τ* induces the identity action on the second Chow group of the intersection of a *τ*-invariant cubic with a *τ*-invariant quadric hypersurface in ℙ4. Let *lτ *and*Πτ *be the one- and two-dimensional components of the fixed locus of the involution *τ*. We describe the generalized Prymian associated with the projection of a *τ*-invariant cubic *𝓵* ⊂ P4 from *lτ *onto *Πτ *in terms of the Prymians *𝓅2 *and *𝓅*3associated with the double covers of two irreducible components, of degree 2 and 3, respectively, of the reducible discriminant curve. This gives a precise description of the induced action of the involution *τ* on the continuous part of the Chow group CH2 (*𝓵*). The action on the subgroup corresponding to *𝓅*3 is the identity, and the action on the subgroup corresponding to *𝓅*2 is the multiplication by —1.

This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following M. Ballard, D. Favero, L. Katzarkov (ArXiv:1012.0864) and D. Favero, L. Katzarkov (Noether--Lefschetz Spectra and Algebraic cycles, in preparation) we enhance constructions from A. Kuznetsov (arXiv:0904.4330) by introducing Noether--Lefschetz spectra --- an interplay between Orlov spectra (C. Oliva, Algebraic cycles and Hodge theory on generalized Reye congruences, Compos. Math. 92, No. 1 (1994) 1--22) and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where above techniques might apply. We start by constructing a sextic double solid $X$ with 35 nodes and torsion in $H^3(X,\ZZ)$. This is a novelty --- after the classical example of Artin and Mumford (1972), this is the second example of a Fano threefold with a torsion in the 3-rd integer homology group. In particular $X$ is non-rational. We consider other examples as well --- $V_{10}$ with 10 singular points and double covering of quadric ramified in octic with 20 nodal singular points. After analyzing the geometry of their Landau--Ginzburg models we suggest a general non-rationality picture based on Homological Mirror Symmetry and category theory.

We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (‘motivic’) Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products.

Let (X, C) be a germ of a threefold X with terminal singularities along an irreducible reduced complete curve C with a contraction f : (X, C) → (Z, o) such that C = f ^{−1} (o)_red and −K_X is ample. Assume that (X, C) contains a point of type (IC) or (IIB). We complete the classification of such germs in terms of a general member H ∈ |O_X | containing C.