We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group Aut(G) of surjective linear isometries of the Gurarij space G is extremely amenable, and that the canonical action Aut(P) y P is the universal minimal flow of the group Aut(P) of affine homeomorphisms of the Poulsen ...

From the action of an affine algebraic group G on an algebraic variety V , one can construct a representation of its Lie algebra L(G) by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → DerK(K[U ]) for an affine algebraic variety U , one may wonder whether it comes from an algebraic group action on U or on ...

We construct and study in detail various dual pairs acting on some Fock spaces between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the ...

We reduce a case of the hidden subgroup problem (HSP) in several families of finite groups of Lie type, such as SL(2; q), PSL(2; q), and PGL(2; q), to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an “almost” 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup. Our observation ...

Let W be an associative PI affine algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W ) and exp(We) denote the codimension growth of W and of the identity component We, respectively. We prove: exp(W ) ≤ |G| exp(We). This inequality had been conjectured by Bahturin and Zaicev.

Let A be a Coxeter hyperplane arrangement, that is the arrangement of reflecting hyperplanes of an irreducible finite Coxeter group. A deformation of A is an affine arrangement each of whose hyperplanes is parallel to some hyperplane of A. We survey some of the interesting combinatorics of classes of such arrangements, reflected in their characteristic polynomials.

We examine the subgroup D ( G ) of a transitive permutation group which is generated by derangements in . Our main results bound index this subgroup: we conjecture that, if has degree n and not Frobenius group, then | : ? ? 1 ; prove except when primitive affine group. For groups, translate our into an equivalent form regarding H R , where linear on finite vector space elements having eigenvalu...

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin’s algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

Let k[V ] be the space of functions from a finite vector space into the algebraically closure of its field of scalars. This paper describes the lattice of subspaces of k[V ] which are invariant under the affine group AGL(V ). The description provides a simple method for finding the submodule generated by any set of functions given as polynomials in the standard coordinates.