Related papers: Binary Hermitian forms over a cyclotomic field

We report on computations of the cohomology of GL_2(O_D) and SL_2(O_D), where D<0 is a fundamental discriminant. These computations go well beyond earlier results of Vogtmann and Scheutzow. We use the technique of homology of Voronoi…

Let G < SL(V) be a finite group, V is finite dimensional over a field F, p=char F and S(V) is the symmetric algebra of V. We determine when the subring of G-invariants S(V)^G is a polynomial ring. As a consequence, we classify, if F is…

We give a generalization of the Hodge operator to spaces $(V,h)$ endowed with a hermitian or symmetric bilinear form $h$ over arbitrary fields, including the characteristic two case. Suitable exterior powers of $V$ become free modules over…

We classify the bireflections (products of 2 involutions) in the commutator subgroup G an orthogonal group O(V) over a finite field GF(q) of characteristic not 2. We show that every element of G is a bireflection if it is reversible…

Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They…

Let $ G $ be a cyclic group, in this paper, we study the Herbrand quotient and $ 1-$th cohomology group on finitely generated $ G-$modules in some cases. When $ G $ is of order $ 2, $ the order of the cohomology group is explicitly related…

The relationship between associative composition algebras of dimensions 2 and 4 within the context of homogeneous spaces, with a particular focus on Hamiltonian quaternions, is explored. In the special case of Hamiltonian quaternions, the…

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree $n$ binary form. A survey of a…

We compute tables of paramodular forms of degree two and cohomological weight via a correspondence with orthogonal modular forms on quinary lattices.

We compute the integer cohomology rings of the ``polygon spaces'' introduced in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we…

Quasiperiodic patterns described by polyhedral "atomic surfaces" and admitting matching rules are considered. It is shown that the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus…

In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear…

We prove that the 2-hermitean matrix model and the complex-matrix model obey the same loop equations, and as a byproduct, we find a formula for Itzykzon-Zuber's type integrals over the unitary group. Integrals over U(n) are rewritten as…

For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical…

We give a sufficient condition on a pair of (primitive) integral polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral…

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in…

Let $K$ be a number field, ${\mathcal O}_K$ its ring of integers, and $f(x, y) \in {\mathcal O}_K[x, y]$ a binary form with integer coefficents. For any given prime $p \in {\mathcal O}_K$ we determine explicitly a binary form $g$ (resp.…

Generalising the concept of a complete permutation polynomial over a finite field, we define completness to level $k$ for $k\ge1$ in fields of odd characteristic. We construct two families of polynomials that satisfy the condition of high…

We construct a model of the Hermitian unital of order 3 (obtained from the non-degenerate hermitian form in three variables over the field of order 9) inside the octonion algebra over the field of order 2. This construction is invariant…

The Hermite interpolation formulas are based on the interpretation of interpolation nodes as roots of suitable polynomials. Therefore, such formulas belong to the class of algebraic interpolations. The article considers a multidimensional…