Related papers: Picard modular groups generated by complex reflect…
The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for…
A discrete subgroup of the group of isometries of the hyperbolic space is called reflective if up to a finite index it is generated by reflections in hyperplanes. The main result of this paper is a complete classification of the reflective…
This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\mathbf{E}_{\infty}$-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and…
We introduce the Picard group of corings. We extend the well-known exact sequence from algebras and coalgebras over fields to corings. We extend the Aut-Pic property to corings and we give some new examples of corings having this property.…
Constructive algorithms, requiring no more than $2\times 2$ matrix manipulations, are provided for finding the entries of the positive definite factor in the polar decomposition of matrices in sixteen groups preserving a bilinear form in…
In this paper, we showed that the Stable Picard group of $A(n)$ for $n\geq 2$ is $\mathbb{Z}\oplus \mathbb{Z}$ by considering the endotrivial modules over $A(n)$. The proof relies on reductions from a Hopf algebra to its proper Hopf…
A Chevalley type integral basis for the ortho-symplectic Lie superalgebra is constructed. The simple modules of the ortho-symplectic supergroup over an algebraically closed field of prime characteristic not equal to 2 are classified, where…
We are extending results from \cite{B-Hurwitz} by building a parallel theory of simple Hurwitz numbers for the reflection groups $G(m,1,n)$. We also study analogs of the cut-and-join operators. An algebraic description as well as a…
The relations satisfied by period polynomials associated to modular forms yield a way to count dimensions of spaces of cusp forms. After showing how these relations arise from those on the mapping class group $PSL(2, \mathbb{Z})$ of the…
Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, $\Gamma$, in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to…
It is possible to construct representations of the Lorentz group using four-dimensional harmonic oscillators. This allows us to construct three-dimensional wave functions with the usual rotational symmetry for space-like coordinates and…
The simplices and the complexes arsing form the grading of the fundamental (desymmetrized) domain of arithmetical groups and non-arithmetical groups, as well as their extended (symmetrized) ones are described also for oriented manifolds in…
We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is…
We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We…
In this paper we define a two-variable, generic Hecke algebra, H, for each complex reflection group G(b,1,n). The algebra H specializes to the group algebra of G(b,1,n) and also to an endomorphism algebra of a representation of GL(n,q)…
Let $(K,\mathcal O,k)$ be a $p$-modular system and assume $k$ is algebraically closed. We show that if $\Lambda$ is an $\mathcal O$-order in a separable $K$-algebra, then $\textrm{Pic}_{\mathcal O}(\Lambda)$ carries the structure of an…
As a result of our study of the hyperbolicity of the moduli space of polarized manifold, we give a general big Picard theorem for a holomorphic curve on a log-smooth pair $(X,D)$ such that $W=X\setminus D$ admits a Finsler pseudometric that…
We show that the simple modules of the Rouquier blocks of symmetric groups, in characteristic $p$ and having $p$-weight $w$ with $w < p$, have a common complexity $w$, and that when $p$ is odd, $D^{(p+1,1^{p-1})}$ has complexity 1, while…
The parentage between Weyl pairs, generalized Pauli group and unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field R and then switch to the discrete Heisenberg-Weyl group or…
We show local rigidity of hyperbolic triangle groups generated by reflections in pairs of $n$-dimensional subspaces of $R^{2n}$ obtained by composition of the geometric representation in $PGL(2, R)$ with the diagonal embeddings into…