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We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and…

In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups. We also exhibit new examples of finite groups that possess a quotient whose minimal degree is strictly greater than that of the…

Let $V,W$ be representations of a cyclic group $G$ of prime order $p$ over a field $k$ of characteristic $p$. The module of covariants $k[V,W]^G$ is the set of $G$-equivariant polynomial maps $V \rightarrow W$, and is a module over…

We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the…

This paper investigates the structure of Verma modules over the N=1 BMS superalgebra. We provide a detailed classification of singular vectors, establish necessary and sufficient conditions for the existence of subsingular vectors, uncover…

We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically reduced base scheme, by specializing to the standard Cremona transformation and computing its Segre class via mixed volumes…

We prove that formal Fourier Jacobi expansions of degree 2 are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension 2, which were defined by Kudla. A second application is…

Let $\mathcal{W}(b)$ be a class of free Lie conformal algebras of rank $2$ with $\mathbb{C}[\partial]$-basis ${L,H}$ and relations \begin{eqnarray*} [L_\lambda L]=(\partial+2\lambda)L,\ \ [L_\lambda H]=\big(\partial+(1-b)\lambda\big)H, \ \…

We develop an elementary divisor theory for the unimodular and the modular group over quadratic field extensions and quaternion algebras. In particular, we investigate which sets of elementary divisors can occur. Under an additional…

Lowest weight representations of the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ graded superalgebra introduced by Rittenberg and Wyler are investigated. We give a explicit construction of Verma modules over the ${\mathbb Z}_2 \otimes {\mathbb…

A cohomological support, Supp_A(M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. It is proved that if the…

We construct degeneration limits of vertex operators for the Virasoro algebra. Our method relies on the rearranged expansion of compositions of vertex operators together with their integral representations. Using this framework, we obtain a…

We demonstrate the construction of several families of flexible polyhedra by extending Bricard octahedra to form larger composite flexible polyhedra. These flexible polyhedra are of genus 0 and 1, have dihedral angles that are non-constant…

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking…

We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan-Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category…

The Schur orthogonality relations are a cornerstone in the representation theory of groups. We utilize a generalization to weak Hopf algebras to provide a new, readily verifiable condition on the skeletal data for deciding whether a given…

The main object of study in this paper is the module $\Omega$ of K\"ahler differentials of an extension of valuation rings. We show that in the case of pure extensions $\Omega$ has a very good description. Namely, it is isomorphic to the…

We investigate scalar restriction, scalar extension, and scalar coextension functors for graded modules, including their interplay with coarsening functors, graded tensor products, and graded Hom functors. This leads to several…

We prove a new converse theorem for Borcherds' multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of…

(Partial) Gorenstein silting modules are introduced and investigated. It is shown that for finite dimensional algebras of finite CM-type, partial Gorenstein silting modules are in bijection with {\tau}_G-rigid modules; Gorenstein silting…