Related papers: Matrices for the Weil representation

The classical construction of the Weil representation, with complex coefficients, has long been expected to work for more general coefficient rings. This paper exhibits the minimal ring $\mathcal{A}$ for which this is possible, the integral…

We study the theta decomposition of Jacobi forms of nonintegral lattice index for a representation that arises in the theory of Weil representations associated to even lattices, and suggest possible applications.

From personal experience, I report about the development of sheaf representations for algebraic structures between the years 1966 and 1976. Starting with rings, lattice-ordered groups and rings it turns to general algebraic structures.

The Weil representation is a particularly significant linear representation of the metaplectic group, used in the study of theta correspondence. In this paper, I introduce a derived category version of the Weil representation in the local…

In this paper we construct a geometric analogue of the Weil representation over a finite field. Our construction is principally invariant, not choosing any specific realization. This eliminates most of the unpleasant formulas that appear in…

We introduce a generalization of representations of quivers that contains also representations of posets, vectorspace problems and other matrix problems. Many examples, some of which are given in the paper, show that the language of marked…

This is an overview of recent developments regarding the complexity of matrix multiplication, with an emphasis on the uses of algebraic geometry and representation theory in complexity theory.

We show that the Weil representation associated with any discriminant form admits a basis in which the action of the representation involves algebraic integers. The action of a general element of $\operatorname{SL}_{2}(\mathbb{Z})$ on many…

Je retracerai l'histoire des conjectures de Weil sur le nombre de solutions d'\'equations polynomiales dans un corps fini et quelques unes des approches qui ont \'et\'e propos\'ees pour les r\'esoudre. The Weil conjectures: origins,…

To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q:D --> Q/Z, it is possible to associate a representation of either the modular group, SL(2,Z), or its metaplectic cover, Mp(2,Z),…

The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.

Let $r$ be an odd prime and $\mathbb{F}$ a field containing a primitive $r$th root of unity. Then for all $\ell \geq 1$, there is a faithful representation $f: \operatorname{Sp}_{2\ell}(r) \rightarrow \operatorname{GL}_{r^\ell}(\mathbb{F})$…

This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting -1 as an eigenvalue and then to all orthogonal…

For a representation of a Lie algebra, one can construct a diagram of the representation, i. e. a directed graph with edges labeled by matrix elements of the representation. This article explains how to use these diagrams to describe normal…

These are lecture notes that arose from a representation theory course given by the first author to the remaining six authors in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students,…

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost…

Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of $SL_{2}(\mathbb{Z})$, admits a representation $\rho_{M}$, called the Weil…

An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is…

We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $\Gamma_0(4)$ with Kohnen's plus condition and…

This is the first one of a series of articles in which we develop the theory of Jacobi forms of lattice index, their close interplay with the arithmetic theory of lattices and the theory of Weil representations. We hope to publish this…