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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded).

The dimension of some affine Deligne–Lusztig varieties

Eva Viehmann (2006)

Annales scientifiques de l'École Normale Supérieure

The Discrete Series of the Linear Groups SL (n, q).

Gustav I. Lehrer (1972)

Mathematische Zeitschrift

The equations of conjugacy classes of nilpotent matrices.

J. Weyman (1989)

Inventiones mathematicae

The existence of equivariant pure free resolutions

David Eisenbud, Gunnar Fløystad, Jerzy Weyman (2011)

Annales de l’institut Fourier

Let $A = K [x_{1}, \dots, x_{m}]$ be a polynomial ring in $m$ variables and let $d = (d_{0} < \dots < d_{m})$ be a strictly increasing sequence of $m + 1$ integers. Boij and Söderberg conjectured the existence of graded $A$ -modules $M$ of finite length having pure free resolution of type $d$ in the sense that for $i = 0, \dots, m$ the $i$ -th syzygy module of $M$ has generators only in degree $d_{i}$ .This paper provides a construction, in characteristic zero, of modules with this property that are also $G L (m)$ -equivariant. Moreover, the construction works over rings of the form $A \otimes_{K} B$ where $A$ is a polynomial...

The Faithful Linear Representation of Least Degree of Sn and An over a Field of Characteristic 2.

Ascher Wagner (1976)

Mathematische Zeitschrift

The Farrell cohomology of $Sp (p - 1, ℤ)$ .

Busch, Cornelia (2002)

Documenta Mathematica

The First Cohomology Group of a Line Bundle on G/B.

Henning Haahr Andersen (1979)

Inventiones mathematicae

The Fixed Point Subvarieties of Unipotent Transformations on Generalized Flag Varieties and the Green Functions. Combinatorial and Cohomological Treatments Centering GLn.

R. Hotta, N. Shimomura (1979)

Mathematische Annalen

The Functional Equation. (Short Communication).

R. Larsen (1970)

Aequationes mathematicae

The fundamental theorem of prehomogeneous vector spaces modulo $p^{m}$ (With an appendix by F. Sato)

Raf Cluckers, Adriaan Herremans (2007)

Bulletin de la Société Mathématique de France

For a number field $K$ with ring of integers $𝒪_{K}$ , we prove an analogue over finite rings of the form $𝒪_{K} / 𝒫^{m}$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $𝒫$ is a big enough prime ideal of $𝒪_{K}$ and $m > 1$ . In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113(1998), 237–346] to the functional equation of $L$ -functions...

The garden of quantum spheres

Ludwik Dąbrowski (2003)

Banach Center Publications

A list of known quantum spheres of dimension one, two and three is presented.

The geometric reductivity of the quantum group $S L_{q} (2)$

Michał Kępa, Andrzej Tyc (2011)

Colloquium Mathematicae

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants $A^{G}$ is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...

It is proved that the group Sp10(ℤ) is generated by an involution and an element of order 3.

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The decomposition numbers of Hecke algebras of type F4 with unequal parameters.

The Decomposition numbers of the Hecke algebra of type F4.

The Decomposition of Tensors over Fields of Prime Characteristic.

The defect of weak approximation for homogeneous spaces

The density of representation degrees