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

Gustav I. Lehrer (1972)

Mathematische Zeitschrift

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 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

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The Clebsch-Gordan coefficients with respect to various bases for unitary and orthogonal representations of $SU (2)$ and $SO (3)$ .

The Conjugating Representation of a Semisimple Group.

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 density of representation degrees

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

The existence of equivariant pure free resolutions

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

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

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

The Grothendieck group of G-equivariant modules over coordinate rings of G-orbits

The modular representation theory of q-Schur algebras. II.

The normality of closures of conjugacy classes of matrices.

The representations and generic degrees of the Hecke algeba of type H4.

The Restriction of a Representation of GL2(k) to GL2(o).

The Restriction of Admissible Representations to n.

The ring of upper triangular invariants as a module over the Dickson invariants.

The Schur Multipliers of Sp6(Z),Spin8(Z),Spin7(Z),andF4(Z).

The Steinberg Function of a Finite Lie Algebra.

Currently displaying 321 – 340 of 402