# Determining representations from invariant dimensions

@article{Larsen1990DeterminingRF, title={Determining representations from invariant dimensions}, author={Michael Larsen and Richard Pink}, journal={Inventiones mathematicae}, year={1990}, volume={102}, pages={377-398} }

This paper is motivated by the following "Tannakian" question: to what extent is a complex Lie group, G, and a finite dimensional representation, (p, V) of G, determined by the dimensions of the various invariant spaces W G, where the W are obtained from V by linear algebra? That is, given dim((Sym2(V)a), dim((A 3 V)~), etc., can one determine (G, p)? This problem arises, for instance, in the cohomological study of exponential sums; we intend to apply the below results to the problem of…

## 72 Citations

### Invariant dimensions and maximality of geometric monodromy action

- Mathematics
- 2015

Let X be a smooth separated geometrically connected variety over F_q and f:Y-> X a smooth projective morphism. We compare the invariant dimensions of the l-adic representation V_l and the…

### Poles of Certain Automorphic L-functions

- Mathematics
- 2007

Let G be a classical group defined over a number field k. Let G = G ⋊Wk be the Langlands dual group of G, where Wk is the Weil group. Let A(G/k) be the set of equivalence classes of irreducible…

### On `-independence of Algebraic Monodromy Groups in Compatible Systems of Representations

- Mathematics
- 1992

Consider a profinite group G, and a collection of continuous representations ρ` : G → GLn(Q`), indexed by a set L of rational primes `. Suppose that G is endowed with a dense subset of “Frobenius”…

### Monodromy of Galois representations and equal-rank subalgebra equivalence

- Mathematics
- 2012

We study l-independence of monodromy groups G_l of any compatible system of l-adic representations (in the sense of Serre) of number field K assuming semisimplicity. We prove that the formal…

### Rigidity in the Invariant Theory of Compact Lie Groups

- Mathematics
- 2015

A compact Lie group G and a faithful complex representation V determine the Sato-Tate measure μG,V on C, defined as the direct image of Haar measure with respect to the character map g 7→ tr(g|V ).…

### l-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecture

- Mathematics
- 1998

We prove that the `-adic algebraic monodromy groups associated to a motive over a number field are generated by certain one-parameter subgroups determined by Hodge numbers. In the special case of an…

### Rigidity in the invariant theory of compact groups

- Mathematics
- 2002

A compact Lie group G and a faithful complex representation V determine a Sato-Tate measure, defined as the direct image of Haar measure on G with respect to the character of V. We give a necessary…

### ON THE DIMENSION DATUM OF A SUBGROUP

- Mathematics
- 2014

1.2. Variants. We may consider the same notion when $G$ and $H$ are complex reductive groups, and $\hat{G}$ is taken to be the set of equivalence classes of irreducible rational representations. By…

### On the fundamental automorphic L-functions of SO(2n+1)

- Mathematics
- 2006

The fundamental automorphic L-functions of SO2n+1 are by definition the Langlands automorphic L-functions attached to irreducible cuspidal automorphic representations σ of SO2n+1(A) and the…

## References

SHOWING 1-6 OF 6 REFERENCES

### Introduction to Lie Algebras and Representation Theory

- Mathematics
- 1973

Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.-…

### Groupes et algèbres de Lie

- Philosophy
- 1971

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur…

### Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe

- Mathematics
- 1927

### Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras

- Mathematics
- 1981

### The Classical Groups

- Mathematics
- 1939

: We consider the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan. These generalize the doubling method of Piatetski-Shapiro and Rallis and represent the standard L-function for…