# Quiver varieties and the character ring of general linear groups over finite fields

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 4, page 1375-1455
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topLetellier, Emmanuel. "Quiver varieties and the character ring of general linear groups over finite fields." Journal of the European Mathematical Society 015.4 (2013): 1375-1455. <http://eudml.org/doc/277547>.

@article{Letellier2013,

abstract = {Given a tuple $(\mathcal \{X\}_1,\ldots ,\mathcal \{X\}_k)$ of irreducible characters of $GL_n(F_q)$ we define a star-shaped quiver $\Gamma $ together with a dimension vector $v$. Assume that $(\mathcal \{X\}_1,\ldots ,\mathcal \{X\}_k)$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $\mathcal \{X\}_1\otimes \cdots \otimes \mathcal \{X\}_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we prove our second result: The multiplicity $(\mathcal \{X\}_1 \otimes \cdots \otimes \mathcal \{X\}_k,1)$ is non-zero if and only if $v$ is a root of the Kac-Moody algebra associated with $\Gamma $. This is somehow similar to the connection between Horn’s problem and the representation theory of $GL_n(C)$.},

author = {Letellier, Emmanuel},

journal = {Journal of the European Mathematical Society},

keywords = {quiver varieties; tensor products of irreducible characters; general linear groups over finite fields; character rings; general linear groups over finite fields; character rings; quiver varieties; tensor products of irreducible characters},

language = {eng},

number = {4},

pages = {1375-1455},

publisher = {European Mathematical Society Publishing House},

title = {Quiver varieties and the character ring of general linear groups over finite fields},

url = {http://eudml.org/doc/277547},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Letellier, Emmanuel

TI - Quiver varieties and the character ring of general linear groups over finite fields

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 4

SP - 1375

EP - 1455

AB - Given a tuple $(\mathcal {X}_1,\ldots ,\mathcal {X}_k)$ of irreducible characters of $GL_n(F_q)$ we define a star-shaped quiver $\Gamma $ together with a dimension vector $v$. Assume that $(\mathcal {X}_1,\ldots ,\mathcal {X}_k)$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $\mathcal {X}_1\otimes \cdots \otimes \mathcal {X}_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we prove our second result: The multiplicity $(\mathcal {X}_1 \otimes \cdots \otimes \mathcal {X}_k,1)$ is non-zero if and only if $v$ is a root of the Kac-Moody algebra associated with $\Gamma $. This is somehow similar to the connection between Horn’s problem and the representation theory of $GL_n(C)$.

LA - eng

KW - quiver varieties; tensor products of irreducible characters; general linear groups over finite fields; character rings; general linear groups over finite fields; character rings; quiver varieties; tensor products of irreducible characters

UR - http://eudml.org/doc/277547

ER -