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

Letellier, 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 -