Quiver varieties and the character ring of general linear groups over finite fields
Quiver varieties and the character ring of general linear groups over finite fields
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有限域上一般线性群的箭袋簇和特征环
DOI：
10.4171/jems/395
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发表时间：
2011
影响因子：
2.6
通讯作者：
E. Letellier
中科院分区：
文献类型：
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作者：
E. Letellier
给定一个元组（X1;:::;Xk）的不可约特征标定义了一个星型向量和一个维向量v。我们的第一个结果是一个公式，它表示在张量积X_1 → X_k中平凡特征标的重数作为某个Weyl群对与（; v）相关联的某些（非仿射）簇的交上同调的作用的迹。这样一个簇的存在是受某些条件的限制的。假设这个条件满足，我们证明了我们的第二个结果：多重性hX 1 ∈ Xk; 1 i非零当且仅当v是与相关联的Kac-Moody代数的根.我们猜想，这仍然是真实的独立于存在的各种。这在某种程度上类似于霍恩问题和GL n（C）的表示论之间的联系[20，第8节]。
Given a tuple (X1;:::; Xk) of irreducible characters of GLn(Fq) we define a star-shaped quiver together with a dimension vector v. Assume that (X1;:::; Xk) is generic. Our first result is a formula which expresses the multiplicity of the trivial character i n the tensor productX1��� X k as the trace of the action of some Weyl group on the intersection cohomology of some (non-affi ne) quiver varieties associated to ( ; 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 m ultiplicityhX 1��� X k; 1i is non-zero if and only if v is a root of the Kac-Moody algebra associated with . We conjecture that this remains true independently from the existence of the quiver variety. This is somehow similar to the connection between Horn’s problem and the representation theory of GL n(C) [20, Section 8].