Canonical Bases for Irreducible Representations of Quantum GLn, II
Canonical Bases for Irreducible Representations of Quantum GLn, II
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量子 GLn 不可约表示的规范基础，II
DOI：
10.1112/jlms/51.3.461
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发表时间：
1995
影响因子：
1.2
通讯作者：
J. Du
中科院分区：
文献类型：
--
作者：
J. Du
在文献[11]中，作者通过#-Schur代数的Kazhdan-Lusztig基（见文献[10]）引入了量子线性群不可约表示的某些标准基。本文利用Dipper-James的tf-Weyl模构造[9]对这些基作了更明确的描述，并揭示了这些基与Dipper-James的半标准基之间的某种复杂关系，即[19，12]意义上的IC关系。同上]。这实际上给出了半标准基的一种新的构造。结合Grojnowski-Lusztig [14]的一个结果，我们还将证明在A型情形下，[11]中引入的标准基与Lusztig的标准基[18，20]完全相同。因此，我们最终得到了A型量子化包络代数不可约表示的Lusztig标准基的一个新的描述。我们将本文组织如下。我们首先简要回顾一下第1节中Dipper-James对q-Weyl模的构造。在对#-Weyl模的典范基给出了更有用的描述之后，我们证明了这些基是关于某些适当的对合的某些IC基和§ 2中的“标准基”。由此，引入了g-Weyl模的标准基.在第3节中，我们将证明这些标准基直到纯量都是Dipper-James的半标准基。这实际上证明了半标准基与标准基之间的IC关系。最后，我们将证明我们的典型基同意Lusztig的典型基的类型A.1。q-Weyl模及其半标准基已知量子线性群的不可约表示的研究可以归结为不可约多项式表示的研究（在经典情况下参见[22，第11章]和[13]），或者等价地，#-Schur代数的不可约表示。为了我们的目的，将我们的注意力限制在^-Schur代数的表示论上就足够了。
In [11] the author introduced certain canonical bases for the irreducible representations of quantum linear groups via the Kazhdan-Lusztig bases (see [10]) of#-Schur algebras. In this paper we shall give a more explicit description of these bases by using Dipper-James' construction of tf-Weyl modules [9], and reveal a certain sophisticated relation, that is, the IC relation in the sense [19, 12], between these bases and Dipper-James' semi-standard bases [loc. cit.]. This in fact gives a new construction of the semi-standard bases. By combining this investigation with a result of Grojnowski-Lusztig [14], we shall also prove that the canonical bases introduced in [11] are exactly the same as Lusztig's canonical bases [18, 20] in the case of type A. Therefore we eventually obtain a new description of Lusztig's canonical bases for irreducible representations of the quantized enveloping algebra of type A. We organize this paper as follows. We first revisit briefly Dipper-James' construction of q-Weyl modules in § 1. After giving a more useful description of the canonical bases for the#-Weyl modules, we show that these bases are certain IC bases with respect to some appropriate involution and'standard bases' in § 2. Thus, the standard bases for g-Weyl modules are introduced. In Section 3, we shall prove that these standard bases are, up to scalars, Dipper-James' semi-standard bases. This in fact proves the IC relation between the semi-standard bases and the canonical bases. Finally, we shall prove that our canonical bases agree with Lusztig's canonical bases for type A.1. q-Weyl modules and their semi-standard basesIt is known that the study of irreducible representations of quantum linear groups can be reduced to the study of irreducible polynomial representations (see [22, Chapter 11] and [13] in the classical case), or equivalently, irreducible representations of#-Schur algebras. For our purpose, it suffices to restrict our attention to the representation theory of^-Schur algebras.