On mullineux' conjecture in the representation theory of symmetric groups
On mullineux' conjecture in the representation theory of symmetric groups
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论对称群表示论中的穆利纽猜想
DOI：
10.1080/00927879708825953
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发表时间：
1997
影响因子：
0.7
通讯作者：
Maozhi Xu
中科院分区：
文献类型：
--
作者：
Maozhi Xu
对称群表示论中的一个众所周知的事实是，我们可以分别用Young图和p-行正则Young图来索引普通不可约群和p-模不可约群[J2][J3，§ 11]。设X，是n的划分，如通常一样。我们用SX，SJ'表示Specht模，如果p是p-行正则的，我们用DJ'表示SJ'与它的唯一极大子模SJ' nSwL的商.当我们用交错表示张量DJ'时，我们得到另一个不可约的。它们的指数之间有何关联？Mullineux提出了解决这个问题[穆尔。几年后，詹姆斯建议这个问题应该得到解决[J3]。为了使事情更容易，我们称这个问题为詹姆斯问题，并称这个猜想为穆林诺猜想。它对于几种特殊情况的正确性可以在文献中找到，例如[Mu 21]对于p-核心情况，[XI]对于与钩子相关的情况。然而，直到最近，Mullineux猜想才被Kleshchev [K1][K2][K3]的一系列论文还原为一个纯粹的组合命题，福特和Kleshchev [FK]给出了这个组合命题的一个非常长而复杂的证明。然后Bessenrodt和Olsson通过发明p-行正则表达式的剩余符号[1BO]给出了这个组合命题的一个较短的证明。本文的主要结果是给出了p-行正则Young图之间Mullineux对应的一个新的构造[穆尔.我们的构造更微妙，它可以很容易地用来获得我们所谓的p-行正则图的p-评价。我们将利用这些结果提出另一个更强的猜想。它也是combinatorial，但不同于Kleshchev的。我们的组合猜想蕴涵了Mullineux猜想。如果一个简短的证明，它可以给，然后一个简短的证明Mullineux'猜想将遵循。
A well-known fact in the representation theory of symmetric groups is that we can index ordinary irreducibles and p-modular irreducibles by Young diagrams and p-row regular Young diagrams respectively [J2][J3, § 11]. Let X, be partitions of n as usual. We use SX, SJ'to denote Specht modules and if p is p-row regular, we use DJ'to denote the quotient of SJ'by its unique maximal submodule SJ'n SwL. When we tensor DJ'with the alternating representation, we obtain another irreducible. How do their indices relate to each other? Mullineux conjectured a solution to this problem [Mull. James suggested some years later that this problem should be solved [J3]. To make things easier we call the problem James' problem and the conjecture Mullineux'conjecture. The correctness of it for several special cases can be found in the literature, for example [Mu21 for the p-core cases,[XI for the cases related to hooks. However it is only recently that Mullineux'conjecture has been reduced to a purely combinatorial statement by a series of papers of Kleshchev [Kl][K2][K3], and a very long and complicated proof of this combinatorial statement was given by Ford and Kleshchev [FK]. Then Bessenrodt and Olsson gave a shorter proof of this combinatorial statement by the invention of the residue symbol for p-row regular diagramslBO]. However a complete proof of Mullineux'conjecture is still too long.The main result of this paper is to give a new construction of Mullineux'correspondence among p-row regular Young diagrams [Mull. Our construction is more subtle and it can be easily used to obtain what we called p-evaluations of p-row regular diagrams. We shall use these result to propose another stronger conjecture. It is also cornbinatorial, but is different from Kleshchev's. Our combinatorial conjecture implies Mullineux'conjecture. If a short proof of it could be given, then a short proof of Mullineux'conjecture would follow.