论文标题
对称群体表示的水平，等级和张量增长
Level, rank, and tensor growth of representations of symmetric groups
论文作者
论文摘要
我们开发了一种与有限古典群体的水平理论相似的程度$ n $的对称群体的不可减至表示的水平理论。一个关键的级别的特性是角色的级别，只要它与$ n $相比不太大，它在其程度上具有良好的下限，而且，每个低度的特征本身都是低水平的，或者在张开符号字符后变得如此。此外，如果$ l_1 $和$ l_2 $满足$ n $中的线性上限，则张贴量的最大组成因子$ l_1 $和$ l_2 $ as $ l_1+l_2 $。为了证明所有这些以积极的特征，我们发展了等级的概念，这是有限古典群体的跨表征表示概念的类似物。我们使用模块化的分支规则和退化仿射Hecke代数表明，只要水平不大，水平和等级都同意。我们利用Schur-Weyl二元性，模块化的Littlewood-Richardson系数和倾斜模块，以证明对对称组的Kronecker产品的Murnaghan-littlewood定理的模块化模拟。作为一种应用，我们获得了对称和交替组的普通和模块化表示的表示形式的增长结果，类似于有限的谎言类型。
We develop a theory of levels for irreducible representations of symmetric groups of degree $n$ analogous to the theory of levels for finite classical groups. A key property of level is that the level of a character, provided it is not too big compared to $n$, gives a good lower bound on its degree, and, moreover, every character of low degree is either itself of low level or becomes so after tensoring with the sign character. Furthermore, if $l_1$ and $l_2$ satisfy a linear upper bound in $n$, then the maximal level of composition factors of the tensor product of representations of levels $l_1$ and $l_2$ is $l_1+l_2$. To prove all of this in positive characteristic, we develop the notion of rank, which is an analogue of the notion of rank of cross-characteristic representations of finite classical groups. We show, using modular branching rules and degenerate affine Hecke algebras, that the level and the rank agree, as long as the level is not too large. We exploit Schur-Weyl duality, modular Littlewood-Richardson coefficients and tilting modules to prove a modular analogue of the Murnaghan-Littlewood theorem on Kronecker products for symmetric groups. As an application, we obtain representation growth results for both ordinary and modular representations of symmetric and alternating groups analogous to those for finite groups of Lie type.
