表示论的渐近结果
Asymptotic results for Representation Theory
摘要：
有限群的表示论描绘了群论、代数组合学和概率的奇妙交叉。特别是Plancherel措施是一个概率，自然产生的表示论，在这篇论文中，我们考虑三个分支的渐近问题随机Plancherel分布表示。首先，我们回忆一下对称群的不可约特征标，它们由整数分区索引。我们专注于所谓的“双重方法”，其中分区索引字符现在被认为是变量。通过研究表示矩阵的部分迹与部分和，推广了Kerov关于Plancherel分布特征标渐近性的一个著名结果.我们将这些对象分解成一个主项和一个余项，在每种情况下，我们证明了一个主项的中心极限定理。应用这些结果证明了部分和的一个大数定律。在第二部分中，我们考虑射影表示的对称群转换的多矩形的方法开发的斯坦利的经典情况。特别是，我们提出了一个正猜想的系数的重整化投影字符，写在一组新的坐标多项式。我们证明了多项式的首项的正性。在论文的最后一部分，我们回顾了超字符理论，这是一个替代的工具，表示论时，经典的方法被证明是棘手的。我们表明，推广的Plancherel措施，称为superplancherel措施，自然产生。我们专注于一个特定的超字符理论的上unitriangular群，其中超字符的索引集分区。我们证明了一个极限形状结果的随机集划分根据这个分布。
英文摘要：
Representation theory of finite groups portrays a marvelous crossroad of group theory, algebraic combinatorics, and probability. In particular the Plancherel measure is a probability that arises naturally from representation theory, and in this thesis we consider three ramifications of asymptotic questions for random Plancherel distributed representations. First we recall irreducible characters of the symmetric group, which are indexed by integer partitions. We focus on the so called 'dual approach', in which the partition indexing a character is now considered to be the variable. We extend a famous result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose each of these objects into a main term and a remainder, and in each case we prove a central limit theorem for the main term. We apply these results to prove a law of large numbers for the partial sum. In the second part we consider projective representations of the symmetric group by converting a multirectangular approach developed by Stanley for the classical case. In particular, we present a positivity conjecture for the coefficients of the renormalized projective character, written as a polynomial in a new set of coordinates. We prove the positivity for the leading term of the polynomial. In the last part of the thesis we recall the theory of supercharacters, which is an alternative tool to representation theory when the classical approach proves to be intractable. We show that a generalization of the Plancherel measure, called the superplancherel measure, arises naturally. We focus on a particular supercharacter theory for the upper unitriangular group in which supercharacters are indexed by set partitions. We prove a limit shape result for a random set partition according to this distribution.
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