表示环链上算子之间的多项式关系
Polynomial relations between operators on chains of representation rings
摘要：
给定一个群链$G_0 \le G_1 \le G_2 ... $，我们可以形成它们的表示环的相应链，连同归纳算子和限制算子。我们可以让$\textrm{Res}^l$表示限制$l$步骤的算子，对于$\textrm{Ind}^l$也是类似的。然后观察到$\textrm{Ind}^l \textrm{Res}^l$是从任何特定表示环到其自身的算子。本文的中心问题是：“如果$\textrm{Ind}^l \textrm{Res}^l$算子是$\textrm{Ind} \textrm{Res}$算子中的多项式，会发生什么？".我们表明，链的圈产品$\{H^n \rtimes S_n\}_{n \in \mathbb{N}}$有这个属性，特别是，出现在对称群的情况下的多项式是下降的阶乘多项式。这一事实的应用给出了一个显着的新方法来计算圈积（特别是对称群）的字符使用矩阵乘法。然后，我们考虑任意链的群体，并找到非常严格的限制，这样的链必须满足，以使$\textrm{Ind}^l \textrm{Res}^l$是一个多项式在$\textrm{Ind} \textrm{Res}$。我们的严格限制证明了这种直觉，即这种性质确实是一种非常罕见和特殊的性质。
英文摘要：
Given a chain of groups $G_0 \le G_1 \le G_2 ... $, we may form the corresponding chain of their representation rings, together with induction and restriction operators. We may let $\textrm{Res}^l$ denote the operator which restricts down $l$ steps, and similarly for $\textrm{Ind}^l$. Observe then that $\textrm{Ind}^l \textrm{Res}^l$ is an operator from any particular representation ring to itself. The central question that this paper addresses is: "What happens if the $\textrm{Ind}^l \textrm{Res}^l$ operator is a polynomial in the $\textrm{Ind} \textrm{Res}$ operator?". We show that chains of wreath products $\{H^n \rtimes S_n\}_{n \in \mathbb{N}}$ have this property, and in particular, the polynomials that appear in the case of symmetric groups are the falling factorial polynomials. An application of this fact gives a remarkable new way to compute characters of wreath products (in particular symmetric groups) using matrix multiplication. We then consider arbitrary chains of groups, and find very rigid constraints that such a chain must satisfy in order for $\textrm{Ind}^l \textrm{Res}^l$ to be a polynomial in $\textrm{Ind} \textrm{Res}$. Our rigid constraints justify the intuition that this property is indeed a very rare and special property.
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