论文标题
停车位的一些自然扩展
Some natural extensions of the parking space
论文作者
论文摘要
我们构建了一个由$ c \ in \ {1，\ dots，n \} $索引的$ s_n $模块的家族，并用限制为$ s_ {n-1} $的属性，他们恢复了海曼的经典停车功能代表。这些模块的构建依赖于与一组停车功能密切相关的集合上的$ s_n $。我们计算这些模块的字符，并使用结果描述将它们分类为同构。特别是，我们表明，同构类别的数量等于满足$ d \ neq 2 \：（\！\！\！\！\！\！\ mod 4）$的$ n $ $ n $的数量。在$ c = n $和$ c = 1 $的情况下，我们计算轨道数。根据经验证据，我们猜测，当$ c = 1 $时，我们的代表为$ h $阳性，实际上是Berget和Rhoades构建的停车功能表示的（未分级）扩展。
We construct a family of $S_n$ modules indexed by $c\in\{1,\dots,n\}$ with the property that upon restriction to $S_{n-1}$ they recover the classical parking function representation of Haiman. The construction of these modules relies on an $S_n$-action on a set that is closely related to the set of parking functions. We compute the characters of these modules and use the resulting description to classify them up to isomorphism. In particular, we show that the number of isomorphism classes is equal to the number of divisors $d$ of $n$ satisfying $ d\neq 2 \: (\!\!\!\!\mod 4)$. In the cases $c=n$ and $c=1$, we compute the number of orbits. Based on empirical evidence, we conjecture that when $c=1$, our representation is $h$-positive and is in fact the (ungraded) extension of the parking function representation constructed by Berget and Rhoades.
