论文标题
自由左规带和Q-Analogue的不变理论
Invariant Theory for the free left-regular band and a q-analogue
论文作者
论文摘要
我们从一个不变理论的观点检查了两个具有大对称基团的单体代数。第一个单体是$ n $字母上的自由左规则频段，在所有注射词的集合中定义，即每个字母最多都会出现的单词。该单体具有对称组的作用。第二个MONOID是其$ Q $ - Analogues之一，由K. Brown考虑，采用了有限的通用线性组的动作。在这两种情况下，我们都表明，不变的亚代词是半圣经的代数，并使用stirling和$ q $ stirling数字来表征它们。 然后，我们使用随机步行理论和随机对顶的理论的结果将整个单体代数分解为不可减数，同时作为不变环上的模块和组表示。我们的不可证实的分解是用Désarménien和Wachs引入的扰动对称函数来描述的。
We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers. We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by Désarménien and Wachs.
