Tail Positive Words and Generalized Coinvariant Algebras
Tail Positive Words and Generalized Coinvariant Algebras
复制标题
尾正词和广义协变代数
DOI：
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发表时间：
2017
影响因子：
0.7
通讯作者：
A. Wilson
中科院分区：
文献类型：
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作者：
B. Rhoades;A. Wilson
设$n,k,$和$r$为非负整数，设$S_n$为对称群。在$n$变量中引入多项式环$mathbb{Q}[x_1, dots, x_n]$的商$R_{n,k,r}$，该商具有分级$S_n$-模的结构。当$r geq n$或$k = 0$时，商$R_{n,k,r}$约化为附加在对称群上的经典协不变代数$R_n$。正如$R_n$的代数性质是由$S_n$中置换的组合性质控制的，$R_{n,k,r}$的代数性质是由称为{em尾正词}的对象的组合性质控制的。我们计算了$R_{n,k,r}$的标准单基及其分级$S_n$-同构类型。将$R_{n,k,r}$看作0- hecke代数$H_n(0)$上的一个模，证明了$R_{n,k,r}$是一个射影0- hecke模，并计算了其拟对称和非对称的0- hecke特征。我们推测了商$R_{n,k,r}$与麦克唐纳多项式理论的δ算子之间的关系。
Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $mathbb{Q}[x_1, dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r geq n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called {em tail positive words}. We calculate the standard monomial basis of $R_{n,k,r}$ and its graded $S_n$-isomorphism type. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_n(0)$, prove that $R_{n,k,r}$ is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient $R_{n,k,r}$ and the delta operators of the theory of Macdonald polynomials.