外代数的拟对称调和

Pub Date : 2022-06-04 DOI:10.4153/S0008439523000024
N. Bergeron, K. Chan, F. Soltani, M. Zabrocki
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引用次数: 0

摘要

摘要我们研究了n个反共轭(费米子)变量中的拟对称多项式环。设$R_n$表示n个反交换变量中的多项式环。本文的主要结果表明了关于反交换变量中拟对称多项式的以下有趣事实:(1)$R_n$中的拟对称多项式形成了$R_n]的交换子代数。(2) 存在$R_n$与理想$I_n$的商的基础,理想$I_n$由$R_n$中的准对称多项式生成,该多项式由投票序列索引。商的希尔伯特级数由$$\beargin{align*}\text给出{Hilb}_{R_n/I_n}(q)=\sum_{k=0}^{\lfloor{n/2}\rfloor}f^{(n-k,k)}q^k\,\end{align*}$$其中$f^{。(3) 由打破投票条件的序列索引的拟对称多项式生成的理想有一个基础。
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Quasisymmetric harmonics of the exterior algebra
Abstract We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$ . (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$ . (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.
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