论文标题
在特征$ p $和$ q $ demformed quasi-invariants中迈向明确的希尔伯特一系列准不变多项式
Toward explicit Hilbert series of quasi-invariant polynomials in characteristic $p$ and $q$-deformed quasi-invariants
论文作者
论文摘要
我们研究$ m $ m $ quasi-quasi-quasi-quasi invariant多项式的空间$ s_n $在特征$ p $中。使用对称组的表示理论，我们描述了$ n = 3 $的希尔伯特系列$ q_m $，证明了Ren和Xu的猜想[arxiv：1907.13417]。由此，我们可以推断出希尔伯特多项式的综合性和最高术语，而$ q_m $的freeness则是对称多项式的模块，这些模块是针对一般$ n $的。我们还证明，在$ n = 3 $的情况下，我们可以计算$ m，p $的$ q_m $具有与特征0和特征$ p $不同的hilbert系列，以及在这种情况下，$ q_m $的生成器的程度是什么。我们还将各种结果扩展到空间$ q_ {m，q} $ $ q $ -deformed $ m $ -quasi-invariants，并证明了希尔伯特系列$ q_ {m，q} $的足够条件，与$ q_m $ $ q_m $的Hilbert系列不同。
We study the spaces $Q_m$ of $m$-quasi-invariant polynomials of the symmetric group $S_n$ in characteristic $p$. Using the representation theory of the symmetric group we describe the Hilbert series of $Q_m$ for $n=3$, proving a conjecture of Ren and Xu [arXiv:1907.13417]. From this we may deduce the palindromicity and highest term of the Hilbert polynomial and the freeness of $Q_m$ as a module over the ring of symmetric polynomials, which are conjectured for general $n$. We also prove further results in the case $n=3$ that allow us to compute values of $m,p$ for which $Q_m$ has a different Hilbert series over characteristic 0 and characteristic $p$, and what the degrees of the generators of $Q_m$ are in such cases. We also extend various results to the spaces $Q_{m,q}$ of $q$-deformed $m$-quasi-invariants and prove a sufficient condition for the Hilbert series of $Q_{m,q}$ to differ from the Hilbert series of $Q_m$.
