In this paper, we study the irreducible quotient ${{ℒ}}_{t,c}$ of the polynomial representation of the rational Cherednik algebra ${{\mathscr{H}}}_{t,c}(S_n,𝔥)$ of type $A_{n-1}$ over an algebraically closed field of positive characteristic $p$ where $p|n-1$. In the $t=0$ case, for all $c\ne 0$ we give a complete description of the polynomials in the maximal proper graded submodule $ker{ℬ}$, the kernel of the contravariant form ${ℬ}$, and subsequently find the Hilbert series of the irreducible quotient ${{ℒ}}_{0,c}$. In the $t=1$ case, we give a complete description of the polynomials in $ker{ℬ}$ when the characteristic $p=2$ and $c$ is transcendental over ${𝔽}_2$, and compute the Hilbert series of the irreducible quotient ${{ℒ}}_{1,c}$. In doing so, we prove a conjecture due to Etingof and Rains completely for $p=2$, and also for any $t=0$ and $n\equiv 1(modp)$. Furthermore, for $t=1$, we prove a simple criterion to determine whether a given polynomial $f$ lies in $ker{ℬ}$ for all $n=kp+r$ with $r$ and $p$ fixed.

在本文中，我们研究了不可约商${{ℒ}}_{吨,C}$有理 Cherednik 代数的多项式表示${{\mathscr{H}}}_{吨,C}({\mathrm{小号}}_n,𝔥)$类型${\mathrm{一个}}_{n-1}$在具有正特征的代数闭域上$p$在哪里$p|n-1$. 在里面$吨=0$案例，对于所有人$C\ne 0$我们给出了最大真分级子模块中多项式的完整描述$克尔{ℬ}$, 逆变形式的核${ℬ}$，然后找到不可约商的希尔伯特级数${{ℒ}}_{0,C}$. 在里面$吨=1$在这种情况下，我们给出了多项式的完整描述$克尔{ℬ}$当特征$p=2$和$C$是超越的${𝔽}_2$，并计算不可约商的希尔伯特级数${{ℒ}}_{1,C}$. 这样做，我们完全证明了 Etingof 和 Rains 的猜想$p=2$, 也适用于任何$吨=0$和$n\equiv 1(模组p)$. 此外，对于$吨=1$，我们证明了一个简单的标准来确定一个给定的多项式是否$F$在于$克尔{ℬ}$对所有人$n=ķp+r$和$r$和$p$固定的。