We study the irreducible quotient $\mathcal{ L}_{ t,c}$ of the polynomial representation of the rational Cherednik algebra $\mathcal{ H}_{ t,c}(S_n,\mathfrak{ h})$ 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 $\ker \mathcal{ B}$, the kernel of the contravariant form $\mathcal{ B}$, and subsequently find the Hilbert series of the irreducible quotient $\mathcal{ L}_{ 0,c}$. In the $t=1$ case, we give a complete description of the polynomials in $\ker \mathcal{ B}$ when the characteristic $p=2$ and $c$ is transcendental over $\mathbb{ F}_2$, and compute the Hilbert series of the irreducible quotient $\mathcal{ L}_{ 1,c}$. In doing so, we prove a conjecture due to Etingof and Rains completely for $p=2$, and also for any $t=0$ and $n\equiv 1\pmod{ p}$. Furthermore, for $t=1$, we prove a simple criterion to determine whether a given polynomial $f$ lies in $\ker \mathcal{ B}$ for all $n=kp+r$ with $r$ and $p$ fixed.
Categories: math.RT