Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
131 tokens/sec
GPT-4o
10 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

The Hilbert Series of the Irreducible Quotient of the Polynomial Representation of the Rational Cherednik Algebra of Type $A_{n-1}$ in Characteristic $p$ for $p|n-1$ (1811.04910v2)

Published 12 Nov 2018 in math.RT

Abstract: 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.

Summary

We haven't generated a summary for this paper yet.