# Tail positive words and generalized coinvariant algebras

## TL;DR

This paper introduces a new family of quotients of polynomial rings, called $R_{n,k,r}$, which generalize classical coinvariant algebras and are governed by combinatorics of tail positive words, with detailed algebraic and representation-theoretic analysis.

## Contribution

It defines and analyzes the algebraic structure of $R_{n,k,r}$, including bases, module properties, and connections to 0-Hecke algebras and Macdonald polynomials.

## Key findings

- Calculated the standard monomial basis of $R_{n,k,r}$.
- Established $R_{n,k,r}$ as a projective 0-Hecke module.
- Connected $R_{n,k,r}$ to tail positive words and conjectured links to Macdonald polynomial operators.

## Abstract

Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \geq n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called {\em tail positive words}. We calculate the standard monomial basis of $R_{n,k,r}$ and its graded $S_n$-isomorphism type. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_n(0)$, prove that $R_{n,k,r}$ is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient $R_{n,k,r}$ and the delta operators of the theory of Macdonald polynomials.