Packed Words and Quotient Rings

TL;DR

This paper introduces a new quotient of the polynomial ring based on packed words, connecting it to existing coinvariant rings and exploring its algebraic and combinatorial properties.

Contribution

It defines a novel quotient ring governed by packed words and relates it to known coinvariant rings, expanding the combinatorial framework.

Findings

Established the quotient $S_n$ based on packed words.

Connected $S_n$ to generalized coinvariant rings.

Linked the new quotient to the superspace coinvariant ring.

Abstract

The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\ldots,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length $n$. A word $w = w_1 \dots w_n$ over the positive integers is packed if whenever $i > 2$ appears as a letter of $w$, so does $i-1$. We introduce a quotient $S_n$ of $\mathbb{Q}[x_1,\ldots,x_n]$ which is governed by the combinatorics of packed words. We relate our quotient $S_n$ to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring.