Viennot shadows and graded module structure in colored permutation groups

TL;DR

This paper extends Viennot shadows and graded module structures to colored permutation groups, introducing new algebraic and combinatorial tools, including an ideal-theoretic approach and a standard monomial basis, with implications for longest increasing subsequence analogies.

Contribution

It develops an ideal-theoretic extension of Viennot shadows for colored permutation groups and analyzes the resulting graded module structures and combinatorial analogies.

Findings

Constructed an ideal $I_{\mathfrak{S}_{n,r}}$ associated with colored permutation groups.

Established a standard monomial basis for the quotient algebra.

Explored the graded module structure and extended combinatorial conjectures.

Abstract

Let $\mathbf{x}_{n \times n}$ be a matrix of $n \times n$ variables, and let $\mathbb{C}[\mathbf{x}_{n \times n}]$ be the polynomial ring on these variables. Let $\mathfrak{S}_{n,r}$ be the group of colored permutations, consisting of $n \times n$ complex matrices with exactly one nonzero entry in each row and column, where each nonzero entry is an $r$-th root of unity. We associate an ideal $I_{\mathfrak{S}_{n,r}} \subseteq \mathbb{C}[\mathbf{x}_{n \times n}]$ with the group $\mathfrak{S}_{n,r}$, and use orbit harmonics to give an ideal-theoretic extension of the Viennot shadow line construction to $\mathfrak{S}_{n,r}$. This extension gives a standard monomial basis of $\mathbb{C}[\mathbf{x}_{n \times n}]/I_{\mathfrak{S}_{n,r}}$, and introduces an analogous definition of ``longest increasing subsequence'' to the group $\mathfrak{S}_{n,r}$. We examine the extension of Chen's conjecture to this analogy. We also study the structure of $\mathbb{C}[\mathbf{x}_{n \times n}]/I_{\mathfrak{S}_{n,r}}$ as a graded $\mathfrak{S}_{n,r} \times \mathfrak{S}_{n,r}$ module, which subsequently induces a graded $\mathfrak{S}_{n,r} \times \mathfrak{S}_{n,r}$ module structure on the $\mathbb{C}$-algebra $\mathbb{C}[\mathfrak{S}_{n,r}]$.