Combinatorial Theory

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 \(\mathbb{C}[\mathfrak{S}_{n,r}]\).

Mathematics Subject Classifications: 05E10, 05E16, 05E18, 05E14

Keywords: Viennot's shadow lines, orbit harmonics, ideals, graded modules