2024/01/15 by Moxuan, Liu, Jasper M. · 4 citations
Computer Science · Mathematics · #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2401.07850
openalex publication_date 2024/01/15 · openalex created_date 2024/01/18 · openalex updated_date 2026/07/28
Let xn × n be a matrix of n × n variables, and let ℂ[xn × n] be the polynomial ring on these variables. Let \mathfrakSn,r be the group of colored permutations, consisting of n × 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_\mathfrakSn,r ⊆ ℂ[xn × n] with the group \mathfrakSn,r, and use orbit harmonics to give an ideal-theoretic extension of the Viennot shadow line construction to \mathfrakSn,r. This extension gives a standard monomial basis of ℂ[xn × n]/I_\mathfrakSn,r, and introduces an analogous definition of ``longest increasing subsequence'' to the group \mathfrakSn,r. We examine the extension of Chen's conjecture to this analogy. We also study the structure of ℂ[xn × n]/I_\mathfrakSn,r as a graded \mathfrakSn,r × \mathfrakSn,r module, which subsequently induces a graded \mathfrakSn,r × \mathfrakSn,r module structure on the ℂ-algebra ℂ[\mathfrakSn,r].