2026/07/30 by Yupeng Li, Jasper Liu, Brendon Rhoades
Mathematics · #math.CO #math.AC
34 pages
arxiv created 2026/07/30 · arxiv updated 2026/07/31
Let xn × n be an n × n matrix of variables and let S = \mathbbF[xn × n] be the polynomial ring over these variables where \mathbbF is a field of characteristic zero. Regard S as the coordinate ring of the affine space \mathbbFn × n of n × n \mathbbF-matrices. Let \mathfrakDn ⊆ \mathbbFn × n be the locus of derangement permutation matrices. We study the orbit harmonics quotient ring \bf R(\mathfrakDn) = S/gr I(\mathfrakDn) where gr I(\mathfrakDn) is the associated graded ideal of the vanishing ideal I(\mathfrakDn) ⊆ S. We give an explicit generating set of gr I(\mathfrakDn), relate the Hilbert series of R(\mathfrakDn) to the Foata transformation and the longest increasing subsequence statistic on \mathfrakSn, and give an alternating sum formula for the graded \mathfrakSn-character of R(\mathfrakDn). Our proofs make heavy use of the mapping cone construction of homological algebra.