Séminaire Lotharingien de Combinatoire, 93B.126 (2025), 12 pp.
Jasper M. Liu, Yichen Ma, Brendon Rhoades and {ai Zhu
Matrix Loci, Orbit Harmonics, and Shadow Play
Abstract.
Let xn × n be an n × n matrix of variables and let C[xn × n] be the polynomial ring in these variables. We consider the ideal In ⊂ C[xn × n] generated by all row sums, column sums, and products of variables in the same row or column. We prove Rn = C[xn × n]/In has standard monomial theory governed by the Viennot shadow line avatar of the Schensted correspondence and has Hilbert series given by the longest increasing subsequence distribution on permutations (up to reversal). The ring Rn coincides with the orbit harmonics quotient ring attached to the permutation matrix locus in the space Matn × n(C) of n × n complex matrices. With Rn as motivation, we prove results on orbit harmonics quotients for other matrix loci.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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