Matrix loci and shadow play

Let lis(w) be the length of the longest increasing subsequence of a permutation w in S_n. We describe a graded quotient R_n of the polynomial ring over an n-by-n matrix of variables whose Hilbert series is the generating function of lis, up to reversal. The Gröbner theory of R_n is governed by the Viennot shadow avatar of the Schensted correspondence. The ring R_n is constructed via the orbit harmonics technique of deformation theory. We will give some related results and open problems. Joint with Jasper Liu, Yichen Ma, Jaeseong Oh, and Hai Zhu.

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