Increasing subsequences, matrix loci, and Viennot shadows
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Let 𝐱_n × n be an n × n matrix of variables and let 𝔽[𝐱_n × n] be the polynomial ring in these variables over a field 𝔽. We study the ideal I_n ⊆𝔽[𝐱_n × n] generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient 𝔽[𝐱_n × n]/I_n admits a standard monomial basis determined by Viennot's shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of 𝔽[𝐱_n × n]/I_n is the generating function of permutations in 𝔖_n by the length of their longest increasing subsequence. Along the way, we describe a `shadow junta' basis of the vector space of k-local permutation statistics. We also calculate the structure of 𝔽[𝐱_n × n]/I_n as a graded 𝔖_n ×𝔖_n-module.
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