2025/03/25 by Oh, Jaeseong, Rhoades, Brendon · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.19694
Let xk × p be a k × p matrix of variables and let \mathbbF[xk × p] be the polynomial ring in these variables. Given two weak compositions α,β\models0 n of lengths ℓ(α) = k and ℓ(β) = p, we study the ideal Iα,β ⊆ \mathbbF[xk × ℓ] generated by row sums, column sums, monomials in row i of degree > αi, and monomials in column j of degree > βj. We prove results connecting algebraic properties of the quotient ring Rα,β := \mathbbF[xk × ℓ]/Iα,β with the set Cα,β of α,β-contingency tables. The standard monomial basis of Rα,β with respect to a diagonal term order is encoded by the matrix-ball avatar of the RSK correspondence. We describe the Hilbert series of Rα,β in terms of a zigzag statistic on contingency tables. The ring Rα,β carries a graded action of the product Stab(α) × Stab(β) of symmetry groups of the sequences α= (α1,…,αk) and β= (β1,…,βp); we describe how to calculate the isomorphism type of this graded action. Our analysis regards the set Cα,β as a locus in the affine space Matk × p(\mathbbF) and applies orbit harmonics to this locus.