ScholarWorks@성균관대학교

초록

Let (Formula presented.) be a (Formula presented.) matrix of variables and let (Formula presented.) be the polynomial ring in these variables. Given two weak compositions (Formula presented.) of lengths (Formula presented.) and (Formula presented.), we study the ideal (Formula presented.) generated by row sums, column sums, monomials in row (Formula presented.) of degree (Formula presented.), and monomials in column (Formula presented.) of degree (Formula presented.). We prove results connecting algebraic properties of the quotient ring (Formula presented.) with the set (Formula presented.) of (Formula presented.) -contingency tables. The standard monomial basis of (Formula presented.) with respect to a diagonal term order is encoded by the matrix-ball avatar of the Robinson–Schensted–Knuth correspondence. We describe the Hilbert series of (Formula presented.) in terms of a zigzag statistic on contingency tables. The ring (Formula presented.) carries a graded action of the product (Formula presented.) of symmetry groups of the sequences (Formula presented.) and (Formula presented.); we describe how to calculate the isomorphism type of this graded action. Our analysis regards the set (Formula presented.) as a locus in the affine space (Formula presented.) and applies orbit harmonics to this locus.