Conjectures on the quotient ring by diagonal invariants
This paper considers the quotient ring \(R_ n\) of the polynomial ring in two sets of variables \(\mathbb{Q} [X,Y] = \mathbb{Q} [x_ 1, \dots, x_ n, y_ 1, \dots, y_ n]\) by the ideal \(I\) generated by all \(S_ n\)-invariant polynomials without constant term. \(R_ n\) is proved to be isomorphic to the space \(H_ n\) of diagonal harmonics for \(S_ n\) which is the orthogonal complement of \(I\). Compared to the classical theory of the corresponding ring in a single set of variables \(X\) the theory of \(R_ n\) is not yet much developed. However, it has recently been discovered that there is a strong evidence of deep and surprising connections between \(R_ n\) and some fundamental and much-studied aspects of enumerative combinatorics, algebraic geometry and Lie theory; the related conjectures have been checked for small \(n\) using the computer algebra system MACAULAY. The author gives a comprehensive, profound survey on the current state of knowledge about theorems and conjectures concerning the structure of the ring \(R_ n\) as a doubly graded \(S_ n\)-module. Among other things the following subjects are dealt with: Conjectures and propositions about the Hilbert series and Frobenius series of \(R_ n\); relations to certain analogs of the Lagrange inversion formula; investigations on the structure of the space \(H_ n\) in connection with the operator conjecture (which means that \(H_ n\) is the smallest space containing the discriminant and closed under the action of some \(S_ n\)-invariant polynomial different operators); a geometrical interpretation which relates \(R_ n\) to the variety of pairs of commuting nilpotent matrices; analogs of the ring \(R_ n\) and the conjectures about it for general reflection groups (Coxeter groups, especially Weyl groups; failure of operator conjecture for the group \(B_ 4)\). Besides, there is an appendix with Frobenius and Hilbert series tables of \(R_ n\) for \(n\) up to 4 and 7, respectively.
scientific article; zbMATH DE number 495998
scientific article