Conjectures on the Quotient Ring by Diagonal Invariants

We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring Q[Z1 xn, y1,.. . , yn] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = { x 1 , . . . , xn} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory. Introduction It has recently been discovered, mainly on the basis of evidence obtained using the computer algebra system MACAULAY, that there seem to be unexpected and profound connections between a certain natural ring and some fundamental and much-studied aspects of combinatorics and algebraic geometry. The ring in question is the quotient of the polynomial ring Q[x1, ..., xn, y1, ..., yn] by the ideal generated by all Sn invariant polynomials without constant term. This paper is an attempt to treat in a reasonably comprehensive way a series of conjectures (and a few theorems) concerning the structure of this ring as a doubly graded Sn module. Besides listing the existing conjectures and the current state of knowledge about them, I have tried, especially in the later sections, to outline some of their combinatorial and geometric implications. A number of people were involved in formulating these conjectures and exploring various observations about them related here. I have made some attempt to give credit through notes appended to many sections. I hope I have not inadvertently shortchanged any of the many contributors to this work. 1. Essential definitions 1.1. Sn action on Q[X1, ..., xn, y 1 . . . , yn] We will be concerned with the diagonal action of the permutation group Sn by automorphisms of the polynomial ring Q[X, Y] = Q[X1, ..., xn, y1, . . . , yn] in