, Volume 3, Issue 1, pp 17-76

Conjectures on the Quotient Ring by Diagonal Invariants

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We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring \(\mathbb{Q}[x_1 , \ldots ,x_n ,y_1 , \ldots ,y_n ]\) in two sets of variables by the ideal generated by all S n invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x 1, ..., x n} 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.