Non-commutative Combinatorial Inverse Systems

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Abstract

We introduce the notion of a combinatorial inverse system in non-commutative variables. We present two important examples, some conjectures and results. These conjectures and results were suggested and supported by computer investigations.

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Is Open Access

Vanishing theorems and character formulas for the Hilbert scheme of points in the plane

Mark Haiman (2002)

Earlier we showed that the Hilbert scheme of \(n\) points in the plane can be identified with the Hilbert scheme of regular \(S_n\) orbits on \(C^{2n}\). Using this result, together with a recent theorem of Bridgeland, King and Reid on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author. In particular we prove that the dimension of the space of diagonal harmonics is equal to \((n+1)^{n-1}\).