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Macdonald polynomials and geometry. (English) Zbl 0952.05074
Billera, Louis J. (ed.) et al., New perspectives in algebraic combinatorics. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 38, 207-254 (1999).
Summary: We explain some remarkable connections between the two-parameter symmetric polynomials discovered in 1988 by Macdonald, and the geometry of certain algebraic varieties, notably the Hilbert scheme \(\text{Hilb}^n(\mathbb{C}^2)\) of points in the plane, and the variety \(C_n\) of pairs of commuting \(n\times n\) matrices.
For the entire collection see [Zbl 0927.00011].

MSC:
05E05 Symmetric functions and generalizations
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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