The Moduli Space of Sheaves and a Generalization of MacMahon's Formula
Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 2, pp. 18-26.

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M. Vuletic has recently found a two-parameter generalization of MacMahon's formula. In this paper we show that the coefficients in her formula are the Betti numbers of certain subvarieties in the moduli space of sheaves on the projective plane.
Mots-clés : moduli space, plane partition
Keywords: quiver variety.
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A. Yu. Buryak. The Moduli Space of Sheaves and a Generalization of MacMahon's Formula. Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 2, pp. 18-26. http://geodesic.mathdoc.fr/item/FAA_2013_47_2_a2/

[1] A. Bialynicki-Birula, “Some theorems on actions of algebraic groups”, Ann. of Math. (2), 98 (1973), 480–497 | DOI | MR | Zbl

[2] A. Bialynicki-Birula, “Some properties of the decompositions of algebraic varieties determined by actions of a torus”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 24:9 (1976), 667–674 | MR | Zbl

[3] A. Buryak, “The classes of the quasihomogeneous Hilbert schemes of points on the plane”, Mosc. Math. J., 12:1 (2012), 21–36 ; arXiv: 1011.4459 | DOI | MR | Zbl

[4] M. Ciucu, Plane partitions I: A generalization of MacMahon's formula, arXiv: 9808017

[5] E. Looijenga, “Motivic measures”, Seminaire Bourbaki, vol. 1999/2000, Asterisque, 276, 2002, 267–297 | MR | Zbl

[6] H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Amer. Math. Soc., Providence, RI, 1999 | MR | Zbl

[7] H. Nakajima, K. Yoshioka, “Instanton counting on blowup. I. $4$-dimensional pure gauge theory”, Invent. Math., 162:2 (2005), 313–355 | DOI | MR | Zbl

[8] H. Nakajima, K. Yoshioka, “Lectures on instanton counting”, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, 38, Amer. Math. Soc., Providence, RI, 2004, 31–101 | DOI | MR | Zbl

[9] A. Okounkov, N. Reshetikhin, “Correlation function of Schur process with application to local geometry of a random $3$-dimensional Young diagram”, J. Amer. Math. Soc., 16:3 (2003), 581–603 | DOI | MR | Zbl

[10] J.-P. Serre, “Espaces fibres algebriques”, Exposes de Seminaires 1950-1999, Doc. Math., 1, Soc. Math. France, 2001, 107–139 | MR

[11] R. Stenli, Perechislitelnaya kombinatorika. Derevya, proizvodyaschie funktsii i simmetricheskie funktsii, Mir, M., 2005 | MR

[12] M. Vuletic, “A generalization of MacMahon's formula”, Trans. Amer. Math. Soc., 361:14 (2009), 2789–2804 | MR | Zbl

[13] M. Vuletic, The Shifted Schur Process and Asymptotics of Large Random Strict Plane Partitions, Int. Math. Res. Notices, 2007, 2007 | MR