# zbMATH — the first resource for mathematics

The moduli space of sheaves and a generalization of MacMahon’s formula. (English. Russian original) Zbl 1295.05259
Funct. Anal. Appl. 47, No. 2, 96-103 (2013); translation from Funkts. Anal. Prilozh. 47, No. 2, 18-26 (2013).
In this paper the author gives an algebro-geometric interpretation of a formula of M. Vuletić. In [Trans. Am. Math. Soc. 361, No. 5, 2789–2804 (2009; Zbl 1228.05297)] M. Vuletić proved the formula $\sum_{\pi \in \mathcal{P}} F_{\pi}(q,t) s^{|\pi|} = \prod_{n=1}^{\infty} \prod_{k=0}^{\infty} (\frac{1-t s^{n} q^{k}}{1-s^{n} q^{k}})^{n},$ which should be seen as a generalization of MacMahon’s formula. Here $$\mathcal{P}$$ is the set of plane partitions and $$F_{\pi}$$ is an explicit rational function whose definition we omit.
Theorem 1.2 of the paper provides an algebro-geometric interpretation of $$F_{\pi}(q,0)$$. Let $$\mathcal{M}_{r,n}$$ denote the moduli space of torsion-free sheaves on $$\mathbb{P}^{2}$$ that have rank $$r$$, second Chern class $$n$$, and are equipped with a framing at infinity. The usual action of the 2-dimensional torus $$T= \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$$ induces an action of $$T$$ on $$\mathcal{M}_{r,n}$$. The author shows that the irreducible components of the locus of torus fixed points are enumerated by plane partitions $$\pi$$ s.t. $$|\pi|=n$$ and $$\pi_{0,0} \leq r$$. Writing $$\mathcal{M}^{T}_{r,n}(\pi)$$ for the component corresponding to a partition $$\pi$$, the author constructs natural embeddings $\mathcal{M}^{T}_{1,n}(\pi) \hookrightarrow \mathcal{M}^{T}_{2,n}(\pi) \hookrightarrow \mathcal{M}^{T}_{3,n}(\pi) \hookrightarrow \dots,$ and then defines the “limit space” $$\mathcal{M}^{T}_{\infty, n}$$. Theorem 1.2 states that the associated Poincaré series $P_{q}(\mathcal{M}^{T}_{\infty, n}(\pi))= \sum_{i=1}^{\infty} \dim H^{i}(\mathcal{M}_{\infty, n}^{T}) q^{i/2}$ satisfies $F_{\pi}(q, 0) = P_{q}(\mathcal{M}^{T}_{\infty, n}(\pi)).$ The author derives this result from an explicit expression for $$P_q(\mathcal{M}^{T}_{r,n}(\pi)$$, a result stated as Theorem 1.1.
Using the formula of Vuletić and Theorem 1.2, he deduces Corollary 1.3, which is the identity $\sum_{n \geq 0} P_{q}(\mathcal{M}_{\infty,n}^{T}) t^{n} = \prod_{i=0}^{\infty} \prod_{j = 1}^{\infty} \frac{1}{(1-q^{i} t^{j})^{j}}.$ He also uses the geometry of $$\mathcal{M}_{r,n}$$ to prove an interesting power series identity, a result that is Theorem 1.4.

##### MSC:
 05E05 Symmetric functions and generalizations 14D20 Algebraic moduli problems, moduli of vector bundles
##### Keywords:
moduli space; plane partition; quiver variety
Full Text:
##### References:
  A. Bialynicki-Birula, ”Some theorems on actions of algebraic groups,” Ann. of Math. (2), 98 (1973), 480–497. · Zbl 0275.14007  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. · Zbl 0355.14015  A. Buryak, ”The classes of the quasihomogeneous Hilbert schemes of points on the plane,” Mosc. Math. J., 12:1, 21–36; http://arxiv.org/abs/1011.4459 . · Zbl 1262.14005  M. Ciucu, Plane partitions I: A generalization of MacMahon’s formula, http://arxiv.org/abs/math/9808017 . · Zbl 1292.52021  E. Looijenga, ”Motivic measures,” Seminaire Bourbaki, vol. 1999/2000, Asterisque No. 276 (2002), 267–297. · Zbl 0996.14011  H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Amer. Math. Soc., Providence, RI, 1999. · Zbl 0949.14001  H. Nakajima and K. Yoshioka, ”Instanton counting on blowup. I. 4-dimensional pure gauge theory,” Invent. Math., 162:2 (2005), 313–355. · Zbl 1100.14009  H. Nakajima and K. Yoshioka, ”Lectures on instanton counting,” in: Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, 31–101. · Zbl 1080.14016  A. Okounkov and 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. · Zbl 1009.05134  J.-P. Serre, ”Espaces fibres algebriques,” in: Exposes de Seminaires 1950–1999, Doc. Math. 1, Soc. Math. France, 2001, 107–139.  R. Stanley, Enumerative combinatorics, Cambridge Univ. Press, Cambridge, 1999.  M. Vuletic, ”A generalization of MacMahon’s formula,” Trans. Amer. Math. Soc., 361:5 (2009), 2789–2804. · Zbl 1228.05297  M. Vuletic, The Shifted Schur Process and Asymptotics of Large Random Strict Plane Partitions, Int. Math. Res. Notices, vol. 2007, 2007.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.