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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
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##### References:
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