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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.

05E05 Symmetric functions and generalizations
14D20 Algebraic moduli problems, moduli of vector bundles
Full Text: DOI
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