Norm conjecture for generalized Macdonald integral forms

Let NN be a positive integer, let u=(u1,,uN)\vec{u}=(u_1,\ldots,u_N), and let λ=(λ(1),,λ(N))\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)}) be an NN-tuple of partitions. Write eN(u)=u1u2uNe_N(\vec{u})=u_1u_2\cdots u_N, and let n(λ)n(\lambda) denote the usual partition statistic. Let Nλ,μ(Q)N_{\lambda,\mu}(Q) be the Nekrasov factor. Generalized Macdonald norm conjecture. The integral-form vectors satisfy

KλKλ=(1)NeN(u)λi=1NtNn(λ(i))qNn(λ(i))uiNλ(i)i,j=1NNλ(i),λ(j)(qui/(tuj)).\langle K_{\vec{\lambda}}|K_{\vec{\lambda}}\rangle=(-1)^N e_N(\vec{u})^{|\vec{\lambda}|}\prod_{i=1}^N t^{-Nn(\lambda^{(i)})}q^{Nn(\lambda^{(i)'})}u_i^{N|\lambda^{(i)}|}\prod_{i,j=1}^N N_{\lambda^{(i)},\lambda^{(j)}}(qu_i/(tu_j)).

This predicts that the norm reproduces the corresponding Nekrasov factor. The source marks the equality with a question mark and gives no resolution.

Sources & referencesView supporting material

Primary source

Yusuke Ohkubo, Hidetoshi Awata and Hiroki Fujino, “Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence”, arXiv:1512.08016 (2016).

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