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Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral | | R. Rimányi
; V. Tarasov
; A. Varchenko
; P. Zinn-Justin
; | | Date: |
10 Oct 2011 | | Abstract: | We consider the tensor power $V=(C^N)^{otimes n}$ of the vector
representation of $gl_N$ and its weight decomposition
$V=oplus_{lambda=(lambda_1,...,lambda_N)}V[lambda]$. For $lambda =
(lambda_1 geq ... geq lambda_N)$, the trivial bundle $V[lambda] imes
C^n oC^n$ has a subbundle of q-conformal blocks at level l, where $l =
lambda_1-lambda_N$ if $lambda_1-lambda_N> 0$ and l=1 if
$lambda_1-lambda_N=0$. We construct a polynomial section
$I_lambda(z_1,...,z_n,h)$ of the subbundle. The section is the main object of
the paper. We identify the section with the generating function
$J_lambda(z_1,...,z_n,h)$ of the extended Joseph polynomials of orbital
varieties, defined in [DFZJ05,KZJ09].
For l=1, we show that the subbundle of q-conformal blocks has rank 1 and
$I_lambda(z_1,...,z_n,h)$ is flat with respect to the quantum
Knizhnik-Zamolodchikov discrete connection.
For N=2 and l=1, we represent our polynomial as a multidimensional
q-hypergeometric integral and obtain a q-Selberg type identity, which says that
the integral is an explicit polynomial. | | Source: | arXiv, 1110.2187 | | Services: | Forum | Review | PDF | Favorites | |
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