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Representations of $E_{7}$ , equivalent combinatorics and algebraic varieties

Xiaoping Xu

Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China

School of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, P. R. China

E-mail Address: xiaoping@math.ac.cn

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https://doi.org/10.1142/S0219498818500457Cited by:1 (Source: Crossref)

Abstract

In our earlier work on a new functor from $E_{6}$ -Mod to $E_{7}$ -Mod, we found a one-parameter ( $c$ ) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type $E_{7}$ in $27$ variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector $\vec{a} \in ℂ^{27} \ {\vec{0}}$ , we prove that the space forms an irreducible $E_{7}$ -module for any $c \in ℂ$ if $\vec{a}$ is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of $E_{6}$ over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

Communicated by I. M. Musson

Keywords:

AMSC: 17B10, 17B25, 22E46

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