# Invariant theory and the W_{1+\infty} algebra with negative integral central charge

Andrew R. Linshaw

Arxiv ID: 0811.4067•Last updated: 5/21/2021

The vertex algebra W_{1+\infty,c} with central charge c may be defined as a
module over the universal central extension of the Lie algebra of differential
operators on the circle. For an integer n\geq 1, it was conjectured in the
physics literature that W_{1+\infty,-n} should have a minimal strong generating
set consisting of n^2+2n elements. Using a free field realization of
W_{1+\infty,-n} due to Kac-Radul, together with a deformed version of Weyl's
first and second fundamental theorems of invariant theory for the standard
representation of GL_n, we prove this conjecture. A consequence is that the
irreducible, highest-weight representations of W_{1+\infty,-n} are parametrized
by a closed subvariety of C^{n^2+2n}.

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