Fock Space of Level infinity and Characters of Vertex Operators

2020/08/14 by Rahmati, Mohammad Reza, Flores, Gerardo
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2008.07287

Abstract

We present an extension of the trace of a vertex operator and explain a representation-theoretic interpretation of the trace. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators and compute its trace as a character formula. To do this, we define the Fock space of infinite level \mathfrakF^∞. Then, we prove a duality between \mathfrakgl_∞ and \mathfraka_∞=\widehat\mathfrakgl_∞ of Howe type, which provides a decomposition of \mathfrakF^∞ into irreducible representations with joint highest weight vector for \mathfrakgl_∞ and \mathfraka_∞. The decomposition of the Fock space \mathfrakF^∞ into highest weight representations provides a method to calculate and interpret the extended trace.

Related

• A combinatorial decomposition of higher level Fock spaces
• Rational semistandard tableaux and character formula for the Lie superalgebra \frakgl_∞|∞
• Infinite-level Fock spaces, crystal bases, and tensor product of extremal weight modules of type A_+∞
• Vertex Operator of U_q(\widehatB_l) for Level One
• Notes on Fock space