Abstract
The polynomial ring Br:=Q[e1,…,er] in r indeterminates is a representation of the Lie algebra of all the endomorphism of Q[X] vanishing at Xj for all but finitely many j. We determine the structural formal power series of the Br-representation of gl∞(Q). This is a formal power series in r+2 indeterminates encoding the images of all the basis elements of Br under the action of the generating function of elementary endomorphisms of Q[X]. The obtained expression implies (and improves) a formula by Gatto & Salehyan, which only computes the generating functions for the images of specified basis elements. For sake of completeness, in the last section we construct the structural formal power series of the B=B∞-representation of gl∞(Q). It consists in the evaluation of a bosonic vertex operator against the generating function of the standard Schur basis of B. It provides an alternative description of the bosonic representation of gl∞, due to Date, Jimbo, Kashiwara and Miwa, which does not explicitly involve exponentials of differential operators.
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