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A comment on the combinatorics of the vertex operator \(\Gamma_{(t|X)}\). (English) Zbl 1433.05324

Summary: The Jacobi-Trudi identity associates a symmetric function to any integer sequence. Let \(\Gamma_{(t|X)}\) be the vertex operator defined by \(\Gamma_{(t|X)} s_\alpha =\sum_{n \in \mathbb{Z} } s_{(n,\alpha )} [X] t^n\). We provide a combinatorial proof for the identity \(\Gamma_{(t|X)} s_\alpha = \sigma [tX] s_{\alpha }[x-1/t]\) due to J.-Y. Thibon et al. [J. Phys. A, Math. Gen. 26, No. 24, 7461–7478 (1993; Zbl 0833.20017); Int. J. Algebra Comput. 1, No. 2, 207–221 (1991; Zbl 0723.05117)]. We include an overview of all the combinatorial ideas behind this beautiful identity, including a combinatorial description for the expansion of \(s_{(n,\alpha )} [X]\) in the Schur basis, for any integer value of \(n\).

MSC:

05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
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