The determinantal identities of Hamel and Goulden have recently been shown to apply to a tableau-based ninth variation of skew Schur functions. Here we extend this approach and its results to the analogous tableau-based ninth variation of supersymmetric skew Schur functions. These tableaux are built on entries taken from an alphabet of unprimed and primed numbers and that may be ordered in a myriad of different ways, each leading to a determinantal identity. At the level of the ninth variation the corresponding determinantal identities are all distinct but the original notion of supersymmetry is lost. It is shown that this can be remedied at the level of the sixth variation involving a doubly infinite sequence of factorial parameters. Moreover, it is shown that the resulting factorial supersymmetric skew Schur functions are independent of the ordering of the unprimed and primed entries in the alphabet.
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The first author (AMF) was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). The second author (RCK) is grateful for the hospitality extended to him by Professor Bill Chen at the Center for Applied Mathematics at Tianjin University and for the opportunity to pursue this project while visiting him there. This work was supported by the Canadian Tri-Council Research Support Fund.
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Communicated by Jang Soo Kim
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Foley, A.M., King, R.C. Factorial Supersymmetric Skew Schur Functions and Ninth Variation Determinantal Identities. Ann. Comb. 25, 229–253 (2021). https://doi.org/10.1007/s00026-021-00526-7