Lapointe–Vinet's identification of Macdonald creation operators

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Let PβP_\beta be the Macdonald polynomial indexed by a generalized partition β\beta, let eke_k be the elementary symmetric function acting by multiplication, and let F(κ)F(\kappa) be the diagonal operator defined by

F(κ)P(β1,…,βN)=∏i=1N(tκ+1−iqβi−1;q−1)∞P(β1,…,βN).F(\kappa)P_{(\beta_1,\dots,\beta_N)}=\prod_{i=1}^N(t^{\kappa+1-i}q^{\beta_i-1};q^{-1})_\infty P_{(\beta_1,\dots,\beta_N)}.

Define Fm,κ=F(κ)emF(κ)−1F_{m,\kappa}=F(\kappa)e_mF(\kappa)^{-1}.

Lapointe–Vinet's operator-identification conjecture. The creation operators satisfy

B~k+=F(k)ekF(k)−1=Fk,k.\tilde B_k^+=F(k)e_kF(k)^{-1}=F_{k,k}.

The conjecture identifies the creation operators with conjugated elementary-symmetric-function operators; the supplied text gives no resolution.

References

Primary source

Luc Lapointe and Luc Vinet, “Creation operators for the Macdonald and Jack polynomials”, arXiv:q-alg/9607024 (1996).

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