We investigate the $sin (phi_Lambda-phi_S)$ single-spin asymmetry in the transversely polarized $Lambda$ production in semi-inclusive deeply inelastic scattering process within the framework of the transverse momentum dependent~(TMD) factorization. The asymmetry is contributed by the convolution of the polarizing TMD fragmentation function $D_{1T}^perp$ of the $Lambda$ hyperon and the unpolarized TMD distribution function $f_1$ of the target nucleon. We apply two sets of $D_{1T}^{perp,Lambda^uparrow/q}$, the spectator diquark model result and available parametrization, to numerically estimate the $sin (phi_Lambda-phi_S)$ asymmetry at the kinematical region of electron ion collider (EIC). We take into account the TMD evolution effect in order to include the scale dependence of the TMD distribution functions and fragmentation functions. It is found that different sets of $D_{1T}^{perp,Lambda^uparrow/q}$ lead to different ${sin (phi_Lambda-phi_S)}$ asymmetry, particularly in sign. Future measurements on the $sin (phi_Lambda-phi_S)$ asymmetry with high precision at EIC can provide important cross check on the available $Lambda$ polarizing fragmentation functions as well as constrain them more stringently.
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