Abstract
In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over \({\mathbb{Z}}\) to a finite field \({\mathbb{F}_{q}}\), what is the probability that the Schur function \({s_{\lambda}}\) maps to zero? We show that this probability is always at least 1/q and is asymptotically 1/q. Moreover, we give a complete classification of all shapes that can achieve probability 1/q. In addition, we identify certain families of shapes for which the events that the corresponding Schur functions are sent to zero are independent. We also look into the probability that Schur functions are mapped to nonzero values in \({\mathbb{F}_{q}}\).
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Acknowledgments
This research was carried out as part of the 2016 REU program at the School of Mathematics at University of Minnesota, Twin Cities, and was supported by NSF RTG grant DMS-1148634 and by NSF grant DMS-1351590. The authors would like to thank Joel Lewis and Ben Strasser for their comments and suggestions. The authors are also grateful to Ethan Alwaise who helped get this project started. Lastly, we thank the anonymous reviewer for many helpful suggestions to improve our exposition.
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Anzis, B., Chen, S., Gao, Y. et al. Jacobi-Trudi Determinants over Finite Fields. Ann. Comb. 22, 447–489 (2018). https://doi.org/10.1007/s00026-018-0399-8
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DOI: https://doi.org/10.1007/s00026-018-0399-8