Abstract
We obtain a combinatorial formula related to the shear transformation for semi-invariants of binary forms, which implies the classical characterization of semi-invariants in terms of a differential operator. Then, we present a combinatorial proof of an identity of Hilbert, which leads to a relation of Cayley on semi-invariants. This identity plays a crucial role in the original proof of Sylvester’s theorem on semi-invariants in connection with the Gaussian coefficients. Moreover, we show that the additivity lemma of Pak and Panova which yields the strict unimodality of the Gaussian coefficients for \(n,k \ge 8\) can be deduced from the ring property of semi-invariants.
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Acknowledgements
We wish to thank the referee for invaluable comments. This work was done under the auspices of the National Science Foundation of China.
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Dedicated to Doron Zeilberger on the Occasion of His Seventieth Birthday.
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Chen, W.Y.C., Jia, I.D.D. Semi-invariants of binary forms and Sylvester’s theorem. Ramanujan J 59, 297–311 (2022). https://doi.org/10.1007/s11139-021-00505-9
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DOI: https://doi.org/10.1007/s11139-021-00505-9
Keywords
- Hilbert’s identity
- Sylvester’s theorem
- Binary forms
- Semi-diagrams
- Semi-invariants
- Gaussian coefficients
- Strict unimodality