Semi-invariants of Binary Forms and Sylvester's Theorem

TL;DR

This paper develops combinatorial formulas and proofs related to semi-invariants of binary forms, connecting classical invariant theory with modern combinatorial identities and properties of Gaussian coefficients.

Contribution

It introduces a combinatorial formula for semi-invariants, provides a combinatorial proof of Hilbert's identity, and links semi-invariants to the additivity lemma of Pak and Panova.

Findings

Derived a combinatorial formula for semi-invariants.

Provided a combinatorial proof of Hilbert's identity.

Connected semi-invariants to the unimodality of Gaussian coefficients.

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 $n,k \geq 8$ can be deduced from the ring property of semi-invariants.