Semi-invariants Of Binary Forms And Sylvester's Theorem

Sylvester established the connection between the Gaussian coefficients and semi-invariants of binary forms,and proved the unimodality of the Gaussian coefficients as conjectured by Cayley.Pak and Panova proved the strict unimodality of the Gaussian coefficients using the semigroup property of the Kronecker coefficients in the repre-sentation theory of the symmetric group.The symmetric difference of the Gaussian coefficients(?)was introduced by Reiner and Stanton,and they proved that Fn,k(q)is symmetric and unimodal for k?2 and n even by using the representation theory for Lie algebras.In this thesis,we bring Sylvester's proof of the unimodality of the Gaussian co-efficients to a combinatorial ground by introducing the notion of a semi-diagram of a Young diagram,and furthermore use semi-invariants to solve other problems related to the Gaussian coefficients in combinatorics.This thesis is organized as follows.In Chapter 1,we give an overview of the research background on the classical invariant theory,as well as Sylvester's theorem and related results on the Gaussian coefficients.At the end of this chapter,we outline the main results.In Chapter 2,we provide relevant notations,definitions and properties that will be used frequently,including integer partitions,invariants,semi-invariants and covariants of binary forms.In Chapter 3,based on the notion of a semi-diagram of a Young diagram,we ob-tain a combinatorial formula related to the semi-invariants of binary forms,which im-plies the characterization of semi-invariants in terms of a differential operator.Then we present a combinatorial proof of an identity of Hilbert,leading to a relation of Cayley on semi-invariants.This identity plays a crucial role in the original proof of Sylvester's theorem.In Chapter 4,we find an interpretation of the unimodality of the symmetric dif-ference Fn,k(q)in terms of semi-invariants.Furthermore,we introduce the symmetric difference(?),and established its strict unimodality,ex-cept for the two terms at both ends,when n,r?8,k?r and at least one of n and r is even.Moreover,we show that the additivity lemma of Pak and Panova can be recognized as the ring property of semi-invariants of binary forms.Finally,through a construction of semi-invariants,we obtain an improvement of a lower bound of Pak and Panova for a range of coefficients near the middle.