Schubert kalkülüsü, eşlenik temsil ve moment politoplari

Let $\mathcal{H}^{\nu}$ denote the irreducible representation of the special unitary group$\SU(n)$ corresponding to Young diagram $\nu$ and let $\mathfrak{g} = \su(n)$ denote the Lie algebra of $\SU(n)$.One can show that $\mathcal{H}^{\nu}$ appears in the symmetric algebra $\operatorname{S}^*\mathfrak{g}$ if and onlyif $n$ divides the size of the Young diagram $\nu$. Kostant's problem asks what is the least number $N$ such that$\mathcal{H}^{\nu}$ appear in $\operatorname{S}^N\mathfrak{g}$. The \textit{moment polytope} of the adjointrepresentation is the polytope generated by the normalized weights $\tilde{\nu}$ such that $\mathcal{H}^{\nu}$appears in $\operatorname{S}^*\mathfrak{g}$ and it helps to put lower bounds on number $N$ in the Kostant's problem.In this thesis, we compute the moment polytope of the adjoint representation of $\SU(n)$ for $n \leq 9$ using the solutions of the classical spectral problem and so-called $\nu$-representability problem.