Schubert calculus, adjoint representation and moment polytopes

Abstract

$/nu$ bir Young diyagram{/i} olsun ve $/mathcal{H}^{/nu}$ {/`o}zel uniter grubun $/nu$'ye kar/c{s}{/i}l{/i}k gelenindirgenemez temsilini g{/`o}stersin. Ayr{/i}ca, $/mathfrak{g} = /su(n)$ de $/SU(n)$'nin Lie cebrini g{/`o}stersin.G{/`osterilebilir ki} $/mathcal{H}^{/nu}$'nin $/mathfrak{g}$'nin simetrik cebri $/operatorname{S}^*/mathfrak{g}$'dag{/`o}r{/`u}nmesi i/c{c}in gerek ve yeter bir ko/c{s}ul $n$'nin $/nu$'n{/`u}n boyutunu b{/`o}lmesidir. Kostant'{/i}nproblemi, $/mathcal{H}^{/nu}$'in $/operatorname{S}^N/mathfrak{g}$'de g{/`o}r{/`u}ld{/`u}$/check{/operatorname{g}}${/`u} en k{/`u}/c{c}{/`u}k $N$ de$/check{/operatorname{g}}$erinin ne oldu$/check{/operatorname{g}}$unusorar. E/c{s}lenik temsilin moment politopu, $/mathcal{H}^{/nu}$'n{/`u}n $/operatorname{S}^*/mathfrak{g}$'deg{/`o}r{/`u}ld{/`u}$/check{/operatorname{g}}${/`u} $/nu$'lerin normalize edilmi/c{s} halleriyle gerilir. Momentpolitopu, Kostant'{/i}n probleminde bahsedilen $N$ say{/i}s{/i} i/c{c}in bir alt s{/i}n{/i}r koymaya yard{/i}mc{/i}olur. Bu tezde klasik spektral probleminin ve $/nu$-temsil edilebilirlik problemi ad{/i}yla bilinen bir di$/check{/operatorname{g}}$er spektral problemin/c{c}{/`o}z{/`u}mlerini kullanarak $n /leq 9$ i/c{c}in moment politoplar{/i}n{/i} hesapl{/i}yoruz.

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.