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 Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2015, Issue 10(132), Pages 9–23 (Mi vsgu479)

Mathematics

$U$-projection for the adjoint representation of the group $\mathrm{GL}(n,K)$

K. A. Vyatkina

Samara State University, 1, Acad. Pavlov Street, Samara, 443011, Russian Federation

Abstract: In the paper we study rings and fields of invariants for the adjoint representation of the group $\mathrm{GL}(n,K)$ over the field of zero characteristic. The aim of this paper is to construct the special linear operator, we call it $U$-projector, that maps any polynomial on the matrix algebra to an $U$-invariant rational function. In this paper we present two different constructions of $U$-projector. Using the $U$-projector we obtain the system of generators of the field of $U$-invariants of the adjoint representation of the group $\mathrm{GL}(n,K)$. We obtain the system of generators of the field of $U$-invariants for the restriction of the adjoint representation to the subgroup of block-diagonal matrices.

Keywords: field of invariants, adjoint representation, unitriangular group, solvable group, algebra of invariants, representations of groups, locally nilpotent derivation, system of generators.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-97017-ð_ïîâîëæüå_à14-01-31052-ìîë_à The work of the author is supported by the grants of the Russian Foundations for Basic Research 14-01-97017 and 14-01-31052.

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Citation: K. A. Vyatkina, “$U$-projection for the adjoint representation of the group $\mathrm{GL}(n,K)$”, Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya, 2015, no. 10(132), 9–23

Citation in format AMSBIB
\Bibitem{Vya15} \by K.~A.~Vyatkina \paper $U$-projection for the adjoint representation of the group~$\mathrm{GL}(n,K)$ \jour Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvenno-Nauchnaya Seriya \yr 2015 \issue 10(132) \pages 9--23 \mathnet{http://mi.mathnet.ru/vsgu479} \elib{https://elibrary.ru/item.asp?id=25377150}