Geodesic
Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients
Derksen, Harm1 ; Weyman, Jerzy2
1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02151
2 Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Journal of the American Mathematical Society, Tome 13 (2000) no. 3, pp. 467-479

Voir la notice de l'article provenant de la source American Mathematical Society

Let $Q$ be a quiver without oriented cycles. For a dimension vector $\beta$ let $\operatorname {Rep}(Q, \beta )$ be the set of representations of $Q$ with dimension vector $\beta$. The group $\operatorname {GL}(Q, \beta )$ acts on $\operatorname {Rep}(Q, \beta )$. In this paper we show that the ring of semi-invariants $\operatorname {SI} (Q,\beta )$ is spanned by special semi-invariants $c^V$ associated to representations $V$ of $Q$. From this we show that the set of weights appearing in $\operatorname {SI}(Q,\beta )$ is saturated. In the case of triple flag quiver this reduces to the results of Knutson and Tao on the saturation of the set of triples of partitions for which the Littlewood-Richardson coefficient is nonzero.
DOI : 10.1090/S0894-0347-00-00331-3

Derksen, Harm 1 ; Weyman, Jerzy 2

1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02151
2 Department of Mathematics, Northeastern University, Boston, Massachusetts 02115 
@article{10_1090_S0894_0347_00_00331_3,
     author = {Derksen, Harm and Weyman, Jerzy},
     title = {Semi-invariants of quivers and saturation for {Littlewood-Richardson} coefficients},
     journal = {Journal of the American Mathematical Society},
     pages = {467--479},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {2000},
     doi = {10.1090/S0894-0347-00-00331-3},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-00-00331-3/}
}
TY  - JOUR
AU  - Derksen, Harm
AU  - Weyman, Jerzy
TI  - Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients
JO  - Journal of the American Mathematical Society
PY  - 2000
SP  - 467
EP  - 479
VL  - 13
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-00-00331-3/
DO  - 10.1090/S0894-0347-00-00331-3
ID  - 10_1090_S0894_0347_00_00331_3
ER  - 
%0 Journal Article
%A Derksen, Harm
%A Weyman, Jerzy
%T Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients
%J Journal of the American Mathematical Society
%D 2000
%P 467-479
%V 13
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-00-00331-3/
%R 10.1090/S0894-0347-00-00331-3
%F 10_1090_S0894_0347_00_00331_3
Derksen, Harm; Weyman, Jerzy. Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients. Journal of the American Mathematical Society, Tome 13 (2000) no. 3, pp. 467-479. doi: 10.1090/S0894-0347-00-00331-3

[1] Akin, Kaan, Buchsbaum, David A., Weyman, Jerzy Schur functors and Schur complexes Adv. in Math. 1982 207 278

[2] De Concini, C., Procesi, C. A characteristic free approach to invariant theory Advances in Math. 1976 330 354

[3] Séminaire Bourbaki. Vol. 1997/98 1998

[4] Fulton, William, Harris, Joe Representation theory 1991

[5] Kac, V. G. Infinite root systems, representations of graphs and invariant theory. II J. Algebra 1982 141 162

[6] King, A. D. Moduli of representations of finite-dimensional algebras Quart. J. Math. Oxford Ser. (2) 1994 515 530

[7] Knutson, Allen, Tao, Terence The honeycomb model of 𝐺𝐿_{𝑛}(𝐶) tensor products. I. Proof of the saturation conjecture J. Amer. Math. Soc. 1999 1055 1090

[8] Ringel, Claus Michael Representations of 𝐾-species and bimodules J. Algebra 1976 269 302

[9] Schofield, Aidan Semi-invariants of quivers J. London Math. Soc. (2) 1991 385 395

[10] Schofield, Aidan General representations of quivers Proc. London Math. Soc. (3) 1992 46 64

Cité par Sources :