The combinatorics of quiver representations
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 1061-1131

We give a description of faces, of all codimensions, for the cones spanned by the set of weights associated to the rings of semi-invariants of quivers. For a triple flag quiver and its faces of codimension 1 this description reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically review and develop the necessary methods (exceptional and Schur sequences, orthogonal categories, semi-stable decompositions, GIT quotients for quivers). In an Appendix we include a variant of Belkale’s geometric proof of a conjecture of Fulton that works for arbitrary quivers.

On donne une description des faces, des toutes codimensions, pour les cônes engendrés par l’ensemble des poids associés aux anneaux des semi-invariants des carquois. Pour un carquois de drapeaux triples et ses faces de codimension 1, la description est équivalente à un résultat de Knutson-Tao-Woodward sur les facettes du cône de Klyachko. On donne des nouvelles applications aux coefficients de Littlewood-Richardson, en particulier une formule pour les coefficients qui correspond à des triples de partitions sur un mur du cône de Klyachko. On commence par rappeler les méthodes utilisées (suites de Schur, les suites exceptionnelles, les catégories orthogonaux, les décompositions semi-stables, et les quotients GIT pour les carquois). Dans une appendice, on donne une variante d’une démonstration géométrique de Belkale d’une conjecture de Fulton qui est valable pour un carquois quelconque.

DOI : https://doi.org/10.5802/aif.2636
Classification:  16G20,  05E10,  13A50
Keywords: Quiver representations, Klyachko cone, Littlewood-Richardson coefficients
@article{AIF_2011__61_3_1061_0,
     author = {Derksen, Harm and Weyman, Jerzy},
     title = {The combinatorics of quiver representations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {1061-1131},
     doi = {10.5802/aif.2636},
     mrnumber = {2918725},
     zbl = {1271.16016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_3_1061_0}
}
Derksen, Harm; Weyman, Jerzy. The combinatorics of quiver representations. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 1061-1131. doi : 10.5802/aif.2636. http://www.numdam.org/item/AIF_2011__61_3_1061_0/

[1] Belkale, P. Geometric proofs of Horn and saturation conjectures, J. Algebraic Geom., Tome 15 (2006) no. 1, pp. 133-176 | Article | MR 2177198 | Zbl 1090.14014

[2] Belkale, P. Geometric proof of a conjecture of Fulton, Advances Math., Tome 216 (2007) no. 1, pp. 346-357 | Article | MR 2353260 | Zbl 1129.14063

[3] Buch, A. S. The saturation conjecture (after A. Knutson and T. Tao), With an appendix by William Fulton, Enseign. Math. (2), Tome 46 (2000) no. 1-2, pp. 43-60 | MR 1769536 | Zbl 0979.20041

[4] Chindris, C. Quivers, long exact sequences and Horn type inequalities, J. Algebra, Tome 320 (2008) no. 1, pp. 128-157 | Article | MR 2417982 | Zbl 1207.16011

[5] Chindris, C. Quivers, long exact sequences and Horn type inequalities II, Glasg. Math. J., Tome 51 (2009) no. 2, pp. 201-217 | Article | MR 2500745 | Zbl 1210.16016

[6] Chindris, C.; Derksen, H.; Weyman, J. Non-log-concave Littlewood-Richardson coefficients, Compos. Math., Tome 43 (2007) no. 6, pp. 1545-1557 | MR 2371381 | Zbl 1184.05136

[7] Crawley-Boevey, W. Exceptional sequences of representations of quivers, Canadian Math. Soc. Conf. Proceedings, Tome 14 (1993), pp. 117-124 | MR 1265279 | Zbl 0828.16012

[8] Crawley-Boevey, W. Subrepresentations of general representations of quivers, Bull. London Math. Soc., Tome 28 (1996) no. 4, pp. 363-366 | Article | MR 1384823 | Zbl 0863.16008

[9] Crawley-Boevey, W. On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J., Tome 118 (2003) no. 2, pp. 339-352 | Article | MR 1980997 | Zbl 1046.15013

[10] Derksen, H.; Schofield, A.; Weyman, J. On the number of subrepresentations of a general quiver representation, J. London Math. Soc. (2), Tome 76 (2007) no. 1, pp. 135-147 | Article | MR 2351613 | Zbl 1146.16007

[11] Derksen, H.; Weyman, J. Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, Journal of the AMS, Tome 13 (2000), pp. 467-579 | MR 1758750 | Zbl 0993.16011

[12] Derksen, H.; Weyman, J. On the canonical decomposition of quiver representations, Compositio Math., Tome 133 (2002), pp. 245-265 | Article | MR 1930979 | Zbl 1016.16007

[13] Derksen, H.; Weyman, J. On the Littlewood-Richardson polynomials, J. Algebra, Tome 255 (2002) no. 2, pp. 247-257 | Article | MR 1935497 | Zbl 1018.16012

[14] Derksen, H.; Weyman, J. The combinatorics of quiver representations (arXiv:math/0608288) | MR 2110070

[15] Fulton, W. Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc., Tome 37 (2000) no. 3, pp. 209-249 | Article | MR 1754641 | Zbl 0994.15021

[16] Hille, L. Quivers, cones and polytopes, Linear Algebra Appl., Tome 365 (2003), pp. 215-237 (special issue on linear algebra methods in representation theory) | Article | MR 1987339 | Zbl 1034.52011

[17] Hille, L.; De La Peña, José Stable representations of quivers, J. Pure Appl. Algebra, Tome 172 (2002) no. 2-3, pp. 205-224 | Article | MR 1906875 | Zbl 1040.16011

[18] Horn, A Eigenvalues of sums of Hermitian matrices, Pacific J. Math., Tome 12 (1962), pp. 620-630 | MR 140521 | Zbl 0112.01501

[19] Kac, V Infinite root systems, representations of graphs and Invariant Theory, Invent. Math., Tome 56 (1980), pp. 57-92 | Article | MR 557581 | Zbl 0427.17001

[20] Kac, V Infinite Root Systems, Representations of Graphs and Invariant Theory II, J. Algebra, Tome 78 (1982), pp. 141-162 | Article | MR 677715 | Zbl 0497.17007

[21] King, A. D. Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford (2), Tome 45 (1994), pp. 515-530 | Article | MR 1315461 | Zbl 0837.16005

[22] King, R. C.; Tollu, C.; Toumazet, F. The hive model and the factorisation of Kostka coefficients, Sém. Lothar. Combin., Tome 54A (2005/07), pp. Art. B54Ah, 22 pp. (electronic) | MR 2264935 | Zbl 1178.05101

[23] King, R. C.; Tollu, Ch; Toumazet, F. Factorisation of Littlewood-Richardson coefficients, J. Combin. Theory Ser. A, Tome 116 (2009) no. 2, pp. 314-333 | Article | MR 2475020 | Zbl 1207.05214

[24] Klein, T The multiplication of Schur-functions and extensions of p-modules, J. London Math. Soc., Tome 43 (1968), pp. 280-284 | Article | MR 228481 | Zbl 0188.09504

[25] Klyachko, A. Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.), Tome 4 (1988) no. 3, pp. 419-445 | MR 1654578 | Zbl 0915.14010

[26] Knutson, A.; Tao, T. The honeycomb model of GL n () tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., Tome 12 (1999) no. 4, pp. 1055-1090 | Article | MR 1671451 | Zbl 0944.05097

[27] Knutson, A.; Tao, T.; Woodward, W. The honeycomb model of GL n () tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., Tome 17 (2004) no. 1, pp. 19-48 | Article | MR 2015329 | Zbl 1043.05111

[28] Le Bruyn, L.; Procesi, C. Semisimple representations of quivers, Trans. Amer. Math. Soc., Tome 317 (1990) no. 2, pp. 585-598 | Article | MR 958897 | Zbl 0693.16018

[29] Ressayre, N. Geometric invariant theory and generalized eigenvalue problem II (arXiv:0903.1187) | Zbl 1197.14051

[30] Ressayre, N. GIT cones for quivers (arXiv: 0903.1202)

[31] Ressayre, N. Geometric invariant theory and the generalized eigenvalue problem, Invent. Math., Tome 180 (2010) no. 2, pp. 389-441 | Article | MR 2609246 | Zbl 1197.14051

[32] Ringel, C. M. Representations of K-species and bimodules, J. Algebra, Tome 41 (1976), pp. 269-302 | Article | MR 422350 | Zbl 0338.16011

[33] Ringel, C. M. Tame algebras and integral quadratic forms, Springer, Lecture Notes in Math., Tome 1099 (1984) | MR 774589 | Zbl 0448.16019

[34] Ringel, Claus Michael The braid group action on the set of exceptional sequences of a hereditary Artin algebra, Abelian group theory and related topics (Oberwolfach, 1993), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 171 (1994), pp. 339-352 | MR 1293154 | Zbl 0851.16010

[35] Rudakov, A. Stability for an abelian category, J. Algebra, Tome 197 (1997), pp. 231-245 | Article | MR 1480783 | Zbl 0893.18007

[36] Schofield, A. Semi-invariants of Quivers, J. London Math. Soc., Tome 43 (1991), pp. 383-395 | Article | MR 1113382 | Zbl 0779.16005

[37] Schofield, A. General Representations of Quivers, Proc. London Math. Soc. (3), Tome 65 (1992), pp. 46-64 | Article | MR 1162487 | Zbl 0795.16008

[38] Schofield, A. Birational classification of moduli spaces of representations of quivers, Indag. Math., N.S., Tome 12 (3) (2001), pp. 407-432 | Article | MR 1914089 | Zbl 1013.16005

[39] Schofield, A.; Van Den Bergh, A. Semi-invariants of quivers for arbitrary dimension vectors, Indag. Math., N.S., Tome 12 (1) (2001), pp. 125-138 | Article | MR 1908144 | Zbl 1004.16012

[40] Vakil, R. Schubert Induction, Annals of Math. (2), Tome 164 (2006) no. 2, pp. 489-512 | Article | MR 2247966 | Zbl 1115.14043

[41] Weyl, H. Das asymtotische Verteilungsgesetz der Eigenwerte lineare partieller Differentialgleichungen, Math. Ann., Tome 71 (1912), pp. 441-479 | Article | MR 1511670