Quivers, long exact sequences and Horn type inequalities. (English) Zbl 1207.16011
Summary: We give necessary and sufficient inequalities for the existence of long exact sequences of \(m\) finite Abelian \(p\)-groups with fixed isomorphism types. This problem is related to some generalized Littlewood-Richardson coefficients that we define in this paper. We also show how this problem is related to eigenvalues of Hermitian matrices satisfying certain (in)equalities. When \(m=3\), we recover the Horn type inequalities that solve the saturation conjecture for Littlewood-Richardson coefficients and Horn’s conjecture.
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
Keywords:
Horn type inequalities; partitions; Littlewood-Richardson coefficients; long exact sequences; finite Abelian groups; Hermitian matrices; semi-invariants of quiversReferences:
| [1] | C. Chindris, Eigenvalues of Hermitian matrices and cones arising from quivers, Int. Math. Res. Not. (2006), in press, preprint available at http://www.math.umn.edu/ chindris/papers.html; C. Chindris, Eigenvalues of Hermitian matrices and cones arising from quivers, Int. Math. Res. Not. (2006), in press, preprint available at http://www.math.umn.edu/ chindris/papers.html · Zbl 1145.15008 |
| [2] | C. Chindris, The cone of effective weights for quivers and Horn type problems, PhD thesis, University of Michigan, 2005; C. Chindris, The cone of effective weights for quivers and Horn type problems, PhD thesis, University of Michigan, 2005 |
| [3] | Crawley-Boevey, W.; Geiss, Ch., Horn’s problem and semi-stability for quiver representations, (Representations of Algebra, vols. I, II (2002), Beijing Norm. Univ. Press: Beijing Norm. Univ. Press Beijing), 40-48 · Zbl 1086.16514 |
| [4] | Derksen, H.; Weyman, J., Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc., 13, 3, 467-479 (2000) · Zbl 0993.16011 |
| [5] | Derksen, H.; Weyman, J., The combinatorics of quiver representations (2006), preprint, arXiv: |
| [6] | Fulton, W., Young Tableaux. With Applications to Representation Theory and Geometry, London Math. Soc. Stud. Texts, vol. 35 (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0878.14034 |
| [7] | Fulton, W., Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.), 37, 3, 209-249 (2000) · Zbl 0994.15021 |
| [8] | Fulton, W., Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients, Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem. Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem, Linear Algebra Appl., 319, 1-3, 23-36 (2000) · Zbl 0968.15010 |
| [9] | Horn, A., Eigenvalues of sums of Hermitian matrices, Pacific J. Math., 12, 225-241 (1962) · Zbl 0112.01501 |
| [10] | James, G.; Lyle, S.; Mathas, A., Rouquier blocks, Math. Z., 252, 3, 511-531 (2006) · Zbl 1096.20007 |
| [11] | Kac, V. G., Infinite root systems, representations of graphs and invariant theory II, J. Algebra, 78, 1, 141-162 (1982) · Zbl 0497.17007 |
| [12] | Klein, T., The multiplication of Schur-functions and extensions of \(p\)-modules, J. London Math. Soc., 43, 280-284 (1968) · Zbl 0188.09504 |
| [13] | Klyachko, A., Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.), 4, 3, 419-445 (1998) · Zbl 0915.14010 |
| [14] | Knutson, A.; Tao, T., The honeycomb model of \(gl_n(C)\) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., 12, 4, 1055-1090 (1999) · Zbl 0944.05097 |
| [15] | Knutson, A.; Tao, T.; Woodward, Ch., The honeycomb model of \(gl_n(C)\) tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., 17, 1, 19-48 (2004) · Zbl 1043.05111 |
| [16] | Leclerc, B.; Miyachi, H., Some closed formulas for canonical bases of Fock spaces, Represent. Theory, 2002, 6, 290-312 (2002) · Zbl 1030.17014 |
| [17] | Schofield, A., Semi-invariants of quivers, J. London Math. Soc. (2), 43, 3, 385-395 (1991) · Zbl 0779.16005 |
| [18] | Schofield, A., General representations of quivers, Proc. London Math. Soc. (3), 65, 1, 46-64 (1992) · Zbl 0795.16008 |
| [19] | Schofield, A.; van den Bergh, M., Semi-invariants of quivers for arbitrary dimension vectors, Indag. Math. (N.S.), 12, 1, 125-138 (2001) · Zbl 1004.16012 |
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