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 Tr. Mat. Inst. Steklova, 2003, Volume 241, Pages 68–89 (Mi tm388)

Discrete Convexity and Hermitian Matrices

V. I. Danilov, G. A. Koshevoy

Central Economics and Mathematics Institute, RAS

Abstract: The question (Horn problem) about the spectrum of the sum of two real symmetric (or complex Hermitian) matrices with given spectra is considered. This problem was solved by A. Klyachko. We suggest a different formulation of the solution to the Horn problem with a significantly more elementary proof. Our solution is that the existence of the required triple of matrices $(A,B,C)$ for given spectra $(\alpha,\beta,\gamma)$ is equivalent to the existence of a so-called discrete concave function on the triangular grid $\Delta(n)$ with boundary increments $\alpha$,$\beta$, and $\gamma$. In addition, we propose a hypothetical explanation for the relation between Hermitian matrices and discrete concave functions. Namely, for a pair $(A,B)$ of Hermitian matrices, we construct a certain function $\phi (A,B;\cdot)$ on the grid $\Delta(n)$. Our conjecture is that this function is discrete concave, which is confirmed in several special cases.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 241, 58–78

Bibliographic databases:

UDC: 512.643
Received in November 2002

Citation: V. I. Danilov, G. A. Koshevoy, “Discrete Convexity and Hermitian Matrices”, Number theory, algebra, and algebraic geometry, Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich, Tr. Mat. Inst. Steklova, 241, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 68–89; Proc. Steklov Inst. Math., 241 (2003), 58–78

Citation in format AMSBIB
\Bibitem{DanKos03} \by V.~I.~Danilov, G.~A.~Koshevoy \paper Discrete Convexity and Hermitian Matrices \inbook Number theory, algebra, and algebraic geometry \bookinfo Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich \serial Tr. Mat. Inst. Steklova \yr 2003 \vol 241 \pages 68--89 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm388} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2024044} \zmath{https://zbmath.org/?q=an:1071.15019} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 241 \pages 58--78 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Danilov, G. A. Koshevoy, “Discrete convexity”, J. Math. Sci. (N. Y.), 133:4 (2006), 1418–1421
2. Danilov V.I., Koshevoy G.A., “Discrete convexity and unimodularity. I”, Adv. Math., 189:2 (2004), 301–324
3. V. I. Danilov, G. A. Koshevoy, “Arrays and the combinatorics of Young tableaux”, Russian Math. Surveys, 60:2 (2005), 269–334
4. Speyer D.E., “Horn's problem, Vinnikov curves, and the hive cone”, Duke Math. J., 127:3 (2005), 395–427
5. V. I. Danilov, G. A. Koshevoy, “The Robinson–Schensted–Knuth correspondence and the bijections of commutativity and associativity”, Izv. Math., 72:4 (2008), 689–716
6. Murota K., “Recent Developments in Discrete Convex Analysis”, Research Trends in Combinatorial Optimization, 2009, 219–260
7. S. Yu. Orevkov, Yu. P. Orevkov, “The Agnihotri–Woodward–Belkale Polytope and Klyachko Cones”, Math. Notes, 87:1 (2010), 96–101
8. Bercovici H., Li W.S., Timotin D., “A family of reductions for Schubert intersection problems”, J Algebraic Combin, 33:4 (2011), 609–649
9. Kumar Sh., “a Survey of the Additive Eigenvalue Problem”, Transform. Groups, 19:4 (2014), 1051–1148
10. Fujishige S., Goemans M., Harks T., Peis B., Zenklusen R., “Congestion Games Viewed From M-Convexity”, Oper. Res. Lett., 43:3 (2015), 329–333
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