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

This article is cited in 10 scientific papers (total in 10 papers)

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
\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.
\jour Proc. Steklov Inst. Math.
\yr 2003
\vol 241
\pages 58--78

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    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  mathnet  crossref  mathscinet  zmath  elib  elib
    2. Danilov V.I., Koshevoy G.A., “Discrete convexity and unimodularity. I”, Adv. Math., 189:2 (2004), 301–324  crossref  mathscinet  zmath  isi  scopus
    3. V. I. Danilov, G. A. Koshevoy, “Arrays and the combinatorics of Young tableaux”, Russian Math. Surveys, 60:2 (2005), 269–334  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Speyer D.E., “Horn's problem, Vinnikov curves, and the hive cone”, Duke Math. J., 127:3 (2005), 395–427  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Murota K., “Recent Developments in Discrete Convex Analysis”, Research Trends in Combinatorial Optimization, 2009, 219–260  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. S. Yu. Orevkov, Yu. P. Orevkov, “The Agnihotri–Woodward–Belkale Polytope and Klyachko Cones”, Math. Notes, 87:1 (2010), 96–101  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. Bercovici H., Li W.S., Timotin D., “A family of reductions for Schubert intersection problems”, J Algebraic Combin, 33:4 (2011), 609–649  crossref  mathscinet  zmath  isi  scopus
    9. Kumar Sh., “a Survey of the Additive Eigenvalue Problem”, Transform. Groups, 19:4 (2014), 1051–1148  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  isi  elib  scopus
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