The Horn cone associated with symplectic eigenvalues

Paradan, Paul-Emile

Comptes Rendus. Mathématique, Tome 360 (2022) no. G10, pp. 1163-1168.

In this note, we show that the Horn cone associated with symplectic eigenvalues admits the same inequalities as the classical Horn cone, except that the equality corresponding to $\mathrm{Tr}\left(C\right)=\mathrm{Tr}\left(A\right)+\mathrm{Tr}\left(B\right)$ is replaced by the inequality corresponding to $\mathrm{Tr}\left(C\right)\ge \mathrm{Tr}\left(A\right)+\mathrm{Tr}\left(B\right)$.

Classification : 00X99

[1] Belkale, Prakash Local systems on $\mathbb{P}-S$ for $S$ a finite set, Compos. Math., Volume 129 (2001) no. 1, pp. 67-86 | DOI | MR | Zbl

[2] Bhatia, Rajendra; Jain, Tanvi Variational principles for symplectic eigenvalues, Can. Math. Bull., Volume 64 (2021) no. 3, pp. 553-559 | DOI | MR | Zbl

[3] Brion, Michel Restrictions de representations et projections d’orbites coadjointes (d’après Belkale, Kumar et Ressayre), 2011 (Séminaire Bourbaki, http://www.bourbaki.ens.fr/TEXTES/1043.pdf) | Zbl

[4] Friedland, Shmuel Finite and infinite dimensional generalizations of Klyachko’s theorem, Linear Algebra Appl., Volume 319 (2000) no. 1-3, pp. 3-22 | DOI | MR | Zbl

[5] Fulton, William Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Am. Math. Soc., Volume 37 (2000) no. 3, pp. 209-249 | DOI | MR | Zbl

[6] Fulton, William Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients, Linear Algebra Appl., Volume 319 (2000) no. 1-3, pp. 23-36 | DOI | MR | Zbl

[7] Hiroshima, Tohya Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs, Phys. Rev. A, Volume 73 (2006) no. 1, 012330, 9 pages | DOI | Zbl

[8] Jain, Tanvi; Mishra, Hemant K. Derivatives of symplectic eigenvalues and a Lidskii type theorem, Can. J. Math., Volume 74 (2020) no. 2, pp. 457-485 | DOI | MR | Zbl

[9] Kirwan, Frances Convexity properties of the moment mapping III, Invent. Math., Volume 77 (1984), pp. 547-552 | DOI | MR | Zbl

[10] Klyachko, Alexander A. Stable bundles, representation theory and Hermitian operators, Sel. Math., New Ser., Volume 4 (1998) no. 3, pp. 419-445 | DOI | MR | Zbl

[11] Knutson, Allen; Tao, Terence; Woodward, Christopher The honeycomb model of ${GL}_n\left(ℂ\right)$ tensor products II: Puzzles determine facets of the Littlewood-Richardson cone, J. Am. Math. Soc., Volume 17 (2004) no. 1, pp. 19-48 | DOI | MR | Zbl

[12] Lerman, Eugene; Meinrenken, Eckhard; Tolman, Sue; Woodward, Christopher Non-Abelian convexity by symplectic cuts, Topology, Volume 37 (1998) no. 2, pp. 245-259 | DOI | Zbl

[13] Paneitz, Stephen M. Invariant convex cones and causality in semisimple Lie algebras and groups, J. Funct. Anal, Volume 43 (1981), pp. 313-359 | DOI | MR | Zbl

[14] Paneitz, Stephen M. Determination of invariant convex cones in simple Lie algebras, Ark. Mat., Volume 21 (1983), pp. 217-228 | DOI | MR | Zbl

[15] Paradan, Paul-Emile Horn problem for quasi-hermitian Lie groups, J. Inst. Math. Jussieu (2022), pp. 1-27 | DOI

[16] Vinberg, Èrnest B. Invariant convex cones and orderings in Lie groups, Funct. Anal. Appl., Volume 14 (1980), pp. 1-10 | DOI | MR | Zbl

[17] Weinstein, Alan Poisson geometry of discrete series orbits and momentum convexity for noncompact group actions, Lett. Math. Phys., Volume 56 (2001) no. 1, pp. 17-30 | DOI | MR | Zbl

[18] Williamson, John On the algebraic problem concerning the normal forms of linear dynamical systems, Am. J. Math., Volume 58 (1936), pp. 141-163 | DOI | MR | Zbl