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Mat. Zametki, 2010, Volume 87, Issue 1, Pages 101–107 (Mi mz8551)  

The Agnihotri–Woodward–Belkale Polytope and Klyachko Cones

S. Yu. Orevkovab, Yu. P. Orevkovc

a Steklov Mathematical Institute, Russian Academy of Sciences
b Laboratoire Emile Picard, Université Paul Sabatier
c M. V. Lomonosov Moscow State University

Abstract: The Agnihotri–Woodward–Belkale polytope $\Delta$ (resp., the Klyachko cone $\mathscr K$) is the set of solutions of the multiplicative (resp., additive) Horn problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) $n\times n$ matrices satisfying $AB=C$ (resp. $A+B=C$). The set $\mathscr K$ is the tangent cone of $\Delta$ at the origin. The group $G=\mathbb Z_n\oplus\mathbb Z_n$ acts naturally on $\Delta$. In this note, we report on a computer calculation showing that $\Delta$ coincides with the intersection of $g\mathscr K$, $g\in G$, for $n\le 14$ but does not coincide with it for $n=15$. Our motivation was an attempt to understand how to solve the multiplicative Horn problem in practice for given conjugacy classes in $SU(n)$.

Keywords: unitary matrix, Weyl chamber, Horn problem, conjugacy class, Schubert calculus, Gromov–Witten invariants, Littlewood–Richardson coefficients, Klyachko cone

DOI: https://doi.org/10.4213/mzm8551

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English version:
Mathematical Notes, 2010, 87:1, 96–101

Bibliographic databases:

UDC: 514.748
Received: 13.05.2008

Citation: S. Yu. Orevkov, Yu. P. Orevkov, “The Agnihotri–Woodward–Belkale Polytope and Klyachko Cones”, Mat. Zametki, 87:1 (2010), 101–107; Math. Notes, 87:1 (2010), 96–101

Citation in format AMSBIB
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\paper The Agnihotri--Woodward--Belkale Polytope and Klyachko Cones
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