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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.
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
Linking options:
https://www.mathnet.ru/eng/mzm8551https://doi.org/10.4213/mzm8551 https://www.mathnet.ru/eng/mzm/v87/i1/p101
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Abstract page: | 538 | Full-text PDF : | 194 | References: | 84 | First page: | 17 |
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