

Colloquium of the Steklov Mathematical Institute of Russian Academy of Sciences
April 6, 2017 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)






The Horn conjecture and the Littlewood–Richardson rule
E. Yu. Smirnov^{ab} ^{a} Independent University of Moscow
^{b} National Research University "Higher School of Economics" (HSE), Moscow

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Abstract:
In 1912 Hermann Weyl asked the following question: what can be said
about the spectrum of the sum of two Hermitian matrices $A$ and $B$ with
given eigenvalues? In 1962 Alfred Horn produced a list of inequalities
on the spectra of $A$, $B$ and $A+B$, which he conjectured to be necessary and
sufficient. This conjecture was proven by A. Klyachko, A. Knutson and
T. Tao in 1999. Knutson and Tao proposed a nice combinatorial description
of Horn's inequalities by certain combinatorial diagrams, known as
honeycombs.
These diagrams are directly related to the Littlewood–Richardson
problem of decomposing the tensor product of two irreducible
representations of $GL(n)$ and to Schubert calculus on Grassmannians. They
provide a new symmetric formulation of the Littlewood–Richardson rule by
means of the socalled KnutsonTao puzzles: tilings of a equilateral
triangle by certain tiles. I will explain the relations among these
problems and, time permitting, discuss some other problems featuring
honeycombs and puzzles.

