## Appleby, Glenn and Whitehead, Tamsen - Honeycombs from Hermitian Matrix Pairs

dmtcs:2451 - Discrete Mathematics & Theoretical Computer Science, January 1, 2014, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Honeycombs from Hermitian Matrix Pairs

Authors: Appleby, Glenn and Whitehead, Tamsen

Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators.

Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Section: Proceedings
Published on: January 1, 2014
Submitted on: November 21, 2016
Keywords: Hermitian matrices,Honeycombs,crystal graphs,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]