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Honeycombs from Hermitian matrix pairs, with interpretations of path operators and \(SL_n\) crystals. (English. French summary) Zbl 1393.15046

Proceedings of the 26th international conference on formal power series and algebraic combinatorics, FPSAC 2014, Chicago, IL, USA, June 29 – July 3, 2014. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 899-910 (2014).
Summary: A. Knutson and T. Tao’s work on the Horn conjectures [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] 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 \(n \times n\) 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 \(c_{\mu \nu}^\lambda \) 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 more simply from the perspective of rotations than that of path operators.
For the entire collection see [Zbl 1298.05004].

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory

Citations:

Zbl 0944.05097
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