An Excursion into Grassmannian Spectra
Abstract
We revisit the classical spectral problem for the Laplace-Beltrami and Laplace-Bochner operators on complex Grassmannians. After reviewing the standard approach based on non-commutative harmonic analysis, we develop two complementary methods originating from the quantization of the corresponding one-dimensional sigma model, both with and without a magnetic monopole field. The first is based on SU(n)-equivariant first-order GLSM formulation and gives explicit eigenfunctions in terms of projectors and Plücker coordinates. The second uses a spin-chain realization of free particles on manifolds and describes the space of square-integrable functions, or sections in the magnetic case, through limiting tensor products of long rectangular SU(n) representations. Together, these approaches provide a more explicit picture of the spectral geometry of Grassmannians and clarify the relation between harmonics, Young diagrams, magnetic bundles, and Laplacian spectra.
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BibTeX
@misc{bykov2026excursion,
title = {{An Excursion into Grassmannian Spectra}},
author = {Dmitri Bykov and Viacheslav Krivorol},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04613},
eprint = {2611.04613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Excursion into Grassmannian Spectra
AU - Dmitri Bykov
AU - Viacheslav Krivorol
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04613
UR - https://arxiv.org/abs/2611.04613
AB - We revisit the classical spectral problem for the Laplace-Beltrami and Laplace-Bochner operators on complex Grassmannians. After reviewing the standard approach based on non-commutative harmonic analysis, we develop two complementary methods originating from the quantization of the corresponding one-dimensional sigma model, both with and without a magnetic monopole field. The first is based on SU(n)-equivariant first-order GLSM formulation and gives explicit eigenfunctions in terms of projectors and Plücker coordinates. The second uses a spin-chain realization of free particles on manifolds and describes the space of square-integrable functions, or sections in the magnetic case, through limiting tensor products of long rectangular SU(n) representations. Together, these approaches provide a more explicit picture of the spectral geometry of Grassmannians and clarify the relation between harmonics, Young diagrams, magnetic bundles, and Laplacian spectra.
ER -