Degeneracy graph determinants and higher-dimensional Young diagrams
Abstract
Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators C1,…, CL, constructed from a degeneracy graph which records how the eigenspaces of C1 are successively split by C2,…, CL. They showed that the basis property is equivalent to the non-vanishing of the determinant of the matrix of values of these monomials on the projector basis, and conjectured a factorised formula for this determinant, as a product of powers of eigenvalue differences. We prove this formula for any number of generators. The proof uses graph analogues of Newton polynomials. Their evaluation matrix is not triangular in general, but expanding the rows of the children of each node of the penultimate layer in a Newton basis of their eigenvalues makes it block triangular, with diagonal blocks given by smaller degeneracy graphs in which the final layer is removed. The formula then follows by induction on the number of layers. We show that the exponents in the determinant are overlap volumes of higher-dimensional Young diagrams attached to the nodes of the graph: intervals, ordinary Young diagrams, plane partitions and their higher analogues. Finally, the block triangular structure gives an explicit recursive inversion of the evaluation matrix, expressing the projectors in the monomial basis.
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BibTeX
@misc{ramgoolam2026degeneracy,
title = {{Degeneracy graph determinants and higher-dimensional Young diagrams}},
author = {Sanjaye Ramgoolam and Michał Studziński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20368},
eprint = {2610.20368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20368},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Degeneracy graph determinants and higher-dimensional Young diagrams
AU - Sanjaye Ramgoolam
AU - Michał Studziński
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20368
UR - https://arxiv.org/abs/2610.20368
AB - Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators C_1,…, C_L, constructed from a degeneracy graph which records how the eigenspaces of C_1 are successively split by C_2,…, C_L. They showed that the basis property is equivalent to the non-vanishing of the determinant of the matrix of values of these monomials on the projector basis, and conjectured a factorised formula for this determinant, as a product of powers of eigenvalue differences. We prove this formula for any number of generators. The proof uses graph analogues of Newton polynomials. Their evaluation matrix is not triangular in general, but expanding the rows of the children of each node of the penultimate layer in a Newton basis of their eigenvalues makes it block triangular, with diagonal blocks given by smaller degeneracy graphs in which the final layer is removed. The formula then follows by induction on the number of layers. We show that the exponents in the determinant are overlap volumes of higher-dimensional Young diagrams attached to the nodes of the graph: intervals, ordinary Young diagrams, plane partitions and their higher analogues. Finally, the block triangular structure gives an explicit recursive inversion of the evaluation matrix, expressing the projectors in the monomial basis.
ER -