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arXiv:2610.20368

Degeneracy graph determinants and higher-dimensional Young diagrams

Sanjaye Ramgoolam, Michał Studziński

hep-thmath-phmath.COmath.RTquant-phSubmitted Oct 14, 2026• Announced Oct 15, 2026
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Abstract

Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators \(C_1,\ldots, C_L\), constructed from a degeneracy graph which records how the eigenspaces of \( C_1\) are successively split by \( C_2,\ldots, C_L\). 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.

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53 pages,

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Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators (C_1,\ldots, C_L), constructed from a degeneracy graph which records how the eigenspaces of ( C_1) are successively split by ( C_2,\ldots, 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.
Questions at the intersection of the AdS/CFT correspondence and quantum information theory motivate the study of projectors in sequences of subalgebras of finite-dimensional commutative associative semisimple algebras A\mathcal{A}A, obtained by incrementally adjoining one generator at each step to produce a non-linear generating set for A\mathcal{A}A. We define degeneracy graphs, which are finite layered tree graphs whose nodes represent projectors in the successive subalgebras. Using combinatorial properties of the degeneracy graph, we give a simple formula for constructing a linear basis of A\mathcal{A}A in terms of monomials in the generators. The nodes can be labelled by formal variables corresponding to the eigenvalues of the generators added at each layer. We prove that the construction is compatible with the required counting of projectors in A\mathcal{A}A, and give explicit constructions of the projectors in terms of the monomials, in the cases of one- and two-layer degeneracy graphs with arbitrary numbers of nodes. More generally, we provide extensive computational evidence for the invertibility of the matrix relating the proposed monomial basis to the projector basis, by evaluating its determinant. In the 1-layer case, this is a Vandermonde determinant. A simple formula for the non-vanishing determinant in the general layer case is conjectured and supported by the computational data. The construction is illustrated with examples including centres of symmetric group algebras and maximally commuting subalgebras generated by JucysMurphy elements. We outline applications of the monomial basis to algorithms for constructing matrix units in non-commutative semisimple algebras, with relevance to orthogonal bases of multi-matrix gauge-invariant operators and to quantum information theory.
In contrast to the determinant, no algorithm is known for the exact determination of the permanent of a square matrix that runs in time polynomial in its dimension. Consequently, noninteracting fermions are classically efficiently simulatable while noninteracting bosons are not, underpinning quantum supremacy arguments for sampling the output distribution of photon interferometer arrays. This work introduces a graph-theoretic framework that bridges both the determinant and permanent. The only nonzero eigenvalues of a sparse non-Hermitian operator $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$ for nnn spin-1/21/21/2 particles are the nthn\mathrm{th}nth roots of the permanent or determinant of an $n\ifmmode\times\else\texttimes\fi{}n$ matrix MMM, interpreting basis states as bosonic or fermionic occupation states, respectively. This operator can be used to design a simple and straightforward method for the classical determination of the permanent that matches the efficiency of the best-known algorithm. Gauss-Jordan elimination for the determinant of MMM is then equivalent to the successive removal of the generalized zero eigenspace of the fermionic $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$, equivalent to the deletion of some nodes and reweighting of the remaining edges in the graph such that only nnn nodes survive after the last step. In the bosonic case, the successive removal of generalized zero eigenspaces for $\stackrel{\ifmmode \u\else \textasciibreve \fi{}}{M}$ is also equivalent to node deletion, but new edges are added during this process, which gives rise to the higher complexity of computing the permanent. Our analysis may point the way to new strategies for classical and quantum evaluation of the permanent.
The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s = 1/2 Heisenberg chain at roots of unity are investigated. Two conjectures are formulated both of which are proved for N = 3 and are verified numerically for several examples with N > 3. The first conjecture identifies an Abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeros of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N = 3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra . This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.
We construct (2n)2×(2n)2 unitary braid matrices R̂ for n⩾2 generalizing the class known for n=1. A set of (2n)×(2n) matrices (I,J,K,L) is defined. R̂ is expressed in terms of their tensor products (such as K⊗J), leading to a canonical formulation for all n. Complex projectors P± provide a basis for our real, unitary R̂. Baxterization is obtained. Diagonalizations and block diagonalizations are presented. The loss of braid property when R̂ (n>1) is block diagonalized in terms of R̂ (n=1) is pointed out and explained. For odd dimension (2n+1)2×(2n+1)2, a previously constructed braid matrix is complexified to obtain unitarity. R̂LL and R̂TT algebras, chain Hamiltonians, potentials for factorizable S matrices, and complex noncommutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning entanglements.
Abstract We consider the problem of construction of determinant formulas for the partition function of the six-vertex model with domain wall boundary conditions that depend on two sets of spectral parameters. In the pioneering works of Korepin and Izergin a determinant formula was proposed and proved using a recursion relation. In later works, equivalent determinant formulas were given by Kostov for the rational case and by Foda and Wheeler for the trigonometric case. Here, we develop an approach in which the recursion relation is replaced by a system of algebraic equations with respect to one of the two sets of spectral parameters. We prove that this system has a unique solution. The result can be easily given as a determinant parametrized by an arbitrary basis of polynomials. In particular, the choice of the basis of Lagrange polynomials with respect to the remaining set of spectral parameters leads to the Izergin–Korepin representation, and the choice of the monomial basis leads to the Kostov and Foda–Wheeler representations.
Abstract Monodromy matrices of the model are known to satisfy a Yang–Baxter equation with a six-vertex R -matrix as the intertwiner. The commutation relations of the elements of the monodromy matrices are completely determined by this R -matrix. We show the reason why in the superintegrable case the eigenspace is degenerate, but not in the general case. We then show that the eigenspaces of special CSOS models descending from the chiral Potts model are also degenerate. The existence of an quantum loop algebra (or subalgebra) in these models is established by showing that the Serre relations hold for the generators. The highest weight polynomial (or the Drinfeld polynomial) of the representation is obtained by using the method of Baxter for the superintegrable case. As a byproduct, the eigenvalues of all such CSOS models are given explicitly.