Gauge-string duality, monomial bases and graph determinants

TL;DR

This paper develops a combinatorial framework using degeneracy graphs to construct monomial bases for finite-dimensional algebras, linking algebraic structures to graph theory with applications in quantum information.

Contribution

It introduces degeneracy graphs and a monomial basis construction for algebras, providing explicit formulas and computational evidence for invertibility, with applications in quantum information and algebraic algorithms.

Findings

Derived a simple formula for algebra bases using degeneracy graphs

Proved the basis construction is compatible with projector counting

Supported invertibility conjecture with computational evidence

Abstract

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 $\mathcal{A}$, obtained by incrementally adjoining one generator at each step to produce a non-linear generating set for $\mathcal{A}$. 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 $\mathcal{A}$ 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 $\mathcal{A}$, 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.