--- title: Permutation-Invariant Gaussian Matrix Models url: https://www.emergentmind.com/topics/permutation-invariant-gaussian-matrix-models type: topic --- # Permutation-Invariant Gaussian Matrix Models Permutation-invariant Gaussian matrix models are Gaussian probability measures on matrix spaces whose action, observables, and correlation structure are invariant under relabeling of indices by a permutation group, most commonly the diagonal action of the symmetric group \(S_D\) on \(D\times D\) matrices. In the basic matrix setting, the symmetry acts by simultaneous permutation of row and column labels, \(M_{ij}\mapsto M_{\sigma(i)\sigma(j)}\), and the resulting theory organizes linear, quadratic, and higher-degree observables into a finite set of invariant channels determined by the representation theory of \(S_D\). The general model for unconstrained real matrices has two linear and eleven quadratic parameters, giving a 13-parameter Gaussian family, while closely related variants arise for symmetric zero-diagonal matrices, multi-matrix systems, tensor models, and gauged matrix quantum mechanics [1809.07559]. ## 1. Symmetry, invariance, and the graph basis of observables The defining symmetry of a permutation-invariant matrix model is invariance under simultaneous relabeling of matrix indices. For a \(D\times D\) matrix \(M\), a permutation-invariant function \(f(M)\) obeys \[ f(M)=f(U_\sigma M U_\sigma^T) \] for all \(\sigma\in S_D\), where \(U_\sigma\) is the permutation matrix. Equivalently, at the level of entries, \[ f(M_{ij})=f(M_{\sigma(i)\sigma(j)}). \] This discrete symmetry differs from the continuous orthogonal or unitary invariance of classical Gaussian ensembles, and it preserves index-level structure rather than reducing observables to eigenvalue data alone [2202.06829]. Permutation-invariant polynomial observables admit a complete graph-theoretic classification. Summed indices are represented as vertices, and each matrix entry \(M_{ij}\) is represented as a directed edge \(i\to j\). Distinct contraction patterns correspond to distinct directed multigraphs, and these graphs form a basis of independent invariants. A standard family is \[ P_\sigma(M)=\sum_{i_1,\dots,i_p}\prod_{a=1}^{p} M_{i_a\, i_{\sigma(a)}}, \] with examples including \(\mathrm{Tr}(M)\), \(\mathrm{Tr}(M^2)\), \(\mathrm{Tr}(M)^2\), \(\sum_{i,j}M_{ij}^2\), \(\sum_{i,j}M_{ij}M_{jj}\), and \(\sum_{i,j,k}M_{ij}M_{jk}M_{ki}\). In the unrestricted matrix case there are 13 independent invariants up to quadratic order, 52 at cubic order, and 256 at quartic order, all organized by graph basis [2202.06829]. The same graph principle persists in related models, but the graph class changes with constraints. For symmetric zero-diagonal correlation matrices, permutation-invariant polynomials are labeled by undirected loop-less graphs, and the linear/quadratic basis collapses to one linear invariant and three quadratic invariants, reflecting the reduced physical subspace [2306.04569]. For two-matrix models, observables correspond to directed colored graphs, with one color for each matrix species, and refined counting is expressed through double cosets determined by local graph structure [2104.03707]. A further structural refinement identifies permutation-invariant matrix observables of degree \(k\) with equivalence classes in the partition algebra \(P_k(N)\). In that formulation, degree-\(k\) observables are associated with diagram classes under \(S_k\)-conjugation, and the algebraic product of diagrams induces an inner product and large-\(N\) factorization statements for the observable space [2112.00498]. ## 2. Representation-theoretic construction and the 13-parameter Gaussian family The representation-theoretic origin of the 13-parameter model is the decomposition of the matrix space \(V_D\otimes V_D\) under the diagonal action of \(S_D\). Writing \(V_D\simeq V_0\oplus V_H\), with \(V_0\) the trivial representation and \(V_H\) the hook representation of dimension \(D-1\), one obtains \[ V_D\otimes V_D \cong 2V_0\oplus 3V_H\oplus V_2\oplus V_3. \] The multiplicities \(2,3,1,1\) imply two linearly independent invariant linear combinations and eleven independent quadratic couplings, because \[ \frac{2\cdot 3}{2}+\frac{3\cdot 4}{2}+\frac{1\cdot 2}{2}+\frac{1\cdot 2}{2}=11. \] Including the two linear parameters gives the 13 free parameters of the general permutation-invariant Gaussian matrix model [1809.07559]. In an irrep-adapted basis, the Gaussian action takes block-diagonal form. In the notation of the solved model, one has two linear sources \(\mu_1,\mu_2\), a symmetric \(2\times 2\) block \(\Lambda_{V_0}\), a symmetric \(3\times 3\) block \(\Lambda_{V_H}\), and scalars \(\Lambda_{V_2},\Lambda_{V_3}\). The action is \[ S(M)= - \mu_1 J_1(M) - \mu_2 J_2(M) + \frac12 \sum_{a=1}^{11} g_a I_a(M), \] or equivalently in block form, \[ S(M)=\frac12 X_{V_0}^T \Lambda_{V_0} X_{V_0} + \frac12 X_H^T \Lambda_{V_H} X_H + \frac12 \Lambda_{V_2}\|X_{V_2}\|^2 + \frac12 \Lambda_{V_3}\|X_{V_3}\|^2 - \mu^T X_{V_0}. \] The two linear invariants are \[ J_1(M)=\sum_i M_{ii},\qquad J_2(M)=\sum_{i,j}M_{ij}, \] and the eleven quadratic invariants include \(\sum_{i,j}M_{ij}M_{ij}\), \(\sum_{i,j}M_{ij}M_{ji}\), \(\sum_{i,j,k}M_{ij}M_{jk}\), \(\sum_i M_{ii}^2\), \(\sum_{i,j}M_{ii}M_{jj}\), and \(\sum_{i,j,k}M_{ii}M_{jk}\) [1912.10839]. A useful simplified covariance illustrating permutation symmetry is \[ E[M_{ij}] = 0,\qquad E[M_{ij}M_{kl}] = \alpha\,\delta_{ik}\delta_{jl}+\beta\,\delta_{il}\delta_{jk}+\gamma\,\delta_{ij}\delta_{kl}, \] which is the minimal three-tensor decomposition invariant under index permutations. The full 13-parameter model refines this by distinguishing diagonal and off-diagonal sectors and by resolving multiple graph channels through the \(V_0\), \(V_H\), \(V_2\), and \(V_3\) blocks [2202.06829]. The same decomposition principle reappears in other settings. For symmetric zero-diagonal matrices, the physical space reduces to \[ V^{\mathrm{phys}}\simeq V_0\oplus V_H\oplus V_2, \] so the most general Gaussian model has four parameters: one linear source in \(V_0\) and three quadratic couplings \(\tau_{V_0},\tau_{V_H},\tau_{V_2}\) [2306.04569]. For two-matrix models, each irrep block is enlarged to include species-mixing terms, producing 4 linear and 37 quadratic invariants for the most general Gaussian two-matrix action [2104.03707]. For rank-3 tensor models, the decomposition \[ V_D\otimes V_D\otimes V_D \cong 5V_0\oplus 10V_H\oplus 6V_2\oplus 6V_3\oplus V_4\oplus 2V_5\oplus V_6 \] provides the analogous representation-basis diagonalization of the Gaussian measure [2312.09205]. ## 3. Gaussian expectations, Wick factorization, and exact computation Once the linear and quadratic blocks are fixed, all higher invariant moments are obtained by Wick’s theorem. In the simplified covariance model, \[ E[\mathrm{Tr}(M^2)] = \beta D^2 + (\alpha+\gamma)D, \] \[ E\Big[\sum_{i,j}M_{ij}^2\Big] = \alpha D^2 + (\beta+\gamma)D, \] \[ E[(\mathrm{Tr}M)^2] = \gamma D^2 + (\alpha+\beta)D. \] Higher-order graph observables factorize into sums over pairings of edges in the corresponding directed graphs, and the full 13-parameter theory gives analytic formulas for cubic and quartic invariants in terms of the inverse irrep blocks [2202.06829]. The solved general model provides explicit formulas for all quadratic graph-basis invariants and a selection of cubic and quartic invariants. For example, the quadratic expectation \[ \sum_{i,j}\langle M_{ij}M_{ij}\rangle \] is expressed as a polynomial in \(\tilde\mu_1,\tilde\mu_2\), the entries of \(\Lambda_{V_0}^{-1}\), the entries of \(\Lambda_{V_H}^{-1}\), and the scalars \(\Lambda_{V_2}^{-1},\Lambda_{V_3}^{-1}\). The cubic triangle \[ \sum_{i,j,k}\langle M_{ij}M_{jk}M_{ki}\rangle \] and quartic invariants such as \(\sum_{i,j}M_{ij}^4\) and \(\sum_{i,j,k,l}M_{ij}M_{jk}M_{kl}M_{li}\) are then recovered by Wick expansion and projector identities [1809.07559]. Partition algebras furnish an alternative but compatible computational language. In the zero-dimensional formulation and in related large-\(N\) analyses, the connected two-point function can be written in terms of invariant tensors \(Q^{\Lambda,\alpha\beta}_{ij;kl}\), which are matrix units of the partition algebra image in \(\mathrm{End}_{S_N}(V_N^{\otimes 2})\). Higher correlators are generated by Wick contraction in the representation basis, then reassembled in diagrammatic form. This gives a direct algebraic explanation for the large-\(N\) behavior of invariant observables and for the role of partition algebras as commutants of the \(S_N\) action [2311.10213]. For the symmetric zero-diagonal model relevant to correlation matrices, the one-point and connected two-point functions are written explicitly in terms of projectors \(P^{\mathrm{phys};V_0}\), \(P^{\mathrm{phys};V_H}\), and \(P^{\mathrm{phys};V_2}\). The linear and three quadratic empirical moments determine the four couplings, and cubic and quartic expectations follow from Wick’s theorem with nonzero mean. The resulting closed forms are polynomials in \(\widetilde{\mu}_{V_0}^2\), \(\tau_{V_0}^{-1}\), \(\tau_{V_H}^{-1}\), and \(\tau_{V_2}^{-1}\), with \(D\)-dependent coefficients [2306.04569]. ## 4. Algebraic structure, large-\(N\) behavior, and relations to other Gaussian ensembles Permutation-invariant Gaussian matrix models admit a hidden-algebra description in terms of partition algebras and permutation centralizer algebras. For degree-\(k\) observables, \(P_k(N)\) appears as the commutant of the diagonal \(S_N\) action on \(V_N^{\otimes k}\), and this establishes a one-to-one correspondence between permutation-invariant matrix observables and equivalence classes of partition diagrams. The resulting inner product at a special \(O(N)\)-enhanced point can be written as \[ \big\langle:\mathcal{O}_{d_1}::\mathcal{O}_{d_2}:\big\rangle = \sum_{\gamma\in S_k}\mathrm{Tr}_{V_N^{\otimes k}}(d_1\,\gamma\,d_2^T\,\gamma^{-1}), \] and large-\(N\) factorization follows from the join structure of diagrams [2112.00498]. The observable algebra of multi-matrix Gaussian models with permutation symmetry is further organized by permutation centralizer algebras. For fixed content \((n_1,\dots,n_m)\), the algebra \[ \mathcal{A}(n_1,\dots,n_m)=\{x\in \mathbb{C}[S_n]\mid xh=hx\ \forall h\in H\}, \qquad H=S_{n_1}\times\cdots\times S_{n_m}, \] has a Wedderburn–Artin decomposition into matrix blocks labeled by symmetric-group irreducibles and Littlewood–Richardson multiplicities. Restricted Schur operators give a basis that diagonalizes Gaussian two-point functions, while the center of the algebra yields an efficient route to correlators and graph counting [1601.06086]. In the \(d=2\) specialization, this framework reduces to the center of \(\mathbb{C}[S_n]\), recovering the representation basis of ordinary matrix multi-traces [1708.03524]. Relative to GOE, GUE, and related continuous-symmetry ensembles, the central distinction is the symmetry group. Continuous ensembles are invariant under orthogonal or unitary conjugation, and their invariant observables reduce to spectral quantities such as \(\mathrm{Tr}(M^k)\). Permutation-invariant ensembles are invariant only under discrete relabelings, so observables include mixed diagonal/off-diagonal contractions such as \(\sum_{ij}M_{ij}^2\) and \(\sum_{ij}M_{ij}M_{jj}\), and the covariance separates into index-coincidence channels rather than purely spectral sectors [2202.06829]. This suggests that permutation-invariant models are better adapted to data where matrix entries retain semantic, relational, or role-specific meaning and where diagonalization would discard the relevant structure. The same symmetry ideas extend to Gaussian models invariant under a subgroup \(\Gamma\leq S_p\). In complete RCOP models, centered Gaussian covariance or precision matrices satisfy \(g\Sigma g^T=\Sigma\) or \(gKg^T=K\) for all \(g\in\Gamma\), and representation theory gives block decompositions, exact MLE distributions, and Diaconis–Ylvisaker conjugate priors with closed-form normalizing constants. This provides a broader statistical setting in which permutation-invariant Gaussian models appear as symmetry-constrained Gaussian families rather than only as matrix ensembles [2004.03503]. A related graphical extension imposes both sparsity and symmetry, yielding homogeneous-cone models for precision matrices on homogeneous graphs with permutation symmetries [2207.13330]. ## 5. Variants: correlation matrices, multi-matrix systems, tensor models, and gauged quantum mechanics A notable constrained variant arises for symmetric zero-diagonal matrices, motivated by daily foreign-exchange correlation matrices. In that setting \(M_{ij}=M_{ji}\) and \(M_{ii}=0\), and the most general permutation-invariant Gaussian action is \[ S = \mu \sum_{i,j} M_{ij} + \tau_1 \sum_{i,j} M_{ij}M_{ij} + \tau_2 \sum_{i,j,k} M_{ij}M_{jk} + \tau_3 \sum_{i,j,k,l} M_{ij}M_{kl}. \] The representation-theoretic description reduces the model to four parameters associated with \(V_0\), \(V_H\), and \(V_2\), and analytic formulas are available for linear, quadratic, cubic, and quartic invariant expectations [2306.04569]. The two-matrix Gaussian model generalizes the one-matrix theory by introducing two real \(D\times D\) matrices \(M\) and \(N\), diagonal \(S_D\) symmetry on both species, and mixed invariants encoded by directed colored graphs. The most general action contains 4 linear and 37 quadratic invariants, and the quadratic form splits into \(V_0\), \(V_H\), \(V_2\), and \(V_3\) blocks for each species together with inter-species couplings. The paper gives explicit expectation values for observables of degree up to four and a Sage program for arbitrary graph-basis evaluations [2104.03707]. Tensor models inherit the same logic but require higher partition algebras and larger irrep multiplicities. In rank 3, the Gaussian action is diagonalized in a representation basis \(S^{\Lambda,\alpha}_a\), and the connected two-point function in the tensor basis is \[ \big\langle \Phi_{ijk}\Phi_{pqr}\big\rangle_{\mathrm{conn}} = \sum_{\Lambda}\sum_{\alpha,\beta} Q^{\Lambda,\alpha\beta}_{ijk;pqr}\,\tilde g_{\Lambda}^{\alpha\beta}, \] with invariant endomorphism tensors \(Q^{\Lambda,\alpha\beta}\). The matrix case then appears as the rank-2 specialization with multiplicities \(2,3,1,1\), reproducing the 13-parameter counting [2312.09205]. Permutation symmetry can also be gauged in matrix quantum mechanics. For the simplest harmonic oscillator Hamiltonian \[ H=\sum_{i,j=1}^N A_{ij}^\dagger A_{ij}, \] the canonical partition function of the \(S_N\)-gauged theory is obtained by inserting the projector onto invariant states or, equivalently, by a finite-group path integral. The result is a Molien–Weyl sum over conjugacy classes, \[ Z(N,x)=\sum_{p\vdash N}\frac{1}{\tilde p}\,Z(N,p,x), \] with \[ Z(N,p,x)= \prod_i \frac{1}{(1-x^{a_i})^{a_i p_i^2}} \prod_{i