On the Peirce spectral rigidity of decorated incidence algebras
The multiplication operator of an idempotent in a metrized commutative nonassociative algebra is a self-adjoint linear endomorphism whose spectrum reflects the algebraic and geometric structure encoded in the algebra. Although for general algebras the spectra of idempotents can behave arbitrarily, for the most interesting classes -- among them algebras of Clifford and Jordan type arising in geometry -- the spectrum is highly constrained. We consider a family of algebras determined by a partial Steiner triple system (PSTS) with blocks decorated by signs, each block determining an idempotent; even for the simplest PSTS the resulting algebras can be quite complicated. For decorations of the Grassmannian PSTS $G_2(m)$, whose blocks are $3$-subsets of an $m$-element set and whose points parametrize an underlying incidence geometry, the spectra behave rigidly: the spectrum with multiplicity of a block idempotent is a function of a single combinatorial parameter, the number of negatively signed Pasch configurations -- quadrilateral subconfigurations -- containing the block. The resulting block spectra organize into a spectral profile, recovered by binomial inversion from a hierarchy of moment invariants. For a distinguished family of decorations, the associated algebras are axial algebras satisfying a fusion law independent of $m$, and for these algebras there is given an alternative explicit matrix model exhibiting the rigidity directly; the even part of this model is a direct sum containing a polar algebra summand, linking the construction to polar algebras and symmetric Clifford systems.