Date:
Mon, 23/06/202511:00-13:00
Location:
Ross 70
Title: Isospectrality of Geometries over Finite Rings
Abstract:
The projective geometry associated with a field F in finite dimension d is classically defined by considering the collection of its (non-trivial) subspaces as vertices, with edges corresponding to incidence relations. A well-known example is the Fano plane, the projective plane of dimension 2 over the field with 2 elements. In this work, we generalize this framework to the setting of quotients of Dedekind domains.
Let R be a Dedekind domain and p an ideal of R with finite index p. For any positive integer r, define O_r as R modulo p to the r-th power. Let P^d_fr(O_r) denote the flag complex on all (non-trivial) free submodules of O_r to the (d+1)-th power, with incidence relation. Two key examples to keep in mind are the integers modulo p^r, and the polynomial ring over the field with p elements modulo t^r. The case r = 1 is the classical projective finite geometry P^d(F_p), which appears as the local structure of vertices in the Bruhat-Tits buildings of PGL_d over the p-adic numbers and PGL_d over formal Laurent series over F_p. The complex P^d_fr(O_r) arises naturally when considering geodesic r-spheres in these buildings.
Let P^d_{m,n}(O_r) be the subgraph induced by the free submodules of ranks m and n. In the first part of the talk, we analyze the adjacency matrix of the square-graph of P^d_{1,n}(O_r), and discover a structure that allows us to calculate their eigenvalues and, in particular, their spectral expansion. These eigenvalues turn out to be independent of the specific choice of R and p, and depend only on the numbers p, r, and d. Hence, for example, P^d_{1,n}(Z mod p^r) and P^d_{1,n}(F_p[t] mod t^r) are isospectral, and their square-graphs are even isomorphic. However, are the graphs themselves isomorphic?
In the second part of the talk, we discuss this question and show how to distinguish between two such graphs using their automorphism groups. We prove a generalization of the Fundamental Theorem of Projective Geometry and find the automorphism groups of P^d_fr(O_r) and P^d_{m,n}(O_r). We then use size considerations of these automorphism groups to show that P^d_fr(Z mod p^r) and P^d_fr(F_p[t] mod t^r) are not isomorphic, except in the case where p = r = 2, in which additional work is required.
Abstract:
The projective geometry associated with a field F in finite dimension d is classically defined by considering the collection of its (non-trivial) subspaces as vertices, with edges corresponding to incidence relations. A well-known example is the Fano plane, the projective plane of dimension 2 over the field with 2 elements. In this work, we generalize this framework to the setting of quotients of Dedekind domains.
Let R be a Dedekind domain and p an ideal of R with finite index p. For any positive integer r, define O_r as R modulo p to the r-th power. Let P^d_fr(O_r) denote the flag complex on all (non-trivial) free submodules of O_r to the (d+1)-th power, with incidence relation. Two key examples to keep in mind are the integers modulo p^r, and the polynomial ring over the field with p elements modulo t^r. The case r = 1 is the classical projective finite geometry P^d(F_p), which appears as the local structure of vertices in the Bruhat-Tits buildings of PGL_d over the p-adic numbers and PGL_d over formal Laurent series over F_p. The complex P^d_fr(O_r) arises naturally when considering geodesic r-spheres in these buildings.
Let P^d_{m,n}(O_r) be the subgraph induced by the free submodules of ranks m and n. In the first part of the talk, we analyze the adjacency matrix of the square-graph of P^d_{1,n}(O_r), and discover a structure that allows us to calculate their eigenvalues and, in particular, their spectral expansion. These eigenvalues turn out to be independent of the specific choice of R and p, and depend only on the numbers p, r, and d. Hence, for example, P^d_{1,n}(Z mod p^r) and P^d_{1,n}(F_p[t] mod t^r) are isospectral, and their square-graphs are even isomorphic. However, are the graphs themselves isomorphic?
In the second part of the talk, we discuss this question and show how to distinguish between two such graphs using their automorphism groups. We prove a generalization of the Fundamental Theorem of Projective Geometry and find the automorphism groups of P^d_fr(O_r) and P^d_{m,n}(O_r). We then use size considerations of these automorphism groups to show that P^d_fr(Z mod p^r) and P^d_fr(F_p[t] mod t^r) are not isomorphic, except in the case where p = r = 2, in which additional work is required.