Alladi Ramakrishnan Hall

A weighted adjacency matrix associated with the projective geometry B(q,n)

Murali K Srinivasan

IIT Bombay

The Boolean algebra B(n) is the poset of subsets of a set of cardinality n and the projective geometry B(q,n) is the poset of subspaces of an n dimensional vector space over a finite field of order q.

The graph C(n), called the n-cube, is the Hasse diagram of B(n) and the graph C(q,n), the q-analog of the n-cube, is the Hasse diagram of B(q,n).

The adjacency matrix of C(n) is a fundamental object in algebraic combinatorics having an elegant spectral theory with several applications (Markov chains, orthogonal polynomials, coding theory, spanning trees etc.).

This talk is motivated by the problem of finding a q-analog of the adjacency matrix of C(n), from the spectral viewpoint. The obvious candidate, namely, the adjacency matrix of C(q,n), does not work. We show that a certain weighted adjacency matrix of C(q,n) is the solution. We discuss the spectral theory of this matrix.

This is joint work with Subhajit Ghosh.