The Goldman-Rota identity and the Grassmann scheme

TL;DR

This paper constructs an explicit orthogonal eigenbasis for the Bose-Mesner algebra of the Grassmann scheme by interpreting the Goldman-Rota recurrence through linear algebra, revealing recursive structures and eigenvalues.

Contribution

It provides a new linear algebraic interpretation of the Goldman-Rota recurrence and explicitly constructs an orthogonal eigenbasis for the Grassmann scheme's Bose-Mesner algebra.

Findings

Explicit orthogonal eigenbasis for Grassmann scheme

Recursive structure of the up operator on subspaces

Singular values of the Jordan chains

Abstract

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. The main step is a constructive, linear algebraic interpretation of the Goldman-Rota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using this we inductively construct an explicit orthogonal symmetric Jordan basis with respect to the up operator and write down the singular values, i.e., the ratio of the lengths of the successive vectors in the Jordan chains. The collection of all vectors in this basis of a fixed rank forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.