### The Goldman-Rota Identity and the Grassmann Scheme

#### Abstract

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme.

The key 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 the interpretation above 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 $m$ forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme of $m$-dimensional subspaces. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.

The key 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 the interpretation above 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 $m$ forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme of $m$-dimensional subspaces. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.

#### Keywords

Grassmann Scheme; Goldman-Rota Identity; Symmetric Jordan Basis