Character tables of association schemes based on attenuated spaces

TL;DR

This paper computes the character tables of association schemes based on attenuated spaces, generalizing several known schemes and including the m-flat association scheme as a special case.

Contribution

It introduces a method to compute character tables for these schemes and establishes their relation to the m-flat association scheme.

Findings

Computed character tables of association schemes based on attenuated spaces.

Showed that these schemes generalize Grassmann, bilinear forms, and non-binary Johnson schemes.

Included the m-flat association scheme as a special case.

Abstract

The set of subspaces of a given dimension in an attenuated space has a structure of a symmetric association scheme and this association scheme is called an association scheme based on an attenuated space. Association schemes based on attenuated spaces are generalizations of Grassmann schemes and bilinear forms schemes, and also $q$-analogues of non-binary Johnson schemes. Wang, Guo and Li computed the intersection numbers of association schemes based on attenuated spaces. The aim of this paper is to compute character tables of association schemes based on attenuated spaces using the method of Tarnanen, Aaltonen and Goethals. Moreover, we also prove that association schemes based on attenuated spaces include as a special case the $m$-flat association scheme, which is defined on the set of cosets of subspaces of a constant dimension in a vector space over a finite field.