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We assume that the diameter\n D<\/jats:italic>\n of\n \n \n $$\\Gamma $$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is at least 3. Fix adjacent vertices\n \n \n $$x,y \\in X$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n \u2208<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In our first main result, we introduce an equitable partition of\n X<\/jats:italic>\n that has\n \n \n $$6D-2$$<\/jats:tex-math>\n \n \n 6<\/mml:mn>\n D<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n subsets and the following feature: for every subset in the equitable partition, the vertices in the subset are equidistant to\n x<\/jats:italic>\n and equidistant to\n y<\/jats:italic>\n . This equitable partition is called the (\n x<\/jats:italic>\n ,\u00a0\n y<\/jats:italic>\n )-partition of\n X<\/jats:italic>\n . By definition, the subconstituent algebra\n \n \n $$T=T(x)$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n =<\/mml:mo>\n T<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is generated by the Bose-Mesner algebra of\n \n \n $$\\Gamma $$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and the dual Bose-Mesner algebra of\n \n \n $$\\Gamma $$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n with respect to\n x<\/jats:italic>\n . As we will see, for the (\n x<\/jats:italic>\n ,\u00a0\n y<\/jats:italic>\n )-partition of\n X<\/jats:italic>\n the characteristic vectors of the subsets form a basis for a\n T<\/jats:italic>\n -module\n \n \n $$U=U(x,y)$$<\/jats:tex-math>\n \n \n U<\/mml:mi>\n =<\/mml:mo>\n U<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In our second main result, we decompose\n U<\/jats:italic>\n into an orthogonal direct sum of irreducible\n T<\/jats:italic>\n -modules. This sum has five summands: the primary\n T<\/jats:italic>\n -module and four irreducible\n T<\/jats:italic>\n -modules that have endpoint one. We show that every irreducible\n T<\/jats:italic>\n -module with endpoint one is isomorphic to exactly one of the nonprimary summands.\n <\/jats:p>","DOI":"10.1007\/s00373-026-03033-9","type":"journal-article","created":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T13:47:07Z","timestamp":1774360027000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An equitable partition for the distance-regular graph of the bilinear forms"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0942-5489","authenticated-orcid":false,"given":"Paul","family":"Terwilliger","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8697-5997","authenticated-orcid":false,"given":"Jason","family":"Williford","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"3033_CR1","doi-asserted-by":"publisher","DOI":"10.1515\/9783110630251","author":"E Bannai","year":"2021","unstructured":"Bannai, E., Bannai, E., Ito, T., Tanaka, R.: Algebraic Combinatorics. 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